CN109962778A - Multi-party homomorphic cryptography method based on integer - Google Patents

Abstract

The multi-party homomorphic cryptography method based on integer that the invention discloses a kind of, for solving the technical problem of the existing homomorphic cryptography method applicability difference based on integer.Technical solution is to generate encryption side and the respective public and private key of decryption side by key generation centre first, encryption side carries out encryption to data using its public key and forms ciphertext, then ciphertext is transferred to cloud computing platform to be calculated, handled by encryption side, is finally decrypted by decryption side using the private key that key generation centre generates.The present invention extends encryption side using stochastic transformation method, constructs one based on approximate greatest common divisor difficult problem and adds close side's homomorphic cryptography method based on integer.Reduce computation complexity by the extremely efficient to encryption side so that the homomorphic cryptography method based on integer is more adaptable in adding the environment that close side participates in, whole system it is more efficient, and be capable of the data-privacy safety of effective protection user.

Description

Multi-party homomorphic cryptography method based on integer

Technical field

The homomorphic cryptography method based on integer that the present invention relates to a kind of, in particular to a kind of multi-party homomorphism based on integer add Decryption method.

Background technique

Document " Fully Homomorphic Encryption over the Integers:Lecture Notes in Computer Science,Advances in Cryptology–EUROCRYPT 2010:29th Annual International Conference on the Theory and Applications of Cryptographic Techniques, French Riviera, May 30-June 3,2010.Proceedings, pp.24-43. " disclose one kind Homomorphic cryptography method based on integer.This method is transported on the basis of the homomorphic cryptography method of part using the integer mould in number theory It can be regarded as the basis for algorithm, shorten keys sizes and ciphertext length, compared to the homomorphic cryptography method effect of lattice operation construction It increases in rate.Meanwhile using homomorphic decryption, Re-encryption Technology, it is complete same that original part homomorphic cryptography method, which is expanded, State encryption method realizes the calculating to ciphertext arbitrary number of times.But in cloud computing environment, this method needs more in a side It is inefficient in the case where secondary upload data.Also, this method is directed to a case where side encrypts side decryption, however practical A decryption side needs to correspond to multiple encryption sides, such as cloud medical environment in situation, and a doctor generally requires to correspond to multiple trouble Person, and need for calculating task to be contracted out to third party cloud platform, this method can only provide a patient and carry out with a doctor Interaction, so the applicability in network environment this method of multi-user is not strong, and the efficiency of system is relatively low.

Summary of the invention

In order to overcome the shortcomings of that the existing homomorphic cryptography method applicability based on integer is poor, the present invention provides a kind of based on whole Several multi-party homomorphic cryptography methods.This method generates encryption side and the respective public and private key of decryption side by key generation centre first, Encryption side carries out encryption to data using its public key and forms ciphertext, and then ciphertext is transferred to cloud computing platform to count by encryption side It calculates, processing, is finally decrypted by decryption side using the private key that key generation centre generates.The present invention utilizes stochastic transformation method Encryption side is extended, one is constructed based on approximate greatest common divisor difficult problem and close side homomorphic cryptography side is added based on integer Method.Reduce computation complexity by the extremely efficient to encryption side, so that the homomorphic cryptography method based on integer is being added It is more adaptable in the environment that close side participates in, whole system it is more efficient, and be capable of the data-privacy of effective protection user Safety.

The technical solution adopted by the present invention to solve the technical problems: a kind of multi-party homomorphic cryptography method based on integer, Its main feature is that the following steps are included:

Step 1: key generation centre according to the security parameter λ of system generate decryption side S public private key pair, wherein λ= 2^k, k is positive integer；

The private key sk of S is

Sk=p (1)

The public key pk of S is

Pk={ x₀,x₁,...,x_τ} (2)

In formula (1), sk indicates that the private key of S, p are by key generation centre from section [2^η-1,2^η)∩(2Z⁺+ 1) random on One η odd positive integers of selection, i.e.,Wherein η indicates the positive integer of η >=1,Generation Table random selection, [2^η-1,2^η) indicate from 2^η-1To 2^ηBetween the set that constitutes of all odd positive integers, and include 2^η-1Not comprising 2^η；Z⁺Indicate Positive Integer Set, (2Z⁺+ 1) odd Positive Integer Set is indicated.

