FR3140728A1 - Cryptographic processing method, electronic device and associated computer program - Google Patents

Abstract

Cryptographic processing method implemented by an electronic device and comprising:i) an initialization phase (P1) comprising:- recording (E120) of a root private key in a memory of the electronic device,ii) a phase generating (P2) keys comprising: - calculating (E220) a derived private key equal to f(s, k) where f is a derivation function, s is a subset of the root private key, and k is a secret scalar, - the calculation (E240) of a derived public key associated with the derived private key, iii) a proof generation phase (P3) comprising: - the generation (E320) of a proof for disclosure zero knowledge that the derived public key is associated with a scalar equal to f(s, k), iv) a transmission phase (P3) comprising: - transmission (E420) of the proof with zero knowledge disclosure to a device of verification. Figure for abstract: fig. 3.

Description

Cryptographic processing method, electronic device and associated computer program

The present invention relates to a cryptographic processing method, as well as an associated computer program and electronic device.

In asymmetric cryptography, the generation of a public key and a secret key, also called a private key, is necessary before signing or encrypting messages by an electronic device.

The public key and the private key may be generated directly by the electronic device, or be loaded into said electronic device after being generated by another device.

It is often required that the generation of these keys be carried out directly by the electronic device, in particular to ensure that the private key is known only to it.

However, despite this requirement, keys are sometimes generated externally to the electronic device and then loaded into it without this being able to be detected during subsequent use of the electronic device.

An aim of the invention is to improve security for the generation of keys by an electronic device and the associated audit.

To this end, the present invention proposes a cryptographic processing method implemented by an electronic device, the electronic device cooperating with a verification device, and said method comprising:
i) a phase of initialization of a set of at least one first secret scalar forming a root private key associated with a root public key, the initialization phase comprising the following step:
- recording of the root private key in a memory of the electronic device,
And
ii) a phase for generating a derived private key and a derived public key, comprising the following steps:
- calculation of a second secret scalar forming the derived private key equal to f(s, k) where f is a derivation function, s is a non-empty subset of the set of at least one first secret scalar, and k is a third secret scalar,
- calculation of the derived public key associated with the second secret scalar,
Then
iii) a proof generation phase comprising the following step:
- generation of a zero-disclosure proof of knowledge that the derived public key is associated with a scalar equal to f(u, l), u being the subset s of the set of at least one first secret scalar forming the root private key associated with the root public key and l being equal to k,
And
iv) a transmission phase comprising the following step:
- transmission of the zero-knowledge proof to the verification device.

The method may also include the following optional features, taken alone or in combination whenever technically possible.

The electronic device further cooperates with a configuration device during the initialization phase, and the root public key is certified by the configuration device.

The cooperation between the electronic device and the configuration device is the sending by the electronic device to the configuration device, and/or the reception from the configuration device by the electronic device, of at least a first secret scalar of the root private key and/or the associated root public key.

The certification of the root public key consists of the registration of the root public key in a list of at least one trusted public key, or in the cryptographic signature of said root public key, for example with an issued certification private key. by a trusted authority.

The initialization phase also includes the following steps:
- calculation, before the recording step, of all the first secret scalars of the set forming the root private key, at least one first secret scalar of the root private key being calculated by random drawing or by application of a function unclonable physics, then
- calculation of the root public key from the root private key.

The configuration device has a fourth secret scalar forming a master private key associated with the root public key, and the initialization phase further comprises the following step before the recording step:
- reception, from the configuration device, of at least one first secret scalar of the root private key, said at least one first secret scalar received from the configuration device being calculated from the master private key.

The electronic device receives from the configuration device all the first secret scalars of the set forming the root private key.

When the root private key is formed from a single first secret scalar, the first secret scalar received from the configuration device is the master private key.

When the root private key is formed of two first secret scalars, the electronic device receives said first two secret scalars from the configuration device. Typically, one of the first two secret scalars is determined by the configuration device by algebraic calculation from the fourth secret scalar and the other of the first two secret scalars, and said other of the first two secret scalars is determined by the configuration device by random selection or by application of a non-clonable physical function (“PUF” or “physical unclonable function” according to the commonly used Anglo-Saxon name).

The configuration device further has a fifth secret scalar forming an additional private key associated with an additional public key, and said at least one first secret scalar received from the configuration device is calculated from the master private key and the key additional private.

The root private key is formed from a single first secret scalar.

The root private key is formed from a plurality of first secret scalars.

The initialization phase further includes the following steps before the registration step:
- determination of a first secret scalar of the root private key, then
- homomorphic encryption of the first determined secret scalar, then
- transmission to the cipher configuration device of the first determined secret scalar,
and during the reception step of the initialization phase, the electronic device:
- receives another first secret scalar of the root private key, encrypted in the same homomorphic manner, the cipher of said other first secret scalar having been determined from the master private key and the cipher transmitted by the electronic device to the configuration device, And
- decrypts the ciphertext of the other first secret scalar to obtain said at least one first secret scalar received from the configuration device.

The determination of a first secret scalar of the root private key is by random drawing or by application of a non-clonable physical function.

The third secret scalar k is determined by the electronic device by random selection or by application of a non-clonable physical function.

The initialization phase is implemented at an initialization site, and the phase for generating a derived private key and a derived public key is implemented at another site distinct from the initialization site.

The zero-knowledge proof is a non-interactive proof.

The zero-knowledge proof is an interactive proof.

The generation of the zero-knowledge proof, and the transmission of the zero-knowledge proof to a verification device, are in the form of a challenge-response protocol.

The invention also proposes a computer program comprising instructions executable by a processor and adapted for the implementation of a cryptographic processing method as defined above, when these instructions are executed by the processor.

The invention also proposes an electronic device comprising means adapted for implementing a cryptographic processing method as defined above.

The invention relates in particular to an electronic cryptographic processing device adapted to cooperate with a verification device and comprising:
i) an initialization module configured to carry out an initialization phase of a set of at least one first secret scalar forming a root private key associated with a root public key, the initialization phase comprising the following step:
- recording of the root private key in a memory of the electronic device,
ii) a key generation module configured to carry out a generation phase of a derived private key and a derived public key, comprising the following steps:
- calculation of a second secret scalar forming the derived private key equal to f(s, k) where f is a derivation function, s is a non-empty subset of the set of at least one first secret scalar, and k is a third secret scalar,
- calculation of the derived public key associated with the second secret scalar,
iii) a proof generation module configured to carry out a proof generation phase comprising the following step:
- generation of a zero-disclosure proof of knowledge that the derived public key is associated with a scalar equal to f(u, l), u being the subset s of the set of at least one first secret scalar forming the root private key associated with the root public key and l being equal to k,
iv) a transmission module configured to carry out a transmission phase comprising the following step:
- transmission of the zero-knowledge proof to the verification device.

This electronic device can be configured for the implementation of each of the possibilities of realization envisaged for the cryptographic processing method as defined previously.

Of course, the different characteristics, variants and embodiments of the invention can be associated with each other in various combinations as long as they are not incompatible or exclusive of each other.

Other characteristics and advantages of the present invention will emerge from the description given below, with reference to the appended figures which illustrate examples of embodiment devoid of any limiting character.

In the figures:

schematically represents the main elements of an electronic device within which the invention is implemented, as well as a verification device and a configuration device with which the electronic device can cooperate according to the invention;

represents an example of an elliptic curve and a group of points of the elliptic curve;

represents in flowchart form the main steps of a cryptographic processing method according to a first mode of implementation of the invention;

represents in flowchart form the main steps of a cryptographic processing method according to a second mode of implementation of the invention;

represents in flowchart form the main steps of a cryptographic processing method according to a third mode of implementation of the invention;

represents in flowchart form the main steps of a cryptographic processing method according to a fourth mode of implementation of the invention.

Unless otherwise indicated, elements common or similar to several figures bear the same reference signs and have identical or similar characteristics, so that these common elements are generally not described again for the sake of simplicity.

In the context of this description, the qualifiers “first”, “second”, “third”, “fourth”, “fifth” or “sixth” are only for information purposes to distinguish the elements that they qualify, but do not imply order between them.

There schematically represents the main elements of an electronic device within which the invention is implemented, as well as a verification device and a configuration device with which the electronic device can cooperate according to the invention.

There thus schematically represents an electronic device 2 comprising a processor 4 (for example a microprocessor), a storage block 6, a RAM 8 and a communication block 10.

The RAM 8 and the storage block 6 are each linked to the processor 4 so that the processor 4 can read or write data in the storage block 6 and/or the RAM 8.

The storage block 6 stores computer program instructions, some of which are designed to implement a cryptographic processing method such as at least one of those described with reference to Figures 3 to 6 when these instructions are executed by the processor 4.

The storage block 6 is for example in practice a hard disk or a non-volatile memory, possibly rewritable, for example of the EEPROM type (for "Electrically Erasable and Programmable Read-Only Memory" according to the Anglo-Saxon name commonly used) .

In addition, the storage block 6 and/or the RAM 8 can memorize at least some of the elements (in particular the scalars and the points of an elliptical curve) manipulated during the different treatments carried out during at least one of the processes described below.

Memory in the remainder of the description is any one of the storage block 6 and the RAM 8.

The electronic device 2 also includes several modules not shown.

Typically, the electronic device 2 comprises an initialization module, a key generation module, a proof generation module and a transmission module.

These modules can in practice be produced by a combination of hardware elements and software elements.

Each module is configured to carry out the steps of a phase described in the methods conforming to the invention and explained below, and therefore has a functionality described in the methods conforming to the invention and explained below. Thus, for each module, the electronic device 2 memorizes for example software instructions executable by the processor 4 of the electronic device 2 in order to use a hardware element (for example a communication interface or a memory) and thus implement the functionality offered by the module.

According to one possible embodiment, the computer program instructions stored in the storage block 6 have for example been received (for example from a remote computer) during an operating phase of the electronic device 2 prior to the methods described in reference to Figures 3 to 6.

The communication block 10 is connected to the processor 4 so as to allow the processor 4 to receive data from another electronic device and/or to transmit data to another electronic device (for example a device configuration 22 and/or a verification device 12 as described below). In certain embodiments, the processor 4 can thus receive data m from the other electronic device, for example a first secret scalar coming from a configuration device, as described with reference to Figures 3 and 6, and/ or emit other data, typically a zero-knowledge proof as also described with reference to Figures 3 and 6.

There also schematically represents a verification device 12 comprising a processor 14 (for example a microprocessor), a storage block 16, a RAM 18 and a communication block 20.

The RAM 18 and the storage block 16 are each linked to the processor 14 so that the processor 14 can read or write data in the storage block 16 and/or the RAM 18.

The storage block 16 stores computer program instructions, some of which are designed to cooperate with the electronic device 2 during a cryptographic processing method such as at least one of those described with reference to Figures 3 to 6 when these instructions are executed by processor 14.

The storage block 16 may also store computer program instructions designed to cooperate with a configuration device.

The storage block 16 is for example in practice a hard disk or a non-volatile memory, possibly rewritable, for example of the EEPROM type (for "Electrically Erasable and Programmable Read-Only Memory" according to the Anglo-Saxon name commonly used). .

The RAM 18 can memorize at least some of the elements (in particular scalars and points of an elliptical curve) manipulated during the different processing operations carried out to cooperate with the electronic device 2 during at least one of the methods described below.

According to one possible embodiment, the computer program instructions stored in the storage block 16 were for example received (for example from a remote computer) during an operating phase of the verification device 12 prior to the methods described with reference to Figures 3 to 6.

The communication block 20 is connected to the processor 14 so as to allow the processor 14 to receive data from another electronic device and/or to transmit data to another electronic device (for example a device configuration 22 as described below / or the electronic device 2). In certain embodiments, the processor 14 can thus receive data from the other electronic device, for example a zero-disclosure proof of knowledge coming from the electronic device 2, as described with reference to Figures 3 and 6, and/ or transmit other data, typically a challenge scalar as also described with reference to Figures 3 and 6.

A configuration device 22 is also represented schematically in the . The configuration device 22 comprises a processor 24 (for example a microprocessor), a storage block 26, a RAM 28 and a communication block 30.

The RAM 28 and the storage block 26 are each linked to the processor 24 so that the processor 24 can read or write data in the storage block 26 and/or the RAM 28.

The storage block 26 stores computer program instructions, some of which are designed to cooperate with the electronic device 2 during a cryptographic processing method such as at least one of those described with reference to Figures 3 to 6 when these instructions are executed by processor 24.

