CN113114461B - N-time public key compression method for integer homomorphic encryption - Google Patents
N-time public key compression method for integer homomorphic encryption Download PDFInfo
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
- H04L9/008—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols involving homomorphic encryption
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
- H04L9/30—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy
- H04L9/3006—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy underlying computational problems or public-key parameters
- H04L9/3033—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy underlying computational problems or public-key parameters details relating to pseudo-prime or prime number generation, e.g. primality test
Abstract
The invention provides an n-time public key compression method for integer homomorphic encryption, which expands the generation of a public key in an integer homomorphic encryption scheme to an n-time form, and reduces the number of integer elements of a stored public key. The invention generates the public key required by encryption by calculating the public key integer element in the public key vector pk, namely the actually stored public key integer element is not used in encryption but the public key used in real encryption is obtained by calculation, thus reducing the number of the public key integer elements during storage and realizing the compression of the size of the public key. Public key passing calculation in encryption
Description
Technical Field
The invention relates to an n-time public key compression method, in particular to an n-time public key compression method suitable for integer homomorphic encryption.
Background
The document "Coron J S, mandal A, naccache D, et al. Fully homomorphic encryption over the interpeneters with short public key [ C]//Proc of the 31 st International Conference on Advances in cryptography. Berlin: springer,2011, 487-504 "discloses a public key compression technical scheme for optimizing the size of an integer homomorphic encryption public key. The scheme provides a public key compression scheme aiming at the problems of overlarge size and low efficiency of the existing full homomorphic encryption scheme on the integer, the scheme changes the integer element of the public key from a linear form to a quadratic form, and the length of the public key is compressed to the length of the public key through proper parameter selection
Wherein λ is a safety parameter, λ =2 k And k is a positive integer. The scheme compresses the public key length of the integer homomorphic encryption scheme, but the compression degree is limited, and the public key size is compressed to
Still too large, therefore, the operation efficiency is not greatly improved, and the application in practice is still difficult。
Disclosure of Invention
In order to solve the problem of low scheme operation efficiency caused by overlarge size of a public key of an integer homomorphic encryption scheme, the invention provides an n-time public key compression method of integer homomorphic encryption.
The technical solution of the present invention is described below:
the n-time public key compression method for integer homomorphic encryption comprises the following steps of:
step 1: generating a private key p and an encryption modulus x 0 ：
Generating a private key p: p is a randomly generated large prime number of length eta bits, p is in the range of [2 ] η-1 ，2 η )；
Generating an encryption modulus x 0 ：x 0 ＝q 0 P, wherein
Is an integer ring, gamma denotes the public key integer element x i The symbol "←" means: the expression "a ← B" denotes the random selection of an element a from the set B;
step 2: generating other public key elements x in a public key vector i，b ：
For 1 ≦ i ≦ β,1 ≦ b ≦ n, generating integer x i，b ：
x i，b ＝r i，b +pq i，b
Where n, beta represent the public key integer element x i，b The number of the (c) is greater than the total number of the (c),
r i，b is random noise interference;
x i，b and the encryption modulus x generated in step 1 0 Together, the public key pk:
pk＝(x 0 ,x 1，1 ，…，x 1，n ，x 2，1， …，x 2，n ，x 3，1 …x 3，n ，…，x β，1 ，…，x β，n )
and step 3: and (3) encryption process:
generating tau public key integers in n-degree form
Wherein 1 is less than or equal to i 1 ，i 2 ，…，i n Beta is less than or equal to, tau and beta satisfy tau = beta n ；
Generating random coefficient vectors
The ciphertext c is represented as:
where m is the plaintext to be encrypted, the encryption is performed in bits, and r is the noise interference.
Further, random noise interference
Where ρ is a noise parameter representing random noise interference r i，b The bit length of (c).
Further, in the random coefficient vector b, i is more than or equal to 1 1 ，i 2 ，…，i n Not more than beta and
α represents the bit length of an integer in the random coefficient vector b, and α · β n And gamma + omega (log lambda) to make the residual hash lemma satisfy the constraint condition of approximate greatest common divisor problem.
Further, the noise interference r is larger than (-2) ρ′ ，2 ρ′ ) ρ 'is a second noise parameter representing the bit length of the noise disturbance r, and ρ' ≧ α + n ρ + ω (log λ), where ω (-) represents the asymptotically tight lower bound.