In formula (2), pk indicates that the public key of S, τ indicate the element number for including in public key pk, each element in the public key Meet condition x_υ=q_υp+2r_υ, wherein υ ∈ { 0,1 ..., τ }, and guarantee in public key set { x₀,x₁,...,x_τIn x₀Number Value is maximum.q_υAnd r_υIt is the randomly selected two groups of integers of key generation centre, i.e., υ∈{0,1,...,τ}.HereRandom selection is represented,Indicate from 0 toBetween wrap Containing 0 and do not includePositive Integer Set；[-2^ρ,2^ρ] indicate from -2^ρWith 2^ρIt include -2 on section^ρWith 2^ρInteger set；Z table Show integer set.

Step 2: key generation centre generates encryption side E according to the security parameter λ of system_iPublic private key pair, wherein λ= 2^k, k is positive integer, i ∈ { 1,2 ..., n }；

E_iPrivate key sk_iFor

sk_i=p_i (3)

E_iPublic key pk_iFor

pk_i={ x '_i,0,x′_i,1,...,x′_i,τ} (4)

In formula (3), sk_iIndicate E_iPrivate key, p_iIt is by the randomly selected η of key generation centre_iThe surprise of position is just whole Number, i.e.,Wherein η i indicates the positive integer of i >=1 η,It represents and randomly chooses, [2^η-1,2^η) It indicates from 2^η-1To 2^ηBetween the set that constitutes of all odd positive integers, and include 2^η-1Not comprising 2^η；Z⁺Indicate Positive Integer Set, (2Z⁺+ 1) odd Positive Integer Set is indicated.

In formula (4), pk_iIndicate E_iPublic key, what wherein i ∈ { 1,2 ..., n }, n were represented is the number of encryption side.It is public Key pk_iIn each element meet condition x '_i,j=x_i,jmodx_i,0.WhereinAnd it needs to guarantee collecting Close { x_i,0,x_i,1,...,x_i,τIn x_i,0Numerical value is maximum, formulaIn q_i,jAnd r_i,jFor in key generation The randomly selected two groups of integers of the heart, i.e.,I ∈ { 1,2 ..., n }, j ∈ {0,1,...,τ}；FormulaInIt is that random permutation is carried out to the element of public key pk in formula (2) Algorithm π_iThe integer of generation, replacement algorithm π_iDetailed process be: key generation centre is to+1 element of τ in the public key pk of S Position carries out stochastic transformation generation

Step 3: encryption side E_iIn its public key pk_iSet X_i={ x '_i,0,x′_i,1,...,x′_i,τIn random selection one Nonvoid subset S_i={ x "_i,0,x″_{I, 1},...,x″_i,t, meetAnd t is the positive integer for meeting t≤τ.Then E_iChoosing Select a random integersE_iFollowing computations are executed, ciphertext c is obtained_i,l:

In formula, m_i,lIt is encryption side E_iThe plaintext m of input_iBinary form, and m_i,l∈ { 0,1 }, m_iIt is encryption side E_i The plaintext of input, binary length L, L are the integer greater than 0.

Step 4: the polynomial computation rule f that cloud computing platform needs to be implemented ciphertext according to decryption side S, converts f to Corresponding Boolean circuit g, then cloud computing platform execution circuit assessment algorithm, calculates multiple ciphertexts, after obtaining assessment Ciphertext C^*:

In formula, Boolean circuit of the g expression cloud platform according to polynomial transformation, Evaluate indication circuit assessment algorithm,Indicate the cloud platform ciphertext encrypted by multiple encryption sides to be calculated.

It is calculated Step 5: decryption side S executes decryption using key sk=p, obtains plaintext M^*:

M^*=(C^*mod p)mod2 (7)

In formula, p indicates that the private key of decryption side, mod indicate modular arithmetic.

The beneficial effects of the present invention are: this method generates encryption side and the respective public affairs of decryption side by key generation centre first Private key, encryption side using its public key to data carry out encryption forms ciphertext, then encryption side by ciphertext transfer to cloud computing platform into Row is calculated, is handled, and is finally decrypted by decryption side using the private key that key generation centre generates.The present invention utilizes stochastic transformation Method extends encryption side, constructs close side's homomorphism of adding based on integer based on approximate greatest common divisor difficult problem and adds Decryption method.Reduce computation complexity by the extremely efficient to encryption side, so that the homomorphic cryptography method based on integer exists Add it is more adaptable in the environment that close side participates in, whole system it is more efficient, and be capable of the data of effective protection user Personal secrets.