The storage block 26 can also store computer program instructions designed to cooperate with the verification device 12.

The storage block 26 is for example in practice a hard disk or a non-volatile memory, possibly rewritable, for example of the EEPROM type (for "Electrically Erasable and Programmable Read-Only Memory" according to the Anglo-Saxon name commonly used). .

The RAM 28 can memorize at least some of the elements (in particular scalars and points of an elliptical curve) manipulated during the different processing operations carried out to cooperate with the electronic device 2 during at least one of the methods described below.

According to one possible embodiment, the computer program instructions stored in the storage block 26 were for example received (for example from a remote computer) during an operating phase of the configuration device 22 prior to the methods described with reference to Figures 3 to 6.

The communication block 30 is connected to the processor 24 so as to allow the processor 24 to receive data from another device and/or to transmit data to another electronic device (for example the monitoring device). verification 12 and/or the electronic device 2). In certain embodiments, the processor 24 can thus receive data from the other electronic device, for example a first secret scalar coming from the electronic device 2 or a root public key, as described with reference to Figures 3 and 6, and/or transmit other data, typically a first secret scalar as also described with reference to Figures 3 and 6.

Each figure among Figures 3 and 6 represents a cryptographic processing method according to the invention.

Each of these processes is implemented here by the electronic device 2 (due to the execution of the computer program instructions stored in the storage block 6 as indicated above).

Each of the methods described with reference to Figures 3 and 4, implements operations in a finite group Gr of first order n, of points of an elliptic curve E. This group Gr is provided with an addition operation, noted below +, of two points of the elliptic curve E. A scalar product denoted [x]·Y, is the multiplication of a point Y of the elliptic curve by a scalar x and corresponds to the iteration x-1 times of the addition operation of the group Gr on said point Y: Y + … + Y. This scalar product can be based on the accumulation of multiples of this point. Note that in the particular case of a scalar x worth 0, the result of the scalar product is the point at infinity of the group Gr.

The elliptic curve E is defined by a set of parameters including a point of the group Gr called generator point G, which is public. This generating point is therefore predetermined and can in practice be stored in the electronic device 2, for example in its storage block 6.

Any point in the group Gr can be uniquely associated with a set of z affine coordinates in a coordinate system of a z-dimensional space.

In addition, each of the methods described with reference to Figures 3 and 4 implements algebraic calculations using algebraic operations (addition, product, subtraction) on scalars.

There represents an example of an elliptic curve E in an affine coordinate system of dimension 2, and an example of a group Gr of points of the elliptic curve E defined on a body K.

The invention is not limited to this example of an elliptic curve and group Gr. According to other examples, the elliptic curve E can have another shape in this reference, or the elliptic curve E can be defined in another reference and /or another space, in particular of dimension greater than 2.

There represents in flowchart form the main steps of a cryptographic processing method according to a first mode of implementation of the invention.

This method is implemented by the electronic device 2 in cooperation with a verification device, typically the verification device 12.

According to a recording step E120, the processor 4 records in a memory of the electronic device 2 a root private key r.

This step is in a phase of the process called the initialization phase (Phase P1) of a set of at least one first secret scalar, forming the root private key r, said root private key r being associated with a root public key R .

This initialization phase is typically implemented by the initialization module of the electronic device 2.

Here the root private key r is formed from a single first secret scalar r ₁ .

In other words, the set of at least one first secret scalar, forming the root private key r, has a single first secret scalar, r ₁ .

The process is thus simplified. The method allows the use of a single and unique r-root private key for a determined R-root public key. The method also allows the use of a single and unique R root public key for a determined R root private key.

Typically, the root public key R is equal to the scalar product of the generating point G by the first secret scalar r ₁ : R = [r ₁ ]·G.

According to one example, when the method is implemented by each electronic device of a plurality of electronic devices, the same first secret scalar and the same root public key R are used for each of the electronic devices of said plurality. Thus, verification is simplified because it requires the use of a single and unique root public key R, which is common to all electronic devices of said plurality.

According to another example, when the method is implemented by each electronic device of a plurality of electronic devices, each of the electronic devices of said plurality uses a first secret scalar and a root public key R which are specific to it, which makes the more secure process.

During this recording step (step E120), the processor 4 therefore records the first secret scalar r ₁ in a memory of the electronic device 2.

The initialization phase (Phase P1) may include the following complementary steps (steps E110, E112 and E114).

Thus the method can comprise, before the recording step (step E120), a step of calculation, by the processor 4, of all the first secret scalars of the set forming the root private key r (step E110), c that is to say here of the first secret scalar r ₁ .

Typically, at least a first secret scalar of the root private key, here the first secret scalar r _{1 ,} can be calculated by random selection or by application of a non-clonable physical function, for example a function as described in the document ' A PUF taxonomy' by Thomas McGrath, Ibrahim E. Bagci, Zhiming M. Wang, Utz Roedig, Robert J. Young, published on 02/12/2019.

The method is thus particularly secure because the electronic device can alone ensure the confidentiality of the root private key r. In addition, the electronic device has its own private root r key.

The method then includes a step of calculating the root public key R from the root private key r (step E112).

Typically, the processor 4 can calculate the root public key R as follows: R = [r ₁ ]·G.

The method can then include a step of sending to a configuration device (step E114), typically the configuration device 22, the calculated root public key R.

Thus, advantageously the configuration device can certify the root public key R.

The method thus allows an authority having the configuration device to ensure that the initialization phase of the electronic device has been carried out under its control.

Certification of the root public key may consist of registering the root public key in a list of at least one trusted public key, or of cryptographic signing of said root public key, for example with a private certification key. issued by a trusted authority.

The step of calculating the root public key R from the root private key r (step E112) and the step of sending the root public key R to a configuration device (step E114) are here executed before the recording step (E120) but the step of calculating the root public key R from the root private key r (step E112) and/or the step of sending to a configuration device (step E114) can be executed after the recording step (E120) as long as the step of sending to a configuration device (step E114) is executed after the step of calculating the root public key R from the key private root r (step E112).

It should also be noted that the step of sending the root public key R to a configuration device (step E114) can be omitted.

According to another possibility, the initialization phase (Phase P1) may include other complementary steps (following steps E111, E112 and E114).

Thus the method can comprise, before the recording step (step E120), a reception step (step E111), from the configuration device, of at least a first secret scalar of the root private key r, c that is to say here of the first secret scalar r ₁ , said at least one first secret scalar received from the configuration device being calculated from a master private key.

Typically, the configuration device has a fourth secret scalar forming the master private key.

The method is thus particularly secure because the determination of the root private key requires cooperation from the configuration device.

Here the root private key r is formed from a single first secret scalar r ₁ , the electronic device therefore receives from the configuration device, all the first secret scalars of the set forming the root private key.

Thus, the efficiency of the process is increased by limiting the interactions between the configuration device and the electronic device.

According to one example, the first secret scalar r ₁ received from the configuration device is the master private key.

According to another example, the first secret scalar r ₁ received from the configuration device can be derived by the configuration device from the master private key using any suitable derivation function well known to those skilled in the art.

The root public key R can be calculated by the configuration device.

According to another possibility, the method can also include a step of calculation by the electronic device 2 of the root public key R from the root private key r (step E112) and a step of sending to the configuration device (step E114 ) of the calculated root public key R.

The step of calculating the root public key R from the root private key r (step E112) and the step of sending the root public key R to a configuration device (step E114) are illustrated in the before the recording step (E120), but when they are not omitted after the reception step (step E111), the step of calculating the root public key R from the root private key r ( step E112) and/or the step of sending to a configuration device (step E114) can be executed after the recording step (step E120) as long as the step of sending to a configuration device ( step E114) is executed after the step of calculating the root public key R from the root private key r (step E112).

Advantageously, the configuration device can certify the root public key R.

The method thus allows an authority having the configuration device to ensure that the initialization phase of the electronic device has been carried out under its control.

Certification of the root public key may consist of registering the root public key in a list of at least one trusted public key, or of cryptographic signing of said root public key, for example with a private certification key. issued by a trusted authority.

After the recording step (step E120), the method then comprises a step (step E220) of calculating a second secret scalar forming a derived private key a equal to f(s, k) where f is a function of derivation, s is a non-empty subset of the set of at least one first secret scalar, and k is a third secret scalar: a = f(s, k).

Here, the set of at least one first secret scalar has a single first secret scalar, the first secret scalar r ₁ , and the subset s therefore has a single first secret scalar, the first secret scalar r ₁ .

The third secret scalar k can be determined by the electronic device by random selection or by application of a non-clonable physical function, for example a function as described in the document 'A PUF taxonomy' by Thomas McGrath, Ibrahim E. Bagci, Zhiming M. Wang, Utz Roedig, Robert J. Young, published 02/12/2019.

According to a first example, the differentiation function f can be defined as follows: f(s, k) = k·r ₁ mod n.

According to a second example, the differentiation function f can be defined as follows: f(s, k) = k + r ₁ mod n.

The method then comprises a step of calculating a derived public key A associated with the second secret scalar a (step E240), that is to say with the derived private key a.

Typically, the processor 4 can calculate the derived public key A as follows: A = [a]·G. According to a first other possibility, when the derivation function f used during the step of calculating a second secret scalar (step E220) is defined according to the first example described above, the processor 4 can calculate the derived public key A as follows: A = [k] · R.

According to a second other possibility, when the derivation function f used during the step of calculating a second secret scalar (step E220) is defined according to the second example described above, the processor 4 can calculate the derived public key A as follows: A = R + [k]·G.

The step of calculating a second secret scalar (step E220) and the step of calculating a derived public key A (step E240) are in a phase of the method called the phase of generating a derived private key and 'a derived public key (phase P2).

This phase of generating a derived private key and a derived public key (phase P2) is typically implemented by the key generation module of the electronic device 2.

The method then comprises a step of generating a proof with zero disclosure of knowledge (step E320) that the derived public key A is associated with a scalar equal to f(u, l), u being the subset s of l set of at least one first secret scalar, forming the root private key r associated with the root public key R and l being equal to k.

The step of generating a proof with zero knowledge disclosure (step E320) is in a phase of the method called the proof generation phase (phase P3).

This proof generation phase (phase P3) is typically implemented by the proof generation module of the electronic device 2.

The method then comprises a step of transmitting the zero knowledge disclosure proof to the verification device (step E420), during which the electronic device 2 transmits to the verification device, typically using its communication block 10, the disclosure proof zero knowledge.

The method thus allows the verification device to verify that the derived private key a derives from the subset s of the set of at least one first secret scalar forming the root private key r recorded in the electronic device, said root private key r being associated with the root public key R.

Furthermore, the method allows this verification without disclosing, typically to the verification device, the root private key r, nor the subset s, nor the derived private key a, nor the third secret scalar k.

The step of transmitting the proof with zero knowledge disclosure to the verification device (step E420) is in a phase of the method called the transmission phase (phase P4).

This transmission phase (phase P4) is typically implemented by the transmission module of the electronic device 2.

According to one implementation possibility, the proof with zero knowledge disclosure is a non-interactive proof.

The implementation of the proof generation phase and the transmission phase is thus simplified, in particular by limiting the exchanges between the electronic device and the verification device which can be unidirectional and asynchronous.

When the derivation function f used during the step of calculating a second secret scalar (step E220) is defined according to the first example described above for this step, the processor 4 can perform the following calculations to generate the proof to zero disclosure of non-interactive knowledge during the associated step (step E320).

The processor 4 calculates a first intermediate point as follows: S ₁ = [t ₁ ]·G where t ₁ is a first intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to a data.

The processor 4 also calculates a second intermediate point as follows: S ₂ = [t ₂ ]·R where t ₂ is a second intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to another datum.

The processor 4 then calculates a first condensate of intermediate data by a hash function, for example SHA256 or SHA3, as specified in the FIPS PUB 180-4 document entitled 'Secure Hash Standard' and published in August 2015 by NIST , the intermediate data being the concatenation of the first intermediate point S ₁ , the second intermediate point S ₂ , the root public key R and the derived public key A.

That is, c = Ha(S ₁ || S ₂ ||R||A), where c is the first condensate and Ha is the hash function.

Then the processor 4 respectively calculates a first proof scalar s ₁ and a second proof scalar s ₂ as follows: s ₁ = t ₁ + c·r ₁ mod n and s ₂ = t ₂ + c·k mod n.

The proof with zero knowledge disclosure is thus made up of the first condensate c, the first proof scalar s ₁ and the second proof scalar s ₂ .