Further, q is 0 Is not prime factor included and is less than
For resisting prime number traversal attacks; where λ represents a security parameter of the homomorphic encryption scheme, λ =2 k And k is a positive integer.
Further, the bit length eta of the private key p is larger than or equal to rho theta (lambda log) 2 λ), supporting enough circuits deep homomorphic operation to satisfy the operation of the 'compression decryption circuit'; where Θ (·) represents the progressive tight bound; and eta is greater than or equal to n rho + alpha +2+ nlog beta to ensure correct decryption of the ciphertext.
Further, the noise parameter ρ = ω (log λ) to prevent brute force estimation of public key random integers.
Further, the public key integer element x i The bit length gamma of ≧ omega (eta) 2 log λ) to a lattice-based multi-angle attack on the approximate greatest common divisor problem.
Furthermore, the invention also provides an encryption device, wherein the device adopts integer homomorphic encryption and adopts a public key compression method for n times in the integer homomorphic encryption.
Advantageous effects
The invention has the beneficial effects that: the invention expands the generation of the public key in the encryption scheme based on the integer homomorphism to a form of n times, thereby reducing the number of the integer elements of the stored public key. The public key required by encryption is generated through calculation of the integer element of the public key in the public key vector pk, namely the actually stored integer element of the public key is not used during encryption, but the public key used during real encryption is obtained through calculation, so that the number of the integer elements of the public key is reduced during storage, and the compression of the size of the public key is realized. Public key passing calculation in encryption
When the public key pk is stored, n beta public key elements are used for replacing original tau public key integer elements, so that the purpose of reducing the number of the public key integer elements is achieved, and the size of the public key is reduced. Reference "Coron J S, mandal A, naccache D, et al. Fully homomorphic encryption over the integrators with short public key [ C]//Proc of the 31 st Springer,2011 487-504", the public key is generated in a quadratic form, the generation of the public key is expanded to a n-th-power form, the number of integer elements of the stored public key is reduced, the size of the public key in an integer homomorphic encryption scheme is compressed, and the size of the public key is compressed into a size of the public key
Making it more suitable for use in cloud computing applications.
Additional aspects and advantages of the invention will be set forth in part in the description which follows and, in part, will be obvious from the description, or may be learned by practice of the invention.
Detailed Description
In the following, a detailed description is given of an n-time public key compression method applied to integer homomorphic encryption, where we first define symbols and parameters that may be used in the method, and unless otherwise specified, the algorithm in the method is based on binary operation, and the logarithmic operation is also based on base-2 logarithmic operation.
For a real number x, the number x,
and
respectively representing the rounding-down, rounding-up and rounding-up of a real number x, for a real number z and an integer p, q p (z) denotes the quotient of z divided by p, r p (z) represents the remainder of z divided by p. If f (λ) = O (g (λ) log) k g (λ)), f (λ) and g (λ) are both functions on λ, O (-) represents a functional series, k ∈ N, N is a set of natural numbers,then
Progressive symbols
Representing an infinite number of orders.
λ denotes the security parameter of the homomorphic encryption scheme, λ =2 k K is a positive integer;
gamma denotes the public key integer element x i I is more than or equal to 0 and less than or equal to tau;
τ denotes a public key integer element x i τ is a positive integer;
n, beta represent the public key integer element x i，b I is more than or equal to 1 and less than or equal to beta, b is more than or equal to 1 and less than or equal to n;
η represents the bit length of private key p;
ρ represents a noise parameter representing the random noise interference r when generating the integer element of the public key i，b The bit length of (d);
ρ' is a second noise parameter indicating the bit length of the random noise interference r used by the encryption process;
alpha represents the bit length of an integer in a random coefficient vector b used for increasing the randomness of a public key in the encryption process;
wherein, λ, γ, τ, n β, η, ρ, ρ', α are all positive integers.
In the prior art, the public key vector pk is (x) 0 ，x 1 …，x τ ) Of the form public key integer element x stored in a public key vector i I is more than or equal to 1 and less than or equal to tau, and tau is obtained, the public key integer elements are generated by using n-order form, and the public key integer elements are x i，b I is more than or equal to 1 and less than or equal to beta, b is more than or equal to 1 and less than or equal to n, and only n beta integers x are stored i，b Can generate tau public key integers
As a public key.