Specifically, this method utilize decryption side public key element stochastic transformation, by the homomorphic cryptography method based on integer from Single encryption side, which is extended to, adds close side, and the calculating by cloud computing platform to ciphertext, decryption side can be decrypted correctly.Same item Under part, when sole user needs to upload n times data, document " Fully Homomorphic Encryption over the Integers:Lecture Notes in Computer Science,Advances in Cryptology–EUROCRYPT 2010:29th Annual International Conference on the Theory and Applications of Cryptographic Techniques,French Riviera,May 30–June 3,2010.Proceedings,pp.24- 43. " need to carry out the decryption oprerations of 2n key generation operation and n times, this method need to only carry out n+1 secondary key generate operation and 1 decryption oprerations, reduces the storage overhead of key, while saving key and generating time loss and decryption time loss, mentions Algorithm computational efficiency is risen.Meanwhile this method supports different user to upload the scene of data, and multi-user side is supported to participate in, and is promoted Adaptability in cloud computing environment, improves the efficiency of whole system.

It elaborates with reference to the accompanying drawings and detailed description to the present invention.

Detailed description of the invention

Fig. 1 is the flow chart of the multi-party homomorphic cryptography method the present invention is based on integer.

The flow chart of cloud platform execution circuit assessment algorithm in Fig. 2 Fig. 1.

Specific embodiment

Explanation of nouns:

KGC: key generation centre；

E_i: the running encryption side of algorithm；

S: the running decryption side of algorithm；

Cloud Platform: cloud computing service platform；

λ: the system security parameter generated by KGC, wherein λ=2^k, k is positive integer；

γ: decryption side S public key bit length is positive integer；

γ_i: encryption side E_iPublic key bit length, be positive integer；

η: decryption side S private key bit length is positive integer；

η_i: encryption side E_iPrivate key bit length, be positive integer；

ρ: decryption side S noise bits length is positive integer；

ρ_i: encryption side E_iNoise bits length, be positive integer；

τ: decryption side S public key element number is positive integer；

T: the element number in nonvoid subset chosen in Encryption Algorithm is positive integer；

P: being the private key of decryption side S for randomly selected odd positive integer；

p_i: it is encryption side E for randomly selected odd-integral number_iPrivate key；

q_υ: the integer generated at random for KGC；

r_υ: the integer generated at random for KGC；

For the set for carrying out stochastic transformation generation to public key pk；

x_v: for the component of decryption side S public key, and v=0,1 ..., τ；

x′_i,j: for encryption side E_iThe component of public key, and i=1,2 ..., n, j=0,1 ..., τ；

The integer of stochastic transformation generation is carried out for the element of public key pk；

x″_i,j: the element in nonvoid subset selected in Encryption Algorithm for encryption side, and i=1,2 ..., n, j=0, 1,…,t；

q_i,j: the integer generated at random for KGC；

r_i,j: the integer generated at random for KGC；

m_i: for encryption side E_iThe plaintext for needing to encrypt；

m_i,l: for plaintext m_iThe binary value of each；

L: for plaintext m_iBinary length；

c_i,l: for plaintext m_i,lCiphertext after Encryption Algorithm calculates；

C^*: for the ciphertext after the calculating of Evaluate algorithm；

F: the polynomial computation rule that decryption side needs to carry out ciphertext；

G: the calculative circuit of cloud computing platform, g ∈ F_ε；

F_ε: for door, OR circuit set；

M^*: indicate the plaintext after decipherment algorithm calculates；

X_i: indicate the public key set of encryption side；

S_i: indicate the nonvoid subset randomly selected in encryption Fang Qi public key set；

π_i: indicate the random permutation algorithm carried out to encryption side；

∈: indicate that an element belongs to the symbol of a set；

∩: indicate that two are gathered the operation that seeks common ground；

Mod: modular arithmetic is indicated；

∑: summation operation is indicated；

It indicates arbitrarily to choose element in a set；

Indicate that a set is the subset symbol of another set.