During the step of transmitting the proof with zero knowledge disclosure to the verification device (step E420), the electronic device 2 can then transmit to the verification device the first condensate c, the first proof scalar s ₁ and the second scalar of proof s ₂ .

The verification device can verify the zero-knowledge proof.

Typically the verification device can calculate a third intermediate point S' ₁ as follows: S' ₁ = [s ₁ ]·G - [c]·R.

The verification device can also calculate a fourth intermediate point S' ₂ as follows: S' ₂ = [s ₂ ]·R - [c]·A.

The verification device can then calculate a second condensate of another intermediate data by the hash function, the other intermediate data being the concatenation of the third intermediate point S' ₁ , of the fourth intermediate point S' ₂ , of the public key root R and the derived public key A.

In other words, c' = Ha(S' ₁ || S' ₂ ||R||A), where it is the second condensate.

The verification device can then compare the second condensate c’ with the first condensate c. If the second condensate c' is equal to the first condensate c, the derived private key a associated with the derived public key A, derives from the subset s of the set of at least one first secret scalar forming the root private key r recorded in the electronic device, said root private key r being associated with the root public key R.

The derived public key A and the root public key R are public keys and can therefore be received by the verification device from any other devices, for example from the configuration device 22 and/or from the electronic device 2.

When the derivation function f used during the step of calculating a second secret scalar (step E220) is defined according to the second example described above for this step, the processor 4 can perform the following calculations to generate the proof to zero disclosure of non-interactive knowledge during the associated step (step E320).

The processor 4 calculates a first intermediate point as follows: S ₁ = [t ₁ ]·G where t ₁ is a first intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to a data.

The processor 4 also calculates a second intermediate point as follows: S ₂ = [t ₂ ]·G where t ₂ is a second intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to another datum.

The processor 4 then calculates a first condensate of intermediate data by a hash function, for example SHA256 or SHA3, as specified in the FIPS PUB 180-4 document entitled 'Secure Hash Standard' and published in August 2015 by NIST , the intermediate data being the concatenation of the first intermediate point S ₁ , the second intermediate point S ₂ , the root public key R and the derived public key A.

That is, c = Ha(S ₁ || S ₂ ||R||A), where c is the first condensate and Ha is the hash function.

Then the processor 4 respectively calculates a first proof scalar s ₁ and a second proof scalar s ₂ as follows: s ₁ = t ₁ + c·r ₁ mod n and s ₂ = t ₂ + c·k mod n.

The proof with zero knowledge disclosure is thus made up of the first condensate c, the first proof scalar s ₁ and the second proof scalar s ₂ .

During the step of transmitting the proof with zero knowledge disclosure to the verification device (step E420), the electronic device 2 can then transmit to the verification device the first condensate c, the first proof scalar s ₁ and the second scalar of proof s ₂ .

The verification device can verify the zero-knowledge proof.

Typically the verification device can calculate a third intermediate point S' ₁ as follows: S' ₁ = [s ₁ ]·G - [c]·R.

The verification device can also calculate a fourth intermediate point S' ₂ as follows: S' ₂ = [s ₂ ] · G - [c] · (AR).

The verification device can then calculate a second condensate of another intermediate data by the hash function, the other intermediate data being the concatenation of the third intermediate point S' ₁ , of the fourth intermediate point S' ₂ , of the public key root R and the derived public key A.

In other words, c' = Ha(S' ₁ || S' ₂ ||R||A), where it is the second condensate.

The verification device can then compare the second condensate c’ with the first condensate c. If the second condensate c' is equal to the first condensate c, the derived private key a associated with the derived public key A, derives from the subset s of the set of at least one first secret scalar forming the root private key r recorded in the electronic device, said root private key r being associated with the root public key R.

The derived public key A and the root public key R are public keys and can therefore be received by the verification device from any other devices, for example from the configuration device and/or the electronic device.

According to another implementation possibility, the proof with zero knowledge disclosure is an interactive proof.

The generation of the zero-knowledge proof, and the transmission of the zero-knowledge proof to a verification device, are for example in the form of a challenge-response protocol.

The method thus allows the verification device to verify that the electronic device has the root private key r and the third secret scalar k, that is to say that the electronic device can carry out at least one processing involving all the first scalars secrets of the set forming the root private key and at least one processing involving the third secret scalar k.

When the derivation function f used during the step of calculating a second secret scalar (step E220) is defined according to the first example described above for this step, the processor 4 can perform the following calculations to generate the proof for disclosure zero interactive knowledge during the associated step (step E320).

The processor 4 calculates a first intermediate point as follows: S ₁ = [t ₁ ]·G where t ₁ is a first intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to a data.

The processor 4 also calculates a second intermediate point as follows: S ₂ = [t ₂ ]·R where t ₂ is a second intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to another datum.

Then the processor 4 respectively calculates a first proof scalar s ₁ and a second proof scalar s ₂ as follows: s ₁ = t ₁ + c r ₁ mod n and s ₂ = t ₂ + c k mod n, where it is a challenge scalar previously received by the electronic device 2 from the verification device.

The proof with zero knowledge disclosure is thus made up of the first intermediate point S ₁ , the second intermediate point S ₂ , the first proof scalar s ₁ and the second proof scalar s ₂ .

During the step of transmitting the proof with zero knowledge disclosure to the verification device (step E420), the electronic device 2 can then transmit to the verification device the first intermediate point S ₁ , the second intermediate point S ₂ , the first proof scalar s ₁ and the second proof scalar s ₂ .

Preferably, to improve security, the challenge scalar c is received by the electronic device 2 from the verification device, prior to the calculation of the first proof scalar s ₁ and the second proof scalar s ₂ , but after the sending by the electronic device 2 to the verification device, of the first intermediate point S ₁ and the second intermediate point S ₂ . The step of generating a proof with zero knowledge disclosure (step E320) and the step of transmitting the proof with zero knowledge disclosure to the verification device (step E420), respectively the proof generation phase and the transmission phase, are then carried out concomitantly (not shown in the figure).

The verification device can verify the zero-knowledge proof.

Typically the verification device can calculate a third intermediate point S' ₁ as follows: S' ₁ = S ₁ + [c]·R.

The verification device can also calculate a fourth intermediate point S' ₂ as follows: S' ₂ = S _{2 +} [c]·A.

The verification device can calculate a fifth intermediate point S'' ₁ as follows: S'' ₁ = [s ₁ ]·G.

The verification device can calculate a sixth intermediate point S'' ₂ as follows: S'' ₂ = [s ₂ ] · R .

The verification device can then respectively compare the third intermediate point _S'1 with the fifth intermediate point S''1 _, and the fourth intermediate point _S'2 with the sixth intermediate point _S''2 .

If the third intermediate point S' ₁ is equal to the fifth intermediate point S'' _1, and if the fourth intermediate point S' ₂ is equal to the sixth intermediate point S'' ₂ , the derived private key a associated with the derived public key A, derives from the subset s of the set of at least one first secret scalar forming the root private key r recorded in the electronic device, said root private key r being associated with the root public key R, and the electronic device has the root private key r and the third secret scalar k.

The derived public key A and the root public key R are public keys and can therefore be received by the verification device from any other devices, for example from the configuration device and/or the electronic device 2.

When the derivation function f used during the step of calculating a second secret scalar (step E220) is defined according to the second example described above for this step, the processor 4 can perform the following calculations to generate the proof for disclosure zero interactive knowledge during the associated step (step E320).

The processor 4 calculates a first intermediate point as follows: S ₁ = [t ₁ ]·G where t ₁ is a first intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to a data.

The processor 4 also calculates a second intermediate point as follows: S ₂ = [t ₂ ]·G where t ₂ is a second intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to another datum.

Then the processor 4 respectively calculates a first proof scalar s ₁ and a second proof scalar s ₂ as follows: s ₁ = t ₁ + c r ₁ mod n and s ₂ = t ₂ + c k mod n, where it is a challenge scalar previously received by the electronic device from the verification device.

The proof with zero knowledge disclosure is thus made up of the first intermediate point S ₁ , the second intermediate point S ₂ , the first proof scalar s ₁ and the second proof scalar s ₂ .

During the step of transmitting the proof with zero knowledge disclosure to the verification device (step E420), the electronic device 2 can then transmit to the verification device the first intermediate point S ₁ , the second intermediate point S ₂ , the first proof scalar s ₁ and the second proof scalar s ₂ .

Preferably, to improve security, the challenge scalar c is received by the electronic device 2 from the verification device, prior to the calculation of the first proof scalar s ₁ and the second proof scalar s ₂ , but after the sending by the electronic device 2 to the verification device, of the first intermediate point S ₁ and the second intermediate point S ₂ . The step of generating a proof with zero knowledge disclosure (step E320) and the step of transmitting the proof with zero knowledge disclosure to the verification device (step E420), respectively the proof generation phase and the transmission phase, are then carried out concomitantly (not shown in the figure).

The verification device can verify the zero-knowledge proof.

Typically the verification device can calculate a third intermediate point S' ₁ as follows: S' ₁ = S ₁ + [c]·R.

The verification device can also calculate a fourth intermediate point S' ₂ as follows: S' ₂ = S _{2 +} [c]·(AR).

The verification device can calculate a fifth intermediate point S'' ₁ as follows: S'' ₁ = [s ₁ ]·G.

The verification device can calculate a sixth intermediate point S'' ₂ as follows: S'' ₂ = [s ₂ ] · G .

The verification device can then respectively compare the third intermediate point _S'1 with the fifth intermediate point S''1 _, and the fourth intermediate point _S'2 with the sixth intermediate point _S''2 .

If the third intermediate point S' ₁ is equal to the fifth intermediate point S'' _1, and if the fourth intermediate point S' ₂ is equal to the sixth intermediate point S'' ₂ , the derived private key a associated with the derived public key A, derives from the subset s of the set of at least one first secret scalar forming the root private key r recorded in the electronic device, said root private key r being associated with the root public key R, and the electronic device has the root private key r and the third secret scalar k.

The derived public key A and the root public key R are public keys and can therefore be received by the verification device from any other devices, for example from the configuration device and/or the electronic device.

A person skilled in the art will understand that the process described with reference to the with a root private key r formed by a single first secret scalar r ₁ , can be adapted for a root private key r formed by a plurality of first secret scalars.

Thus, when the method is implemented by each electronic device of a plurality of electronic devices, each of the electronic devices of said plurality can use a private root key r which is specific to it and the same public root key R. The use of a root r private key specific to each electronic device of the plurality makes the method more secure with respect to side channel attacks aimed at obtaining a root r private key. In addition, verification is simplified because it requires the use of a single and unique root public key R, which is common to all electronic devices of said plurality.

There represents in flowchart form the main steps of a cryptographic processing method according to a second mode of implementation of the invention.

This method is implemented by the electronic device 2 in cooperation with a verification device, typically the verification device 12.

According to a recording step E1120, the processor 4 records in a memory of the electronic device 2 a root private key r.

This step is in a phase of the process called the initialization phase (Phase P1) of a set of at least one first secret scalar, forming the root private key r, said root private key r being associated with a root public key R .

This initialization phase is typically implemented by the initialization module of the electronic device 2.

Here the root private key r is formed from two first secret scalars r₁, r₂.

In other words, the set of at least one first secret scalar, forming the root private key r, has two first secret scalars: r₁And r₂.

Typically, the root public key R has a value defined as follows: R = [r ₁ ] · G + [r ₂ ] · H with H an additional public point of the elliptic curve E, the additional public point H being associated with a supplementary scalar h such that H = [h]·G.

During this recording step (step E1120), the processor 4 therefore records in a memory of the electronic device 2 the first two secret scalars r ₁ and r ₂ .

According to a first possibility, the initialization phase (Phase P1) may include the following complementary steps (steps E1110, E1112 and E1114).

Thus the method can comprise, before the recording step (step E1120), a step of calculation, by the processor 4, of all the first secret scalars of the set forming the root private key r (step E1110), c that is to say here of the first secret scalar r ₁ and the first secret scalar r ₂ .

Typically, the first secret scalar r₁, and or the first secret scalar r₂, can be calculated by random sampling or by application of a non-clonable physical function, for example a function as described in the document 'A PUF taxonomy' by Thomas McGrath, Ibrahim E. Bagci, Zhiming M. Wang, Utz Roedig, Robert J. Young, published 02/12/2019.

The method is thus particularly secure because the electronic device can alone ensure the confidentiality of the root private key r. In addition, the electronic device has its own private root r key.

The method then includes a step of calculating the root public key R from the root private key r (step E1112).