The integer homomorphic encryption method provided by the invention is established on the problem of interference-free approximate maximum common divisorThe common divisor problem, also called the non-interfering approximate maximum common divisor hypothesis, can be expressed as p, p ∈ [2 ] for p η-1， 2 η ) P is a large prime number with the length of eta bit randomly generated, and x is calculated 0 ＝q 0 P of wherein
Is an integer ring, q 0 Is not prime factor included and is less than
Is an integer of (2). For two specified integers p and q 0 Giving a distribution
In formula (1), q ← [0, q) 0 )，
From distribution
In the randomly drawn polynomial samples and x 0 By these sample sums x 0 Solving for large prime numbers p of η bits is extremely difficult and computationally infeasible.
In order to ensure the security of the n-time public key compression method for integer homomorphic encryption, the following restrictions are made on the parameters:
ρ = ω (log λ), ω (-) representing a non-progressive tight lower bound in order to prevent brute force estimation of public key random integers;
η≥ρ·Θ(λlog 2 λ), in order to support enough circuit depth homomorphic operations to satisfy the operation on the "compression decryption circuit", Θ (·) represents a progressive tight bound;
γ≥ω(η 2 log λ), dealing with lattice-based multi-angle attacks that approximate the greatest common divisor problem;
α·β n gamma + omega (log lambda) or more, in order to make the remaining hash lemma satisfy the constraint condition of the approximate greatest common divisor problem;
eta is greater than or equal to n rho + alpha +2+ nlog beta, in order to ensure correct decryption of the ciphertext;
ρ' ≧ α + n ρ + ω (log λ) as a condition satisfied by the second noise parameter.
Based on the above setting, the following four specific processes of key generation, encryption, ciphertext operation and decryption are given:
1. and (3) key generation:
generating a private key firstly: sk = p, p is a large prime number with length of η bits randomly generated, p ∈ [2 ] η-1 ，2 η )。
The public key pk is then generated: generating an encryption modulus x 0 Calculating x 0 ＝q 0 P of wherein
Integer ring, q 0 Is not prime factor included and is less than
Is an integer of (b), to q 0 The limitation of (2) is to be able to resist prime traversal attacks. Wherein "←" means: the expression "a ← B" denotes an element a randomly selected from the set B.
For 1 ≦ i ≦ β,1 ≦ b ≦ n, generating integer x i，b ：
x i，b ＝r i，b +pq i，b (2)
Wherein
Random noise interference
x i，b And a previously generated encryption modulus x 0 Are formed togetherPublic key, then public key pk is expressed as:
pk＝(x 0 ,x 1，1 ，…，x 1，n ，x 2，1， …，x 2，n ，x 3，1 …x 3，n ，…，x β，1 ，…，x β，n ) (3)
from equation (3), it can be seen that the public key pk is in addition to the encryption modulus x 0 Other public key elements than x i，b I is more than or equal to 1 and less than or equal to beta, b is more than or equal to 1 and less than or equal to n, and the number of integer elements of the public key is n beta + 1.
2. Encryption:
m is a plaintext to be encrypted, the plaintext is encrypted according to bits, and the plaintext space is m E {0,1}.
By x i，b I is more than or equal to 1 and less than or equal to beta, b is more than or equal to 1 and less than or equal to n, and calculating tau public key integers
I is more than or equal to 1 in the formula (4) 1 ，i 2 ，…，i n Beta is less than or equal to beta, tau and beta satisfy tau = beta n In the scheme, n = log lambda is taken.
Generating random coefficient vectors
Wherein 1 is less than or equal to i 1 ，i 2 ，…，i n Not more than beta and
τ＝β n noise interference r ∈ (-2) ρ′ ，2 ρ′ )。
And (4) calculating a ciphertext c:
in the formula (5), i is more than or equal to 1 and less than or equal to beta, i is more than or equal to 1b≤n，x 0 ＝q 0 ·p，q 0 Is not prime factor included and is less than
The number of the integer (c) of (d),
p is a randomly generated large prime number of length eta bits, p is in the range of [2 ] η-1 ，2 η )。
3. Ciphertext operation:
given a circuit C (binary circuit) with t input bits and t ciphertexts C i (1. Ltoreq. I. Ltoreq.t) t is a positive integer, c i All addition and multiplication operations are performed on integers by all addition and multiplication gate circuits of circuit C and the final integer C' is returned as a result.