Referring to Fig.1-2.It considers the following scenario, in medical system, there are multiple sufferer E_i, wherein i=1,2 ..., n, Middle n is integer and n >=1, belongs to doctor S treatment, but due to the limitation of hospital internal computing system condition, need by The diagnostic analysis of patient data is carried out in cloud computing service platform Cloud Platform auxiliary doctor, due to patient data requirement The privacy of height, therefore the processing work of sufferer private data can be implemented by means of this method.

The participant that this method is related to are as follows: key generation centre (Key Generation Center, KGC), n sufferer Encryption side E_i, wherein i=1,2 ..., n, wherein n is integer and n >=1, decryption side doctor S and cloud computing service platform (Cloud Platform, CP).

If it is the product of ciphertext a Yu ciphertext b that circuit assessment algorithm is calculative, required for the ciphertext c exported The circuit of operation is as shown in Figure 2.Ciphertext a and ciphertext b is respectively to pass through the ciphertext that Encryption Algorithm is calculated, and cloud computing platform will A and b, by AND gate circuit, exports ciphertext c, ciphertext c is the product of ciphertext a and b as inputting.

Due in Encryption Algorithm, being that obtain after encryption by " position " to encrypting in plain text is an integer.And boolean Circuit is based on binary operation, so the circuit assessment algorithm of homomorphic encryption algorithm uses Boolean circuit, circuit is commented Complexity of the complexity of estimation algorithm dependent on the Boolean circuit of institute's operation.If Boolean circuit is carried out with corresponding power function It compares, its plus operation and multiplication are converted with the corresponding OR circuit of Boolean circuit, "AND" circuit.Boolean's electricity The complexity on road depends on the quantity of gate circuit, therefore these circuits can be converted any form of polynomial computation.

The security parameter of this method are as follows:

λ is the system security parameter generated by KGC, wherein λ=2^k, k is positive integer；

γ is the public key bit length of decryption side S, γ_iIt is encryption side E_iPublic key bit length；

η is the private key bit length of decryption side S, η_iIt is encryption side E_iPrivate key bit length；

ρ is the noise bits length of decryption side S, ρ_iIt is encryption side E_iNoise bits length；

Wherein γ, γ_i, η, η_i, ρ, ρ_iIt is positive integer, while the security parameter that KGC is generated in this method needs to meet such as Lower condition ρ=λ, ρ_i=λ, η=λ+γ, η_i=λ+γ.

The present invention is based on the multi-party homomorphic cryptography method of integer, specific step is as follows:

1, KGC generates the public private key pair of doctor S according to the security parameter λ of system, wherein λ=2^k, k is positive integer；

The private key sk of doctor is

Sk=p (1)

The public key pk of doctor is

Pk={ x₀,x₁,...,x_τ} (2)

In formula (1), sk indicates that the private key of doctor S, p are by key generation centre from section [2^η-1,2^η)∩(2Z⁺+ 1) on Randomly selected one η odd positive integers, i.e.,Wherein η indicates the positive integer of η >=1,It represents and randomly chooses, [2^η-1,2^η) indicate from 2^η-1To 2^ηBetween the set that constitutes of all odd positive integers, and include 2^η-1 Not comprising 2^η；Z⁺Indicate Positive Integer Set, (2Z⁺+ 1) odd Positive Integer Set is indicated.

In formula (2), pk indicates that the public key of doctor S, τ indicate the element number that public key pk includes, each member in the public key Element meets condition x_υ=q_υp+2r_υ, wherein υ ∈ { 0,1 ..., τ }, and guarantee in public key { x₀,x₁,...,x_τIn x₀Numerical value It is maximum.q_υAnd r_υIt is the randomly selected two groups of integers of key generation centre, i.e., υ ∈{0,1,...,τ}.HereRandom selection is represented,Indicate from 0 toBetween comprising 0 and not including's Positive Integer Set；[-2^ρ,2^ρ] indicate from -2^ρWith 2^ρIt include -2 on section^ρWith 2^ρInteger set；Z indicates integer set.