Typically, the processor 4 can calculate the root public key R as follows: R = [r ₁ ] · G + [r ₂ ] · H.

The method can then include a step of sending to a configuration device (step E1114), typically the configuration device 22, the calculated root public key R.

Thus, advantageously the configuration device can certify the root public key R.

The method thus allows an authority having the configuration device to ensure that the initialization phase of the electronic device has been carried out under its control.

Certification of the root public key may consist of registering the root public key in a list of at least one trusted public key, or of cryptographic signing of said root public key, for example with a private certification key. issued by a trusted authority.

The step of calculating the root public key R from the root private key r (step E1112) and the step of sending the root public key R to a configuration device (step E1114) are here executed before the recording step (E1120) but the step of calculating the root public key R from the root private key r (step E1112) and/or the step of sending to a configuration device (step E1114) can be executed after the recording step (E1120) as long as the step of sending to a configuration device (step E1114) is executed after the step of calculating the root public key R from the key private root r (step E1112).

It should also be noted that the step of sending the root public key R to a configuration device (step E1114) can be omitted.

According to a second possibility, the initialization phase (Phase P1) may include another complementary step (step E1111 described below).

Thus the method can comprise, before the recording step (step E1120), a reception step (step E1111), from the configuration device, of at least a first secret scalar of the root private key (r) , for example the first secret scalar r ₂ , said at least one first secret scalar received from the configuration device being calculated from a master private key.

Typically, the configuration device has a fourth secret scalar forming the master private key.

The method is thus particularly secure because the determination of the root private key requires cooperation from the configuration device.

Typically, the master private key is a fourth secret scalar m such that R = [m] · G and the configuration device can calculate the first secret scalar r₂as following : r₂= h^-1·(m - r₁).

The root public key R can be calculated by the configuration device.

According to one implementation, the configuration device may have previously calculated the first secret scalar r _1, for example by random selection. In this implementation, during the reception step (step E1111), the electronic device also receives, from the configuration device, the first secret scalar r ₁ .

According to another implementation, the configuration device may have previously received the first secret scalar r ₁ from the electronic device, said electronic device having previously calculated the first secret scalar r _1, for example by random drawing or by application of a function unclonable physics, for example a function as described in the document 'A PUF taxonomy' by Thomas McGrath, Ibrahim E. Bagci, Zhiming M. Wang, Utz Roedig, Robert J. Young, published on 02/12/2019.

According to a third possibility, the initialization phase (Phase P1) can comprise the following different complementary steps (steps E1116, E1117, E1118 and E1119), before the recording step (step E1120).

Thus the method can include a step of determining a first secret scalar of the root private key (r) (step E1116), for example the first secret scalar r ₁ , typically by random drawing or by application of a non-physical function. clonable, for example a function as described in the document 'A PUF taxonomy' by Thomas McGrath, Ibrahim E. Bagci, Zhiming M. Wang, Utz Roedig, Robert J. Young, published on 02/12/2019.

When the method is implemented by each electronic device of a plurality of electronic devices, the method thus ensures that each of the electronic devices of said plurality uses a root private key r which is specific to it. This makes the process more secure.

The method then comprises a step of homomorphically encrypting (step E1117) the first determined secret scalar, that is to say the first secret scalar r ₁ . Homomorphic encryption is for example a Paillier encryption.

Then the method comprises a step of transmitting to the device for configuring the cipher of the first determined secret scalar (step E1118), that is to say the cipher of the first secret scalar r ₁ .

The method then includes a step of receiving (step E1119), from the configuration device, another first secret scalar of the root private key (r). During this step, the electronic device 2 receives the other first secret scalar of the root private key (r) encrypted in the same homomorphic manner, the cipher of said other first secret scalar having been determined from the master private key and the cipher transmitted by the electronic device to the configuration device, and the electronic device 2 decrypts the cipher of the other first secret scalar.

The method is particularly secure because the determination of the root private key requires cooperation from the configuration device. The security of the method is further increased via the implementation by the electronic device of the determination of a first secret scalar of the root private key r, and the non-disclosure by the electronic device of the first secret scalar determined by said electronic device .

Typically, the master private key is a fourth secret scalar m such that R = [m]·G and the configuration device can determine the other first secret scalar of the root private key (r), encrypted in the same homomorphic manner, as following : hom(r₂) = h^-1·( hom(m)- hom(r₁)) with hom the homomorphic encryption function and hom(r₁) the homomorphically encrypted first secret scalar determined by the electronic device.

For each of these three possibilities, the configuration device can advantageously certify the root public key R.

The method thus allows an authority having the configuration device to ensure that the initialization phase of the electronic device has been carried out under its control.

Certification of the root public key may consist of registering the root public key in a list of at least one trusted public key, or of cryptographic signing of said root public key, for example with a private certification key. issued by a trusted authority.

After the recording step (step E1120), the method then comprises a step (step E1220) of calculating a second secret scalar forming a derived private key a equal to f(s, k) where f is a function of derivation, s is a non-empty subset of the set of at least one first secret scalar, and k is a third secret scalar: a = f(s, k).

Here, the set of at least one first secret scalar has two first secret scalars r ₁ , r ₂ , and the subset s has only one first secret scalar, the first secret scalar r ₁ .

The third secret scalar k can be determined by the electronic device by random selection or by application of a non-clonable physical function, for example a function as described in the document 'A PUF taxonomy' by Thomas McGrath, Ibrahim E. Bagci, Zhiming M. Wang, Utz Roedig, Robert J. Young, published 02/12/2019.

According to a first example, the differentiation function f can be defined as follows: f(s, k) = k·r ₁ mod n.

According to a second example, the differentiation function f can be defined as follows: f(s, k) = k + r ₁ mod n.

The method then includes a step of calculating a derived public key A associated with the second secret scalar a (step E1240), that is to say with the derived private key a.

Typically, the processor 4 can calculate the derived public key A as follows: A = [a]·G.

According to a first other possibility, when the derivation function f used during the step of calculating a second secret scalar (step E1220) is defined according to the first example described above, the processor 4 can calculate the derived public key A as follows: A = [k] · R – [kr ₂ ].H.

According to a second other possibility, when the derivation function f used during the step of calculating a second secret scalar (step E1220) is defined according to the second example described above, the processor 4 can calculate the derived public key A as follows: A = R + [k]·G – [r ₂ ].H.

The step of calculating a second secret scalar (step E1220) and the step of calculating a derived public key A (step E1240) are in a phase of the method called the phase of generating a derived private key and 'a derived public key (phase P2).

This phase of generating a derived private key and a derived public key (phase P2) is typically implemented by the key generation module of the electronic device 2.

The method then comprises a step of generating a proof with zero disclosure of knowledge (step E1320) that the derived public key A is associated with a scalar equal to f(u, l), u being the subset s of l set of at least one first secret scalar, forming the root private key r associated with the root public key R and l being equal to k.

The step of generating a proof with zero knowledge disclosure (step E1320) is in a phase of the method called the proof generation phase (phase P3).

This proof generation phase (phase P3) is typically implemented by the proof generation module of the electronic device 2.

The method then comprises a step of transmitting the zero knowledge disclosure proof to the verification device (step E1420), during which the electronic device 2 transmits to the verification device, typically using its communication block 10, the disclosure proof zero knowledge.

The method thus allows the verification device to verify that the derived private key a derives from the subset s of the set of at least one first secret scalar forming the root private key r recorded in the electronic device, said root private key r being associated with the root public key R.

Furthermore, the method allows this verification without disclosing, typically to the verification device, the root private key r, nor the subset s, nor the derived private key a, nor the third secret scalar k.

The step of transmitting the proof with zero knowledge disclosure to the verification device (step E1420) is in a phase of the method called the transmission phase (phase P4).

This transmission phase (phase P4) is typically implemented by the transmission module of the electronic device 2.

According to one implementation possibility, the proof with zero knowledge disclosure is a non-interactive proof.

The implementation of the proof generation phase and the transmission phase is thus simplified, in particular by limiting the exchanges between the electronic device and the verification device which can be unidirectional and asynchronous.

When the derivation function f used during the step of calculating a second secret scalar (step E1220) is defined according to the first example described above for this step, the processor 4 can perform the following calculations to generate the proof to zero disclosure of non-interactive knowledge during the associated step (step E1320).

The processor 4 calculates a first intermediate point as follows: S ₁ = [t ₁ ] · G + [t ₂ ] · H where t ₁ is a first intermediate scalar determined by the electronic device 2, for example by random selection or by application of a pseudo-random algorithm to a piece of data, and t ₂ is a second intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to a piece of data.

The processor 4 also calculates a second intermediate point as follows: S ₂ = [t ₃ ]·G where t ₃ is a third intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to another datum.

The processor 4 also calculates a third intermediate point as follows: S ₃ = [t ₄ ] · R - [t ₅ ] · H where t ₄ is a fourth intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to another piece of data, and t ₅ is a fifth intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to another piece of data.

The processor 4 then calculates a first condensate of intermediate data by a hash function, for example SHA256 or SHA3, as specified in the FIPS PUB 180-4 document entitled 'Secure Hash Standard' and published in August 2015 by NIST , the intermediate data being the concatenation of the first intermediate point S₁, from the second intermediate point S₂, from the third intermediate point S₃, of the root public key R and the derived public key A.

That is, c = Ha(S ₁ || S ₂ || S ₃ || R || A), where c is the first condensate and Ha is the hash function.

Then the processor 4 respectively calculates a first proof scalar s ₁ , a second proof scalar s ₂ , a third proof scalar s ₃ , a fourth proof scalar s ₄ and a fifth proof scalar s ₅ as follows: s ₁ = t ₁ + c r ₁ mod n, s ₂ = t ₂ + c r ₂ mod n, s ₃ = t ₃ + c r ₁ k mod n, s ₄ = t ₄ + c k mod n and s ₅ = t ₅ + c r ₂ k mod n.

The proof with zero knowledge disclosure is thus made up of the first condensate c, the first proof scalar s ₁ , the second proof scalar s ₂ , the third proof scalar s ₃ , the fourth proof scalar s ₄ and the fifth scalar of proof s ₅ .

During the step of transmitting the proof with zero knowledge disclosure to the verification device (step E1420), the electronic device 2 can then transmit to the verification device the first condensate c, the first proof scalar s ₁ , the second scalar of proof s ₂ , the third scalar of proof s ₃ , the fourth scalar of proof s ₄ and the fifth scalar of proof s _5.

The verification device can verify the zero-knowledge proof.

Typically the verification device can calculate a fourth intermediate point S' ₁ as follows: S' ₁ = [s ₁ ] · G + [s ₂ ] · H - [c] · R.

The verification device can also calculate a fifth intermediate point S' ₂ as follows: S' ₂ = [s ₃ ]·G - [c]·A.

The verification device can also calculate a sixth intermediate point S' ₃ as follows: S' ₃ = [s ₄ ]·R - [s ₅ ]·H - [c]·A.

The verification device can then calculate a second condensate of another intermediate data by the hash function, the other intermediate data being the concatenation of the fourth intermediate point S' ₁ , of the fifth intermediate point S' ₂ , of the sixth intermediate point S' ₃ , of the root public key R and the derived public key A.

In other words, c' = Ha(S' ₁ || S' ₂ || S' ₃ || R || A), where it is the second condensate.

The verification device can then compare the second condensate c’ with the first condensate c. If the second condensate c' is equal to the first condensate c, the derived private key a associated with the derived public key A, derives from the subset s of the set of at least one first secret scalar forming the root private key r recorded in the electronic device, said root private key r being associated with the root public key R.

The derived public key A, the additional public point H and the root public key R are public and can therefore be received by the verification device from any other devices, for example from the configuration device and/or the electronic device 2.

When the derivation function f used during the step of calculating a second secret scalar (step E1220) is defined according to the second example described above for this step, the processor 4 can perform the following calculations to generate the proof for disclosure zero interactive knowledge during the associated step (step E1320).

The processor 4 calculates a first intermediate point as follows: S ₁ = [t ₁ ] · G + [t ₂ ] · H · where t ₁ is a first intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to a piece of data, and t ₂ is a second intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to another piece of data.

The processor 4 also calculates a second intermediate point as follows: S ₂ = [t ₃ ]·G where t ₃ is a third intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to another datum.

The processor 4 also calculates a third intermediate point as follows: S ₃ = [t ₄ ] · G - [t ₂ ] · H where t ₄ is a fourth intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to another piece of data.