2, KGC generates patient E according to the security parameter λ of system_iPublic private key pair, wherein λ=2^k, k is positive integer, i ∈ {1,2,...,n}；

Patient E_iPrivate key sk_iFor

sk_i=p_i (3)

Patient E_iPublic key pk_iFor

pk_i={ x '_i,0,x′_i,1,...,x′_i,τ} (4)

In formula (3), sk_iIndicate patient E_iPrivate key, p_iIt is by the randomly selected η of key generation centre_iThe surprise of position Positive integer, i.e.,Wherein η_iIndicate η_i>=1 positive integer,It represents and randomly chooses, [2^η-1, 2^η) indicate from 2^η-1To 2^ηBetween the set that constitutes of all odd positive integers, and include 2^η-1Not comprising 2^η；Z⁺Indicate positive integer collection It closes, (2Z⁺+ 1) odd Positive Integer Set is indicated.

In formula (4), pk_iIndicate E_iPublic key, that wherein i ∈ { 1,2 ..., n }, n are represented is the patient for participating in encryption Number.Public key pk_iIn each element meet condition x '_i,j=x_i,jmodx_i,0, i ∈ { 1,2 ..., n }, j ∈ { 0,1 ..., τ }； WhereinAnd it needs to guarantee in set { x_i,0,x_i,1,...,x_i,τIn x_i,0Numerical value is maximum, formulaIn q_i,jAnd r_i,jIndicate the randomly selected two groups of integers of key generation centre, i.e.,I ∈ { 1,2 ..., n }, j ∈ { 0,1 ..., τ }；FormulaInIt is that random permutation algorithm π is carried out to the element of public key pk in formula (2)_iThe integer of generation, Replacement algorithm π_iDetailed process be: key generation centre carries out stochastic transformation to the position of+1 element of τ in the public key pk of S It generates

3, patient executes Encryption Algorithm.

The public private key pair for the patient that formula (4) generates is sent to patient, patient E by KGC_iThe public affairs generated according to formula (4) Key, in its public key pk_iSet X_i={ x '_i,0,x′_i,1,...,x′_i,τOne nonvoid subset S of middle random selection_i={ x "_i,0, x″_i,1,...,x″_i,t, meetAnd t is the positive integer for meeting t≤τ.Then E_iSelect a random integersE_iFollowing computations are executed, ciphertext c is obtained_i,l:

Then patient E_iSelect a random integersPatient E_iIt executes following Encryption Algorithm and calculates ciphertext c_i,l:

In formula, m_i,lIt is patient E_iThe plaintext m of input_iBinary form, and m_i,l∈ { 0,1 }, m_iIt is patient E_iInput Plaintext, binary length L, L are the integer greater than 0.

4, cloud computing platform execution circuit assessment algorithm.

The polynomial computation rule that cloud computing platform needs to be implemented ciphertext according to doctor S, converts corresponding boolean for f Circuit g, g ∈ F_ε, wherein F_εFor with door, OR circuit set；The basic principle of conversion be by the addition in f be converted into or door electricity Road, multiplication are converted into AND gate circuit.Then cloud computing platform execution circuit assessment algorithm calculates multiple ciphertexts.It will be public The ciphertext obtained after multiple patient's encryptions in formula (5) exports ciphertext C as input, by the calculating of Boolean circuit^*。

In formula, g indicates cloud computing platform according to the Boolean circuit of polynomial transformation, and the assessment of Evaluate indication circuit is calculated Method,Indicate the cloud computing platform ciphertext encrypted by multiple patients to be calculated.Due to the present invention is based on be Modulo-two operation on integer, so cloud computing platform selection is based on binary Boolean circuit.Specific conversion method be by Ciphertext multiplying be converted to door operation, ciphertext add operation is converted to or door operation, and ciphertext is then input to gate Middle carry out operation, returns the result value, obtains ciphertext C^*。

5, doctor S executes decipherment algorithm.

Ciphertext after calculating in formula (6) is sent to doctor by cloud computing platform, and doctor S executes solution using private key sk=p Close algorithm is decrypted calculating, obtains plaintext M^*:

M^*=(C^*mod p)mod2 (7)

In formula, C^*Indicate the ciphertext obtained after cloud computing platform calculates, p indicates that the private key of S, mod indicate modular arithmetic.