The processor 4 then calculates a first condensate of intermediate data by a hash function, for example SHA256 or SHA3, as specified in the FIPS PUB 180-4 document entitled 'Secure Hash Standard' and published in August 2015 by NIST , the intermediate data being the concatenation of the first intermediate point S₁, from the second intermediate point S₂, from the third intermediate point S₃, of the root public key R and the derived public key A.

That is, c = Ha(S ₁ || S ₂ || S ₃ || R || A), where c is the first condensate and Ha is the hash function.

Then the processor 4 respectively calculates a first proof scalar s ₁ , a second proof scalar s ₂ , a third proof scalar s ₃ and a fourth proof scalar s ₄ as follows: s ₁ = t ₁ + c · r ₁ mod n, s ₂ = t ₂ + c·r ₂ mod n, s ₃ = t ₃ + c·(r ₁ +k) mod n and s ₄ = t ₄ + c·k mod n.

The proof with zero knowledge disclosure is thus made up of the first condensate c, the first proof scalar s ₁ , the second proof scalar s ₂ , the third proof scalar s ₃ and the fourth proof scalar s _4.

During the step of transmitting the proof with zero knowledge disclosure to the verification device (step E1420), the electronic device 2 can then transmit to the verification device the first condensate c, the first proof scalar s ₁ , the second scalar of proof s ₂ , the third scalar of proof s ₃ and the fourth scalar of proof s _4.

The verification device can verify the zero-knowledge proof.

Typically the verification device can calculate a fourth intermediate point S' ₁ as follows: S' ₁ = [s ₁ ] · G + [s ₂ ] · H - [c] · R.

The verification device can also calculate a fifth intermediate point S' ₂ as follows: S' ₂ = [s ₃ ]·G - [c]·A.

The verification device can also calculate a sixth intermediate point S' ₃ as follows: S' ₃ = [s ₄ ] · R - [s ₂ ] · H - [c] · (AR).

The verification device can then calculate a second condensate of another intermediate data by the hash function, the other intermediate data being the concatenation of the fourth intermediate point S' ₁ , of the fifth intermediate point S' ₂ , of the sixth intermediate point S' ₂ , of the root public key R and the derived public key A.

In other words, c' = Ha(S' ₁ || S' ₂ || S' ₃ || R || A), where it is the second condensate.

The verification device can then compare the second condensate c’ with the first condensate c. If the second condensate c' is equal to the first condensate c, the derived private key a associated with the derived public key A, derives from the subset s of the set of at least one first secret scalar forming the root private key r recorded in the electronic device, said root private key r being associated with the root public key R.

The derived public key A, the additional public point H and the root public key R are public and can therefore be received by the verification device from any other devices, for example from the configuration device and/or the electronic device 2.

According to another implementation possibility, the proof with zero knowledge disclosure is an interactive proof.

The generation of the zero-knowledge proof, and the transmission of the zero-knowledge proof to a verification device, are for example in the form of a challenge-response protocol.

The method thus allows the verification device to verify that the electronic device has the root private key r and the third secret scalar k, that is to say that the electronic device can carry out at least one processing involving all the first scalars secrets of the set forming the root private key and at least one processing involving the third secret scalar k.

When the derivation function f used during the step of calculating a second secret scalar (step E1220) is defined according to the first example described above for this step, the processor 4 can perform the following calculations to generate the proof for disclosure zero interactive knowledge during the associated step (step E1320).

The processor 4 calculates a first intermediate point as follows: S ₁ = [t ₁ ] · G + [t ₂ ] · H · where t ₁ is a first intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to a piece of data, and t ₂ is a second intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to another piece of data.

The processor 4 also calculates a second intermediate point as follows: S ₂ = [t ₃ ]·G where t ₃ is a third intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to another datum.

The processor 4 also calculates a third intermediate point as follows: S ₃ = [t ₄ ] · R - [t ₅ ] · H where t ₄ is a fourth intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to another piece of data, and t ₅ is a fifth intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to another piece of data.

Then the processor 4 respectively calculates a first proof scalar s ₁ , a second proof scalar s ₂ , a third proof scalar s ₃ , a fourth proof scalar s ₄ and a fifth proof scalar s ₅ as follows: s ₁ = t ₁ + c r ₁ mod n, s ₂ = t ₂ + c r ₂ mod n, s ₃ = t ₃ + c r ₁ k mod n, s ₄ = t ₄ + c k mod n , s ₅ = t ₅ + c · r ₂ · k mod n where c is a challenge scalar previously received by the electronic device 2 from the verification device.

The zero knowledge disclosure proof thus consists of the first intermediate point S ₁ , the second intermediate point S ₂ , the third intermediate point S ₃ , the first proof scalar s ₁ , the second proof scalar s ₂ , the third scalar of proof s ₃ , of the fourth scalar of proof s ₄ and of the fifth scalar of proof s ₅ .

During the step of transmitting the proof with zero knowledge disclosure to the verification device (step E1420), the electronic device 2 can then transmit to the verification device the first intermediate point S ₁ , the second intermediate point S ₂ , the third intermediate point S ₃ , the first proof scalar s ₁ , the second proof scalar s ₂ , the third proof scalar s ₃ , the fourth proof scalar s ₄ and the fifth proof scalar s ₅ .

Preferably, to improve security, the challenge scalar c is received by the electronic device 2 from the verification device, prior to calculating the first proof scalar s ₁ , the second proof scalar s ₂ , the third proof scalar s 2, proof s ₃ , the fourth proof scalar s ₄ and the fifth proof scalar s ₅ , but after sending by the electronic device 2 to the verification device, the first intermediate point S ₁ , the second intermediate point S ₂ and the third intermediate point S ₃ . The step of generating a proof with zero knowledge disclosure (step E1320) and the step of transmitting the proof with zero knowledge disclosure to the verification device (step E1420), respectively the proof generation phase and the transmission phase, are then carried out concomitantly (not shown in the figure).

The verification device can verify the zero-knowledge proof.

Typically the verification device can calculate a fourth intermediate point S' ₁ as follows: S' ₁ = S ₁ + [c]·R.

The verification device can also calculate a fifth intermediate point S' ₂ as follows: S' ₂ = S _{2 +} [c]·A.

The verification device can also calculate a sixth intermediate point S' ₃ as follows: S' ₃ = S _{3 +} [c]·A.

The verification device can calculate a seventh intermediate point S'' ₁ as follows: S'' ₁ = [s ₁ ]·G + [s ₂ ]·H.

The verification device can calculate an eighth intermediate point S'' ₂ as follows: S'' ₂ = [s ₃ ]·G.

The verification device can calculate a ninth intermediate point S'' ₃ as follows: S'' ₃ = [s ₄ ]·R - [s ₅ ]·H.

The verification device can then compare respectively the fourth intermediate point _S'1 with the seventh intermediate point S''1 _, the fifth intermediate point _S'2 with the eighth intermediate point _S''2 , and the sixth intermediate point _S'3 with the ninth intermediate point S'' ₃ .

If the fourth intermediate point S’₁is equal to the seventh intermediate point S’’₁, if the fifth intermediate point S’₂is equal to the eighth intermediate point S’’₂, and if the sixth intermediate point S’₃is equal to the ninth intermediate point S’’₃, the derived private key a associated with the derived public key A, derives from the subset s of the set of at least one first secret scalar forming the root private key r recorded in the electronic device, said root private key r being associated with the root public key R, and the electronic device has the root private key r and the third secret scalar k.

The derived public key A, the additional public point H and the root public key R are public and can therefore be received by the verification device from any other devices, for example from the configuration device and/or the electronic device 2.

When the derivation function f used during the step of calculating a second secret scalar (step E1220) is defined according to the second example described above for this step, the processor 4 can perform the following calculations to generate the proof for disclosure zero interactive knowledge during the associated step (step E1320).

The processor 4 calculates a first intermediate point as follows: S ₁ = [t ₁ ] · G + [t ₂ ] · H · where t ₁ is a first intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to a piece of data, and t ₂ is a second intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to another piece of data.

The processor 4 also calculates a second intermediate point as follows: S ₂ = [t ₃ ]·G where t ₃ is a third intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to another datum.

The processor 4 also calculates a third intermediate point as follows: S ₃ = [t ₄ ] · R - [t ₂ ] · H where t ₄ is a fourth intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to another piece of data.

Then the processor 4 respectively calculates a first proof scalar s ₁ , a second proof scalar s ₂ , a third proof scalar s ₃ and a fourth proof scalar s ₄ as follows: s ₁ = t ₁ + c · r ₁ mod n, s ₂ = t ₂ + c·r ₂ mod n, s ₃ = t ₃ + c·(r ₁ +k) mod n and s ₄ = t ₄ + c·k mod n where c is a scalar of challenge previously received by the electronic device 2 from the verification device.

The zero knowledge disclosure proof thus consists of the first intermediate point S ₁ , the second intermediate point S ₂ , the third intermediate point S ₃ , the first proof scalar s ₁ , the second proof scalar s ₂ , the third scalar of proof s ₃ and of the fourth scalar of proof s ₄ .

During the step of transmitting the proof with zero knowledge disclosure to the verification device (step E1420), the electronic device 2 can then transmit to the verification device the first intermediate point S ₁ , the second intermediate point S ₂ , the third intermediate point S ₃ , the first proof scalar s ₁ , the second proof scalar s ₂ , the third proof scalar s ₃ and the fourth proof scalar s ₄ .

Preferably, to improve security, the challenge scalar c is received by the electronic device 2 from the verification device, prior to calculating the first proof scalar s ₁ , the second proof scalar s ₂ , the third proof scalar s 2, proof s ₃ and the fourth proof scalar s ₄ , but after sending by the electronic device 2 to the verification device, the first intermediate point S ₁ , the second intermediate point S ₂ and the third intermediate point S ₃ . The step of generating a proof with zero knowledge disclosure (step E1320) and the step of transmitting the proof with zero knowledge disclosure to the verification device (step E1420), respectively the proof generation phase and the transmission phase, are then carried out concomitantly (not shown in the figure).

The verification device can verify the zero-knowledge proof.

Typically the verification device can calculate a fourth intermediate point S' ₁ as follows: S' ₁ = S ₁ + [c]·R.

The verification device can also calculate a fifth intermediate point S' ₂ as follows: S' ₂ = S _{2 +} [c]·A.

The verification device can also calculate a sixth intermediate point S' ₃ as follows: S' ₃ = S _{3 +} [c]·(AR).

The verification device can calculate a seventh intermediate point S'' ₁ as follows: S'' ₁ = [s ₁ ]·G + [s ₂ ]·H.

The verification device can calculate an eighth intermediate point S'' ₂ as follows: S'' ₂ = [s ₃ ]·G.

The verification device can calculate a ninth intermediate point S'' ₃ as follows: S'' ₃ = [s ₄ ]·R - [s ₂ ]·H.

The verification device can then compare respectively the fourth intermediate point _S'1 with the seventh intermediate point S''1 _, the fifth intermediate point _S'2 with the eighth intermediate point _S''2 , and the sixth intermediate point _S'3 with the ninth intermediate point S'' ₃ .

If the fourth intermediate point S’₁is equal to the seventh intermediate point S’’₁, if the fifth intermediate point S’₂is equal to the eighth intermediate point S’’₂, and if the sixth intermediate point S’₃is equal to the ninth intermediate point S’’₃, the derived private key a associated with the derived public key A, derives from the subset s of the set of at least one first secret scalar forming the root private key r recorded in the electronic device, said root private key r being associated with the root public key R, and the electronic device has the root private key r and the third secret scalar k.

The derived public key A, the additional public point H and the root public key R are public and can therefore be received by the verification device from any other devices, for example from the configuration device and/or the electronic device2.

In an implementation of the method described with reference to the , the additional public point H can be the generating point G, the additional scalar h having the value 1.

The process is thus simplified.

In another implementation of the method described with reference to the , the additional scalar h is a fifth secret scalar forming an additional private key and the additional public point H is an associated additional public key. Thus, the configuration device has a fifth secret scalar forming an additional private key h associated with an additional public key H, and at least a first secret scalar received from the configuration device is calculated from the master private key and the additional private key.

Thus, this procedure makes it possible to secure the master private key.

The additional public point H can be recorded in a memory of the electronic device 2, typically during the initialization phase, for example after having received it from the configuration device.

A person skilled in the art will understand that the process described with reference to the with a root private key r formed by two first secret scalars r₁, r₂, can be adapted for a root private key r formed by three, or more, first secret scalars. In addition, those skilled in the art can adapt the process described with reference to the to use a sub-set s, of the set of at least one first secret scalar, comprising a plurality of first secret scalars.

When the process described with reference to the , is implemented by each electronic device of a plurality of electronic devices, each of the electronic devices of said plurality can use a private root key r which is specific to it and the same public root key R. The use of a key private root r specific to each electronic device of the plurality makes the method more secure with respect to attacks by side channels aimed at obtaining a private key root r. In addition, verification is simplified because it requires the use of a single and unique public root key R, which is common to all electronic devices of said plurality.

Each of the implementation modes described above with reference to Figures 3 and 4, implements operations in a finite group Gr of prime order n, of points of an elliptic curve E.

Each of these implementation modes can be adapted to implement operations in a multiplicative subgroup of order q of the multiplicative group of a ring Fp, Fp being a ring of integers modulo p, p being a prime number, and q being a prime factor of p-1. Fp is therefore a finite field.

The finite field and the multiplicative subgroup of order q are defined by a set of parameters including a generator G which is a public integer of the multiplicative subgroup of order q. This generator G is here also predetermined and can in practice be stored in the electronic device 2, for example in its storage block 6.

There represents in flowchart form the main steps of a cryptographic processing method according to a third mode of implementation of the invention, which is a possibility of such an adaptation of the mode of implementation described with reference to the .

Furthermore, like each of the implementation modes described above with reference to Figures 3 and 4, the implementation mode described with reference to , implements algebraic calculations using algebraic operations (addition, product, subtraction) on scalars.

The method according to this third mode of implementation is implemented by the electronic device 2 in cooperation with a verification device, typically the verification device 12.

According to a recording step E2120, the processor 4 records in a memory of the electronic device 2 a root private key r.

This step is in a phase of the process called the initialization phase (Phase P1) of a set of at least one first secret scalar, forming the root private key r, said root private key r being associated with a root public key R .

This initialization phase is typically implemented by the initialization module of the electronic device 2.

Here the root private key r is formed from a single first secret scalar r ₁ .

In other words, the set of at least one first secret scalar, forming the root private key r, has a single first secret scalar, r ₁ .

The process is thus simplified. The method allows the use of a single and unique r-root private key for a determined R-root public key. The method also allows the use of a single and unique R root public key for a determined R root private key.

Typically the root public key R has a value defined as follows : .

According to one example, when the method is implemented by each electronic device of a plurality of electronic devices, the same first secret scalar and the same root public key R are used for each of the electronic devices of said plurality. Thus, verification is simplified because it requires the use of a single and unique root public key R, which is common to all electronic devices of said plurality.

According to another example, when the method is implemented by each electronic device of a plurality of electronic devices, each of the electronic devices of said plurality uses a first secret scalar and a root public key R which are specific to it, which makes the more secure process.

During this recording step (step E2120), the processor 4 therefore records in a memory of the electronic device 2 the first secret scalar r ₁ .

The initialization phase (Phase P1) may include the following complementary steps (steps E2110, E2112 and E2114).

Thus the method can comprise, before the recording step (step E2120), a step of calculation, by the processor 4, of all the first secret scalars of the set forming the root private key r (step E2110), c that is to say here of the first secret scalar r ₁ .

Typically, the first secret scalar r ₁ can be calculated by random drawing or by application of a non-clonable physical function, for example a function as described in the document 'A PUF taxonomy' by Thomas McGrath, Ibrahim E. Bagci, Zhiming M. Wang, Utz Roedig, Robert J. Young, published 02/12/2019.

The method is thus particularly secure because the electronic device can alone ensure the confidentiality of the root private key r. In addition, the electronic device has its own private root r key.

The method then includes a step of calculating the root public key R from the root private key r (step E2112).

Typically, processor 4 can calculate the root public key R as follows: .

The method can then include a step of sending to a configuration device (step E2114), typically the configuration device 22, the calculated root public key R.

Thus, advantageously the configuration device can certify the root public key R.

The method thus allows an authority having the configuration device to ensure that the initialization phase of the electronic device has been carried out under its control.

Certification of the root public key may consist of registering the root public key in a list of at least one trusted public key, or of cryptographic signing of said root public key, for example with a private certification key. issued by a trusted authority.

The step of calculating the root public key R from the root private key r (step E2112) and the step of sending the root public key R to a configuration device (step E2114) are here executed before the recording step (E2120) but the step of calculating the root public key R from the root private key r (step E2112) and/or the step of sending to a configuration device (step E2114) can be executed after the recording step (E2120) as long as the step of sending to a configuration device (step E2114) is executed after the step of calculating the root public key R from the key private root r (step E2112).

It should also be noted that the step of sending the root public key R to a configuration device (step E2114) can be omitted.

According to another possibility, the initialization phase (Phase P1) may include other complementary steps (following steps E2111, E2112 and E2114).

Thus the method can comprise, before the recording step (step E2120), a reception step (step E2111), from the configuration device, of at least a first secret scalar of the root private key (r) , that is to say here of the first secret scalar r ₁ , said at least one first secret scalar received from the configuration device being calculated from a master private key.

Typically, the configuration device has a fourth secret scalar forming the master private key.

The method is thus particularly secure because the determination of the root private key requires cooperation from the configuration device.

Here the root private key r is formed from a single first secret scalar r ₁ , the electronic device therefore receives from the configuration device, all the first secret scalars of the set forming the root private key.

Thus, the efficiency of the process is increased by limiting the interactions between the configuration device and the electronic device.

According to one example, the first secret scalar r ₁ received from the configuration device is the master private key.

According to another example, the first secret scalar r ₁ received from the configuration device can be derived by the configuration device from the master private key using any suitable derivation function well known to those skilled in the art.

The root public key R can be calculated by the configuration device.

According to another possibility, the method can also include a step of calculation by the electronic device 2 of the root public key R from the root private key r (step E2112) and a step of sending to the configuration device (step E2114 ) of the calculated root public key R.

The step of calculating the root public key R from the root private key r (step E2112) and the step of sending the root public key R to a configuration device (step E2114) are illustrated in the before the recording step (E2120) but when they are not omitted after the reception step (step E2111), the step of calculating the root public key R from the root private key r (step E2112) and/or the step of sending to a configuration device (step E2114) can be executed after the recording step (E2120) as long as the step of sending to a configuration device (step E2114 ) is executed after the step of calculating the root public key R from the root private key r (step E2112).

Advantageously, the configuration device can certify the root public key R.

The method thus allows an authority having the configuration device to ensure that the initialization phase of the electronic device has been carried out under its control.

Certification of the root public key may consist of registering the root public key in a list of at least one trusted public key, or of cryptographic signing of said root public key, for example with a private certification key. issued by a trusted authority.

After the recording step (step E2120), the method then comprises a step (step E2220) of calculating a second secret scalar forming a derived private key a equal to f(s, k) where f is a function of derivation, s is a non-empty subset of the set of at least one first secret scalar, and k is a third secret scalar: a = f(s, k).

Here, the set of at least one first secret scalar has a single first secret scalar, the first secret scalar r ₁ , and the subset s therefore has a single first secret scalar, the first secret scalar r ₁ .

The third secret scalar k can be determined by the electronic device by random selection or by application of a non-clonable physical function, for example a function as described in the document 'A PUF taxonomy' by Thomas McGrath, Ibrahim E. Bagci, Zhiming M. Wang, Utz Roedig, Robert J. Young, published 02/12/2019.

According to an example, the differentiation function f can be defined as follows: f(s, k) = k·r ₁ mod q.

The method then comprises a step of calculating a derived public key A associated with the second secret scalar a (step E2240), that is to say with the derived private key a.

Typically, the processor 4 can calculate the derived public key A as follows: .

According to another possibility, when the derivation function f used during the step of calculating a second secret scalar (step E2220) is defined according to the example described above, the processor 4 can calculate the derived public key A as follows: .

The step of calculating a second secret scalar (step E2220) and the step of calculating a derived public key A (step E2240) are in a phase of the method called the phase of generating a derived private key and 'a derived public key (phase P2).

This phase of generating a derived private key and a derived public key (phase P2) is typically implemented by the key generation module of the electronic device 2.

The method then comprises a step of generating a proof with zero disclosure of knowledge (step E2320) that the derived public key A is associated with a scalar equal to f(u, l), u being the subset s of l set of at least one first secret scalar, forming the root private key r associated with the root public key R and l being equal to k.

The step of generating a proof with zero knowledge disclosure (step E2320) is in a phase of the method called the proof generation phase (phase P3).

This proof generation phase (phase P3) is typically implemented by the proof generation module of the electronic device 2.

The method then comprises a step of transmitting the zero knowledge disclosure proof to the verification device (step E2420), during which the electronic device 2 transmits to the verification device, typically using its communication block 10, the disclosure proof zero knowledge.

The method thus allows the verification device to verify that the derived private key a derives from the subset s of the set of at least one first secret scalar forming the root private key r recorded in the electronic device, said root private key r being associated with the root public key R.

Furthermore, the method allows this verification without disclosing, typically to the verification device, the root private key r, nor the subset s, nor the derived private key a, nor the third secret scalar k.

The step of transmitting the proof with zero knowledge disclosure to the verification device (step E2420) is in a phase of the method called the transmission phase (phase P4).

This transmission phase (phase P4) is typically implemented by the transmission module of the electronic device 2.

According to one implementation possibility, the proof with zero knowledge disclosure is a non-interactive proof.

The implementation of the proof generation phase and the transmission phase is thus simplified, in particular by limiting the exchanges between the electronic device and the verification device which can be unidirectional and asynchronous.

When the derivation function f used during the step of calculating a second secret scalar (step E2220) is defined according to the example described above for this step, the processor 4 can perform the following calculations to generate the proof to zero disclosure of non-interactive knowledge during the associated step (step E2320).

The processor 4 calculates a first intermediate element as follows: where t ₁ is a first intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to data.

The processor 4 also calculates a second intermediate element as follows: where t ₂ is a second intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to another piece of data.

The processor 4 then calculates a first condensate of intermediate data by a hash function, for example SHA256 or SHA3, as specified in the FIPS PUB 180-4 document entitled 'Secure Hash Standard' and published in August 2015 by NIST , the intermediate data being the concatenation of the first intermediate element S ₁ , the second intermediate element S ₂ , the root public key R and the derived public key A.

That is, c = Ha(S ₁ || S ₂ ||R||A), where c is the first condensate and Ha is the hash function.

Then the processor 4 respectively calculates a first proof scalar s ₁ and a second proof scalar s ₂ as follows: s ₁ = t ₁ + c r ₁ mod q and s ₂ = t ₂ + c k mod q.

The proof with zero knowledge disclosure is thus made up of the first condensate c, the first proof scalar s ₁ and the second proof scalar s ₂ .

During the step of transmitting the proof with zero knowledge disclosure to the verification device (step E2420), the electronic device 2 can then transmit to the verification device the first condensate c, the first proof scalar s ₁ and the second scalar of proof s ₂ .

The verification device can verify the zero-knowledge proof.

Typically the verification device can calculate a third intermediate element S' ₁ as follows: .

The verification device can also calculate a fourth intermediate element S' ₂ as follows: .

The verification device can then calculate a second condensate of another intermediate data by the hash function, the other intermediate data being the concatenation of the third intermediate element S' ₁ , of the fourth intermediate element S' ₂ , of the public key root R and the derived public key A.

In other words, c' = Ha(S' ₁ || S' ₂ ||R||A), where it is the second condensate.

The verification device can then compare the second condensate c’ with the first condensate c. If the second condensate c' is equal to the first condensate c, the derived private key a associated with the derived public key A, derives from the subset s of the set of at least one first secret scalar forming the root private key r recorded in the electronic device, said root private key r being associated with the root public key R.

The derived public key A and the root public key R are public keys and can therefore be received by the verification device from any other devices, for example from the configuration device and/or the electronic device 2.

According to another implementation possibility, the proof with zero knowledge disclosure is an interactive proof.

The generation of the zero-knowledge proof, and the transmission of the zero-knowledge proof to a verification device, are for example in the form of a challenge-response protocol.

The method thus allows the verification device to verify that the electronic device has the root private key r and the third secret scalar k, that is to say that the electronic device can carry out at least one processing involving all the first scalars secrets of the set forming the root private key and at least one processing involving the third secret scalar k.

When the derivation function f used during the step of calculating a second secret scalar (step E2220) is defined according to the example described above for this step, the processor 4 can perform the following calculations to generate the proof for disclosure zero interactive knowledge during the associated step (step E2320).

The processor 4 calculates a first intermediate element as follows: where t ₁ is a first intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to data.