Claims (1)

1. a kind of multi-party homomorphic cryptography method based on integer, it is characterised in that the following steps are included:

Step 1: key generation centre generates the public private key pair of decryption side S, wherein λ=2 according to the security parameter λ of system^k, k is Positive integer；

The private key sk of S is

Sk=p (1)

The public key pk of S is

Pk={ x₀,x₁,...,x_τ} (2)

In formula (1), sk indicates that the private key of S, p are by key generation centre from section [2^η-1,2^η)∩(2Z⁺+ 1) it is randomly choosed on One η odd positive integers, i.e.,Wherein η indicates the positive integer of η >=1,Represent with Machine selection, [2^η-1,2^η) indicate from 2^η-1To 2^ηBetween the set that constitutes of all odd positive integers, and include 2^η-1Not comprising 2^η；Z⁺ Indicate Positive Integer Set, (2Z⁺+ 1) odd Positive Integer Set is indicated；

In formula (2), pk indicates that the public key of S, τ indicate the element number for including in public key pk, and each element meets in the public key Condition x_υ=q_υp+2r_υ, wherein υ ∈ { 0,1 ..., τ }, and guarantee in public key set { x₀,x₁,...,x_τIn x₀Numerical value most Greatly；q_υAnd r_υIt is the randomly selected two groups of integers of key generation centre, i.e., υ∈ {0,1,...,τ}；HereRandom selection is represented,Indicate from 0 toBetween comprising 0 and not includingJust Integer set；[-2^ρ,2^ρ] indicate from -2^ρWith 2^ρIt include -2 on section^ρWith 2^ρInteger set；Z indicates integer set；

Step 2: key generation centre generates encryption side E according to the security parameter λ of system_iPublic private key pair, wherein λ=2^k, k is Positive integer, i ∈ { 1,2 ..., n }；

E_iPrivate key sk_iFor

sk_i=p_i (3)

E_iPublic key pk_iFor

pk_i={ x '_i,0,x′_i,1,...,x′_i,τ} (4)

In formula (3), sk_iIndicate E_iPrivate key, p_iIt is by the randomly selected η of key generation centre_iThe odd positive integer of position, i.e.,Wherein η_iIndicate η_i>=1 positive integer,It represents and randomly chooses, [2^η-1,2^η) indicate from 2^η-1To 2^ηBetween the set that constitutes of all odd positive integers, and include 2^η-1Not comprising 2^η；Z⁺Indicate Positive Integer Set, (2Z⁺+1) Indicate odd Positive Integer Set；

In formula (4), pk_iIndicate E_iPublic key, what wherein i ∈ { 1,2 ..., n }, n were represented is the number of encryption side；Public key pk_i In each element meet condition x '_i,j=x_i,jmodx_i,0；WhereinAnd it needs to guarantee gathering {x_i,0,x_i,1,...,x_i,τIn x_i,0Numerical value is maximum, formulaIn q_i,jAnd r_i,jFor key generation centre Randomly selected two groups of integers, i.e.,I ∈ { 1,2 ..., n }, j ∈ 0, 1,...,τ}；FormulaInIt is that random permutation algorithm is carried out to the element of public key pk in formula (2) π_iThe integer of generation, replacement algorithm π_iDetailed process be: position of the key generation centre to+1 element of τ in the public key pk of S Carry out stochastic transformation generation

Step 3: encryption side E_iIn its public key pk_iSet X_i={ x '_i,0,x′_i,1,...,x′_i,τOne non-empty of middle random selection Subset S_i={ x "_i,0,x″_i,1,...,x″_i,t, meetAnd t is the positive integer for meeting t≤τ；Then E_iSelection one A random integersE_iFollowing computations are executed, ciphertext c is obtained_i,l:

In formula, m_i,lIt is encryption side E_iThe plaintext m of input_iBinary form, and m_i,l∈ { 0,1 }, m_iIt is encryption side E_iInput Plaintext, binary length L, L are the integer greater than 0；

Step 4: the polynomial computation rule f that cloud computing platform needs to be implemented ciphertext according to decryption side S, converts f to accordingly Boolean circuit g, then cloud computing platform execution circuit assessment algorithm, calculates multiple ciphertexts, close after being assessed Literary C^*:

In formula, Boolean circuit of the g expression cloud platform according to polynomial transformation, Evaluate indication circuit assessment algorithm,Indicate the cloud platform ciphertext encrypted by multiple encryption sides to be calculated；

It is calculated Step 5: decryption side S executes decryption using key sk=p, obtains plaintext M^*:

M^*=(C^*modp)mod2 (7)

In formula, p indicates that the private key of decryption side, mod indicate modular arithmetic.