The processor 4 also calculates a second intermediate element as follows: where t ₂ is a second intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to another piece of data.

Then the processor 4 respectively calculates a first proof scalar s ₁ and a second proof scalar s ₂ as follows: s ₁ = t ₁ + c r ₁ mod q and s ₂ = t ₂ + c k mod q, where it is a challenge scalar previously received by the electronic device from the verification device.

The proof with zero knowledge disclosure is thus made up of the first intermediate element S ₁ , the second intermediate element S ₂ , the first proof scalar s ₁ and the second proof scalar s ₂ .

During the step of transmitting the proof with zero knowledge disclosure to the verification device (step E2420), the electronic device 2 can then transmit to the verification device the first intermediate element S ₁ , the second intermediate element S ₂ , the first proof scalar s ₁ and the second proof scalar s ₂ .

Preferably, to improve security, the challenge scalar c is received by the electronic device 2 from the verification device, prior to the calculation of the first proof scalar s ₁ and the second proof scalar s ₂ , but after the sending by the electronic device 2 to the verification device, of the first intermediate element S ₁ and the second intermediate element S ₂ . The step of generating a proof with zero knowledge disclosure (step E2320) and the step of transmitting the proof with zero knowledge disclosure to the verification device (step E2420), respectively the proof generation phase and the transmission phase, are then carried out concomitantly (not shown in the figure).

The verification device can verify the zero-knowledge proof.

Typically the verification device can calculate a third intermediate element S' ₁ as follows: .

The verification device can also calculate a fourth intermediate element S' ₂ as follows: .

The verification device can calculate a fifth intermediate element S'' ₁ as follows: .

The verification device can calculate a sixth intermediate element S'' ₂ as follows: .

The verification device can then respectively compare the third intermediate element _S'1 with the fifth intermediate element S''1 _, and the fourth intermediate element _S'2 with the sixth intermediate element _S''2 .

If the third intermediate element S' ₁ is equal to the fifth intermediate element S'' _1, and if the fourth intermediate element S' ₂ is equal to the sixth intermediate element S'' ₂ , the derived private key a associated with the derived public key A, derives from the subset s of the set of at least one first secret scalar forming the root private key r recorded in the electronic device, said root private key r being associated with the root public key R, and the electronic device has the root private key r and the third secret scalar k.

The derived public key A and the root public key R are public keys and can therefore be received by the verification device from any other devices, for example from the configuration device and/or the electronic device 2.

A person skilled in the art will understand that the process described with reference to the with a root private key r formed by a single first secret scalar r ₁ , can be adapted for a root private key r formed by a plurality of first secret scalars.

Thus, when the method is implemented by each electronic device of a plurality of electronic devices, each of the electronic devices of said plurality can use a private root key r which is specific to it and the same public root key R. The use of a root r private key specific to each electronic device of the plurality makes the method more secure with respect to side channel attacks aimed at obtaining a root r private key. In addition, verification is simplified because it requires the use of a single and unique root public key R, which is common to all electronic devices of said plurality.

There represents in flowchart form the main steps of a cryptographic processing method according to a fourth mode of implementation of the invention.

In this fourth implementation mode, the method implements operations in a group of unknown order p'·q' with p' and q' two prime numbers such that p = 2·p'+1 and q=2 ·q'+1 are also two prime numbers. The operations are done in a ring of integers modulo N where N = p·q. N is therefore an RSA type module.

A G generator is also used here. This generator G is an element of order p'·q' belonging to a group QR(N) defined as being the group of quadratic residues modulo N. QR(N) is a subgroup of integers modulo N and deprived of zero .

This generator G is here also predetermined and can in practice be stored in the electronic device 2, for example in its storage block 6.

In addition, like each of the processes described above with reference to Figures 3 to 5, the process described with reference to the , implements algebraic calculations using algebraic operations (addition, product, subtraction) on scalars.

This method is implemented by the electronic device 2 in cooperation with a verification device, typically the verification device 12.

According to a recording step E3120, the processor 4 records in a memory of the electronic device 2 a root private key r.

This step is in a phase of the process called the initialization phase (Phase P1) of a set of at least one first secret scalar, forming the root private key r, said root private key r being associated with a root public key R .

This initialization phase is typically implemented by the initialization module of the electronic device 2.

Here the root private key r is formed from a single first secret scalar r ₁ . The first secret scalar r ₁ is an integer belonging to an interval S(v,m) defined as follows: S(v,m) = [2 ^v - 2 ^m + 1, 2 ^v + 2 ^m - 1], with v and m positive integers, that is to say greater than or equal to 0.

In other words, the set of at least one first secret scalar, forming the root private key r, has a single first secret scalar, r ₁ .

The process is thus simplified. The method allows the use of a single and unique r-root private key for a determined R-root public key. The method also allows the use of a single and unique R root public key for a determined R root private key.

Typically R has a value defined as follows : .

According to one example, when the method is implemented by each electronic device of a plurality of electronic devices, the same first secret scalar and the same root public key R are used for each of the electronic devices of said plurality. Thus, verification is simplified because it requires the use of a single and unique root public key R, which is common to all electronic devices of said plurality.

According to another example, when the method is implemented by each electronic device of a plurality of electronic devices, each of the electronic devices of said plurality uses a first secret scalar and a root public key R which are specific to it, which makes the more secure process.

During this recording step (step E3120), the processor 4 therefore records in a memory of the electronic device 2 the first secret scalar r ₁ .

The initialization phase (Phase P1) can also include the following complementary steps (steps E3110, E3112 and E3114).

Thus the method can comprise, before the recording step (step E3120), a step of calculation, by the processor 4, of all the first secret scalars of the set forming the root private key r (step E3110), c that is to say here of the first secret scalar r ₁ .

Typically, the first secret scalar r ₁ can be calculated by random drawing or by application of a non-clonable physical function, for example a function as described in the document 'A PUF taxonomy' by Thomas McGrath, Ibrahim E. Bagci, Zhiming M. Wang, Utz Roedig, Robert J. Young, published 02/12/2019.

The method is thus particularly secure because the electronic device can alone ensure the confidentiality of the root private key r. In addition, the electronic device has its own private root r key.

The method then includes a step of calculating the root public key R from the root private key r (step E3112).

Typically, processor 4 can calculate the root public key R as follows: .

The method can then include a step of sending the calculated root public key R to a configuration device (step E3114), typically the configuration device 22.

Thus, advantageously the configuration device can certify the root public key R.

The method thus allows an authority having the configuration device to ensure that the initialization phase of the electronic device has been carried out under its control.

Certification of the root public key may consist of registering the root public key in a list of at least one trusted public key, or of cryptographic signing of said root public key, for example with a private certification key. issued by a trusted authority.

The step of calculating the root public key R from the root private key r (step E3112) and the step of sending the root public key R to a configuration device (step E3114) are here executed before the recording step (E3120) but the step of calculating the root public key R from the root private key r (step E3112) and/or the step of sending to a configuration device (step E3114) can be executed after the recording step (E3120) as long as the step of sending to a configuration device (step E3114) is executed after the step of calculating the root public key R from the key private root r (step E3112).

It should also be noted that the step of sending the root public key R to a configuration device (step E3114) can be omitted.

According to another possibility, the initialization phase (Phase P1) may include other complementary steps (following steps E3111, E3112 and E3114).

Thus the method can comprise, before the recording step (step E3120), a reception step (step E3111), from the configuration device, of at least a first secret scalar of the root private key (r) , that is to say here of the first secret scalar r ₁ , said at least one first secret scalar received from the configuration device being calculated from a master private key. The master private key is an integer belonging to the interval S(v,m).

Typically, the configuration device has a fourth secret scalar forming the master private key.

The method is thus particularly secure because the determination of the root private key requires cooperation from the configuration device.

Here the root private key r is formed from a single first secret scalar r ₁ , the electronic device therefore receives from the configuration device, all the first secret scalars of the set forming the root private key.

Thus, the efficiency of the process is increased by limiting the interactions between the configuration device and the electronic device.

According to one example, the first secret scalar r ₁ received from the configuration device is the master private key.

According to another example, the first secret scalar r ₁ received from the configuration device can be derived by the configuration device from the master private key using any suitable derivation function well known to those skilled in the art.

The root public key R can be calculated by the configuration device.

According to another possibility, the method can also include a step of calculating the root public key R from the root private key r (step E3112) and a step of sending the public key to the configuration device (step E3114). root R calculated.

The step of calculating the root public key R from the root private key r (step E3112) and the step of sending the root public key R to a configuration device (step E3114) are illustrated in the before the recording step (E3120) but when they are not omitted after the reception step (step E3111), the step of calculating the root public key R from the root private key r (step E3112) and/or the step of sending to a configuration device (step E3114) can be executed after the recording step (E3120) as long as the step of sending to a configuration device (step E3114 ) is executed after the step of calculating the root public key R from the root private key r (step E3112).

Advantageously, the configuration device can certify the root public key R.

The method thus allows an authority having the configuration device to ensure that the initialization phase of the electronic device has been carried out under its control.

Certification of the root public key may consist of registering the root public key in a list of at least one trusted public key, or of cryptographic signing of said root public key, for example with a private certification key. issued by a trusted authority.

After the recording step (step E3120), the method then comprises a step (step E3220) of calculating a second secret scalar forming a derived private key equal to f(s, k) where f is a differentiation function , s is a non-empty subset of the set of at least one first secret scalar, and k is a third secret scalar: a = f(s, k).

Here, the set of at least one first secret scalar has a single first secret scalar, the first secret scalar r ₁ , and the subset s therefore has a single first secret scalar, the first secret scalar r ₁ .

Furthermore, the second secret scalar a and the third secret scalar are integers belonging to the interval S(v,m).

The third secret scalar k can be determined by the electronic device by random selection or by application of a non-clonable physical function, for example a function as described in the document 'A PUF taxonomy' by Thomas McGrath, Ibrahim E. Bagci, Zhiming M. Wang, Utz Roedig, Robert J. Young, published 02/12/2019.

According to an example, the differentiation function f can be defined as follows: f(s, k) = k·r ₁ .

The method then comprises a step of calculating a derived public key A associated with the second secret scalar a (step E3240), that is to say with the derived private key a.

Typically, the processor 4 can calculate the derived public key A as follows: .

According to another possibility, when the derivation function f used during the step of calculating a second secret scalar (step E3220) is defined according to the example described above, the processor 4 can calculate the derived public key A as follows: .

The step of calculating a second secret scalar (step E3220) and the step of calculating a derived public key A (step E3240) are in a phase of the method called the phase of generating a derived private key and 'a derived public key (phase P2).

This phase of generating a derived private key and a derived public key (phase P2) is typically implemented by the key generation module of the electronic device 2.

The method then comprises a step of generating a proof with zero disclosure of knowledge (step E3320) that the derived public key A is associated with a scalar equal to f(u, l), u being the subset s of l set of at least one first secret scalar, forming the root private key r associated with the root public key R and l being equal to k.

The step of generating a proof with zero knowledge disclosure (step E3320) is in a phase of the method called the proof generation phase (phase P3).

This proof generation phase (phase P3) is typically implemented by the proof generation module of the electronic device 2.

The method then comprises a step of transmitting the zero knowledge disclosure proof to the verification device (step E3420), during which the electronic device 2 transmits to the verification device, typically using its communication block 10, the disclosure proof zero knowledge.

The method thus allows the verification device to verify that the derived private key a derives from the subset s of the set of at least one first secret scalar forming the root private key r recorded in the electronic device, said root private key r being associated with the root public key R.

Furthermore, the method allows this verification without disclosing, typically to the verification device, the root private key r, nor the subset s, nor the derived private key a, nor the third secret scalar k.

The step of transmitting the proof with zero knowledge disclosure to the verification device (step E3420) is in a phase of the method called the transmission phase (phase P4).

This transmission phase (phase P4) is typically implemented by the transmission module of the electronic device 2.

According to one implementation possibility, the proof with zero knowledge disclosure is a non-interactive proof.

The implementation of the proof generation phase and the transmission phase is thus simplified, in particular by limiting the exchanges between the electronic device and the verification device which can be unidirectional and asynchronous.

When the derivation function f used during the step of calculating a second secret scalar (step E3220) is defined according to the example described above for this step, the processor 4 can perform the following calculations to generate the proof to zero disclosure of non-interactive knowledge during the associated step (step E3320).

The processor 4 calculates a first intermediate element as follows: where t ₁ is a first intermediate scalar determined by the electronic device 2, for example by random selection or by application of a pseudo-random algorithm to a piece of data, and belonging to an interval S(v,e·(m+k') +2) defines as follows: S(v,e·(m+k')+2) = [2 ^v – 2 ^{e ·(m+k ' )+2} + 1, 2 ^v + 2 ^{e ·(m+ k ' )+2} - 1] with e and k' predetermined parameters such that e is an integer whose value is strictly greater than one and k' is a natural number.

The processor 4 also calculates a second intermediate element as follows: where t ₂ is a second intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to another piece of data, and belonging to the interval S(v,e·(m+k ')+2).

The processor 4 then calculates a first condensate of intermediate data by a hash function, for example SHA256 or SHA3, as specified in the FIPS PUB 180-4 document entitled 'Secure Hash Standard' and published in August 2015 by NIST , the intermediate data being the concatenation of the first intermediate element S ₁ , the second intermediate element S ₂ , the root public key R and the derived public key A.

That is, c = Ha(S ₁ || S ₂ ||R||A), where c is the first condensate and Ha is the hash function.

Then the processor 4 respectively calculates a first proof scalar s ₁ and a second proof scalar s ₂ as follows: s ₁ = t ₁ - c · (r ₁ - 2 ^v ) and s ₂ = t ₂ - c · (k - ^2v ).

The proof with zero knowledge disclosure is thus made up of the first condensate c, the first proof scalar s ₁ and the second proof scalar s ₂ .

During the step of transmitting the proof with zero knowledge disclosure to the verification device (step E3420), the electronic device 2 can then transmit to the verification device the first condensate c, the first proof scalar s ₁ and the second scalar of proof s ₂ .

The verification device can verify the zero-knowledge proof.

Typically the verification device can verify that the first proof scalar s ₁ and the second proof scalar s ₂ received belong to the interval S(v,e·(m+k')+2).

The verification device can also calculate a third intermediate element S' ₁ as follows: .

The verification device can calculate a fourth intermediate element S' ₂ as follows: .

The verification device can then calculate a second condensate of another intermediate data by the hash function, the other intermediate data being the concatenation of the third intermediate element S' ₁ , of the fourth intermediate element S' ₂ , of the public key root R and the derived public key A.

In other words, c' = Ha(S' ₁ || S' ₂ ||R||A), where it is the second condensate.

The verification device can then compare the second condensate c’ with the first condensate c. If the second condensate c' is equal to the first condensate c, the derived private key a associated with the derived public key A, derives from the subset s of the set of at least one first secret scalar forming the root private key r recorded in the electronic device, said root private key r being associated with the root public key R.

The derived public key A and the root public key R are public keys and can therefore be received by the verification device from any other devices, for example from the configuration device and/or the electronic device 2.

According to another implementation possibility, the proof with zero knowledge disclosure is an interactive proof.

The generation of the zero-knowledge proof, and the transmission of the zero-knowledge proof to a verification device, are for example in the form of a challenge-response protocol.

The method thus allows the verification device to verify that the electronic device has the root private key r and the third secret scalar k, that is to say that the electronic device can carry out at least one processing involving all the first scalars secrets of the set forming the root private key and at least one processing involving the third secret scalar k.

When the derivation function f used during the step of calculating a second secret scalar (step E3220) is defined according to the example described above for this step, the processor 4 can perform the following calculations to generate the proof for disclosure zero interactive knowledge during the associated step (step E3320).

The processor 4 calculates a first intermediate element as follows: where t ₁ is a first intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to data, belonging to an interval S(v,e·(m+k')+ 2) defines as follows: S(v,e·(m+k')+2) = [2 ^v – 2 ^{e ·(m+k ' )+2} + 1, 2 ^v + 2 ^{e ·(m+k ' )+2} - 1] with e and k' predetermined parameters such that e is an integer whose value is strictly greater than one and k' is a natural number.

The processor 4 also calculates a second intermediate element as follows: where t ₂ is a second intermediate scalar determined by the electronic device 2, for example by random drawing or by application of a pseudo-random algorithm to another piece of data, belonging to the interval S(v,e·(m+k' )+2).

Then the processor 4 respectively calculates a first proof scalar s ₁ and a second proof scalar s ₂ as follows: s ₁ = t ₁ - c · (r ₁ - 2 ^v ) and s ₂ = t ₁ - c · (k - 2 ^v ), where c is a challenge scalar previously received by the electronic device from the verification device.

The proof with zero knowledge disclosure is thus made up of the first intermediate element S1, the second intermediate element S2, the first proof scalar s1 and the second proof scalar s2.

During the step of transmitting the proof with zero knowledge disclosure to the verification device (step E3420), the electronic device 2 can then transmit to the verification device the first intermediate element S1, the second intermediate element S2, the first scalar of proof s1 and the second proof scalar s2.

Preferably, to improve security, the challenge scalar c is received by the electronic device 2 from the verification device, prior to the calculation of the first proof scalar s ₁ and the second proof scalar s ₂ , but after the sending by the electronic device 2 to the verification device, of the first intermediate element S ₁ and the second intermediate element S ₂ . The step of generating a proof with zero knowledge disclosure (step E3320) and the step of transmitting the proof with zero knowledge disclosure to the verification device (step E3420), respectively the proof generation phase and the transmission phase, are then carried out concomitantly (not shown in the figure).

The verification device can verify the zero-knowledge proof.

Typically the verification device can verify that the first proof scalar s ₁ and the second proof scalar s ₂ received belong to the interval S(v,e·(m+k')+2).

The verification device can also calculate a third intermediate element S' ₁ as follows: .

The verification device can calculate a fourth intermediate element S' ₂ as follows: .

The verification device can calculate a fifth intermediate element S'' ₁ as follows: .

The verification device can calculate a sixth intermediate element S'' ₂ as follows: .

The verification device can then respectively compare the third intermediate element _S'1 with the fifth intermediate element S''1 _, and the fourth intermediate element _S'2 with the sixth intermediate element _S''2 .

If the third intermediate element S' ₁ is equal to the fifth intermediate element S'' _1, and if the fourth intermediate element S' ₂ is equal to the sixth intermediate element S'' ₂ , the derived private key a associated with the derived public key A, derives from the subset s of the set of at least one first secret scalar forming the root private key r recorded in the electronic device, said root private key r being associated with the root public key R, and the electronic device has the root private key r and the third secret scalar k.

The derived public key A and the root public key R are public keys and can therefore be received by the verification device from any other devices, for example from the configuration device and/or the electronic device.

A person skilled in the art will understand that the process described with reference to the with a root private key r formed by a single first secret scalar r ₁ , can be adapted for a root private key r formed by a plurality of first secret scalars.

Thus, when the method is implemented by each electronic device of a plurality of electronic devices, each of the electronic devices of said plurality can use a private root key r which is specific to it and the same public root key R. The use of a root r private key specific to each electronic device of the plurality makes the method more secure with respect to side channel attacks aimed at obtaining a root r private key. In addition, verification is simplified because it requires the use of a single and unique root public key R, which is common to all electronic devices of said plurality.

Advantageously, whatever the mode of implementation of the cryptographic processing method according to the invention, the initialization phase (phase P1) can be implemented at an initialization site, and the phase of generation of a derived private key a and a derived public key A (phase P2) is implemented at another site distinct from the initialization site.

Implementing the initialization phase and the generation phase on separate sites improves the safety of the process.

Claims (13)

Cryptographic processing method implemented by an electronic device, the electronic device cooperating with a verification device, and said method comprising:
i) an initialization phase (P1) of a set of at least one first secret scalar (r1; r2) forming a root private key (r) associated with a root public key (R), the initialization phase including the following step:
- recording (E120; E1120; E2120; E3120) of the root private key (r) in a memory of the electronic device,
And
ii) a generation phase (P2) of a derived private key (a) and a derived public key (A), comprising the following steps:
- calculation (E220; E1220; E2220; E3220) of a second secret scalar forming the derived private key (a) equal to f(s, k) where f is a derivation function, s is a non-empty subset of the set of at least one first secret scalar, and k is a third secret scalar,
- calculation (E240; E1240; E2240; E3240) of the derived public key (A) associated with the second secret scalar (a),
Then
iii) a proof generation phase (P3) comprising the following step:
- generation (E320; E1320; E2320; E3320) of a zero-disclosure proof of knowledge that the derived public key (A) is associated with a scalar equal to f(u, l), u being the subset s of the set of at least one first secret scalar (r _{1 ;} r ₂ ) forming the root private key (r) associated with the root public key (R) and l being equal to k,
And
iv) a transmission phase (P4) comprising the following step:
- transmission (E420; E1420; E2420; E3420) of the proof with zero knowledge disclosure to the verification device. Cryptographic processing method according to the preceding claim in which
- the electronic device further cooperates with a configuration device during the initialization phase, and
- the root public key (R) is certified by the configuration device. Cryptographic processing method according to any one of the preceding claims in which the initialization phase further comprises the following steps:
- calculation (E110; E1110; E2110; E3110), before the recording step, of all the first secret scalars of the set forming the root private key (r), at least one first secret scalar of the root private key (r) being calculated by random selection or by application of a non-clonable physical function, then
- calculation (E112; E1112; E2112; E3112) of the root public key (R) from the root private key (r). Cryptographic processing method according to claim 2 in which the configuration device has a fourth secret scalar forming a master private key associated with the root public key (R), and in which the initialization phase further comprises the step following before the recording step:
- reception (E111; E1111; E1119; E2111; E3111), from the configuration device, of at least one first secret scalar of the root private key (r), said at least one first secret scalar received from the configuration device being calculated from the master private key. Cryptographic processing method according to claim 4 in which the configuration device further has a fifth secret scalar forming an additional private key (h) associated with an additional public key (H), and in which said at least one first secret scalar received from the configuration device is calculated from the master private key and the additional private key. Cryptographic processing method according to any one of claims 1 to 4 in which the root private key (r) is formed from a single first secret scalar (r ₁ ). Cryptographic processing method according to any one of claims 1 to 5 in which the root private key (r) is formed of a plurality of first secret scalars (r _{1 ;} r ₂ ). Cryptographic processing method according to claim 7 when dependent on claim 4, in which the initialization phase further comprises the following steps before the recording step:
- determination (E1116) of a first secret scalar of the root private key (r), then
- encryption (E1117) in a homomorphic manner of the first determined secret scalar, then
- transmission (E1118) to the cipher configuration device of the first determined secret scalar,
and in which during the reception step (E1119) of the initialization phase, the electronic device:
- receives another first secret scalar of the root private key (r), encrypted in the same homomorphic manner, the cipher of said other first secret scalar having been determined from the master private key and the cipher transmitted by the electronic device to the device configuration, and
- decrypts the ciphertext of the other first secret scalar to obtain said at least one first secret scalar received from the configuration device. Cryptographic processing method according to claim 8 in which the determination (E1116) of a first secret scalar of the root private key (r) is by random drawing or by application of a non-clonable physical function. Cryptographic processing method according to any one of the preceding claims in which the third secret scalar k is determined by the electronic device by random drawing or by application of a non-clonable physical function. Cryptographic processing method according to any one of the preceding claims in which
- the initialization phase is implemented at an initialization site, and
- the phase of generating a derived private key (a) and a derived public key (A) is implemented at another site distinct from the initialization site. Computer program comprising instructions executable by a processor (4) and adapted to implement a method according to any one of claims 1 to 11 when these instructions are executed by the processor (4). Electronic cryptographic processing device adapted to cooperate with a verification device and comprising:
i) an initialization module configured to carry out an initialization phase of a set of at least one first secret scalar (r₁ ; r₂) forming a root private key (r) associated with a root public key (R), the initialization phase comprising the following step:
- recording of the root private key (r) in a memory of the electronic device,
ii) a key generation module configured to carry out a generation phase of a derived private key (a) and a derived public key (A), comprising the following steps:
- calculation of a second secret scalar forming the derived private key (a) equal to f(s, k) where f is a derivation function, s is a non-empty subset of the set of at least one first secret scalar, and k is a third secret scalar,
- calculation of the derived public key (A) associated with the second secret scalar (a),
iii) a proof generation module configured to carry out a proof generation phase comprising the following step:
- generation of a zero-knowledge proof that the derived public key (A) is associated with a scalar equal to f(u, l), u being the subset s of the set of at least one first secret scalar (r_{1 ;}r₂) forming the root private key (r) associated with the root public key (R) and l being equal to k,
iv) a transmission module configured to carry out a transmission phase comprising the following step:
- transmission of the zero-knowledge proof to the verification device.