4. And (3) decryption:
outputting the decrypted plaintext m:
m＝(c′mod p)mod 2 (6)
in the formula (5), c' is a ciphertext obtained by ciphertext operation, p is a private key and is a large prime number with the length of eta bit generated in the key generation stage, and p belongs to [2 ] η-1 ，2 η ). The final plaintext results are as follows:
although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention, and that variations, modifications, substitutions and alterations can be made in the above embodiments by those of ordinary skill in the art without departing from the principle and spirit of the present invention.
Claims (9)
1. A method of n-time public key compression for integer homomorphic encryption, characterized by: the method comprises the following steps:
step 1: generating privacyKey p and encryption modulus x 0 ：
Generating a private key p: p is a randomly generated large prime number of length eta bits, p belongs to [2 ] η-1 ，2 η )；
Generating an encryption modulus x 0 ：x 0 ＝q 0 P, wherein
Is an integer ring, gamma denotes the public key integer element x i The symbol "←" means: the expression "a ← B" denotes the random selection of an element a from the set B;
step 2: generating other public key elements x in a public key vector i，b ：
For 1. Ltoreq. I.ltoreq.beta.1. Ltoreq. B.ltoreq.n, the integer x is generated i，b ：
x i， b＝r i，b +pq i，b
Where n, beta represent the public key integer element x i，b The number of the (c) is,
r i，b is random noise interference;
x i，b and the encryption modulus x generated in step 1 0 Together, the public key pk:
pk＝(x 0 ，x 1，1 ，...，x 1，n ，x 2，1 ，...，x 2，n ，x 3，1 ...x 3，n ，...，x β，1 ，...，x β，n )
and step 3: and (3) encryption process:
generating tau public key integers in n-degree form
Wherein 1 is less than or equal to i 1 ，i 2 ，...，i n Beta is less than or equal to, tau and beta satisfy tau = beta n ；
Generating random coefficient vectors
The ciphertext c is represented as:
where m is the plaintext to be encrypted, the encryption is performed in bits, and r is the noise interference.
2. The n-time public key compression method for integer homomorphic encryption according to claim 1, characterized in that: random noise interference
Where ρ is a noise parameter representing random noise interference r i，b The bit length of (c).
3. A method of n-time public key compression for integer homomorphic encryption according to claim 1 or 2, characterized by: in the random coefficient vector b, i is more than or equal to 1 1 ，i 2 ，...，i n Not more than beta and
α represents the bit length of an integer in the random coefficient vector b, and α · β n ≥γ+ω(logλ)。
4. A method of n-time public key compression for integer homomorphic encryption according to claim 3, characterized in that: noise interference r epsilon (-2) ρ′ ，2 ρ′ ) And ρ' is a second noise parameter representing the bit length of the noise disturbance rAnd ρ' ≧ α + n ρ + ω (log λ), where ω (-) represents the non-progressive tight lower bound.
5. The n-time public key compression method for integer homomorphic encryption according to claim 4, wherein: q. q.s 0 Is not inclusive of prime factors and is less than
Where λ represents a security parameter of the homomorphic encryption scheme, λ =2 k And k is a positive integer.
6. The n-time public key compression method for integer homomorphic encryption according to claim 5, characterized in that: bit length eta of private key p is larger than or equal to rho theta (lambda log) 2 λ), where Θ (·) represents the progressive tight bound and η ≧ n ρ + α +2+ nlog β.
7. The n-time public key compression method for integer homomorphic encryption according to claim 6, wherein: the noise parameter ρ = ω (log λ) to prevent brute force estimation of public key random integers.
8. The n-time public key compression method for integer homomorphic encryption according to claim 7, characterized in that: public key integer element x i The bit length gamma of ≧ omega (eta) 2 logλ)。
9. An encryption apparatus, characterized in that: the device adopts integer homomorphic encryption, and adopts the n-time public key compression method of claim 1 in the integer homomorphic encryption.
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| CN103916248A (en) * | 2014-04-10 | 2014-07-09 | 东南大学 | Fully homomorphic encryption public key space compression method |
| CN111585743A (en) * | 2020-04-28 | 2020-08-25 | 西安电子科技大学 | Homomorphic encryption public key compression method for many-to-one on integer |
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| KR102019159B1 (en) * | 2013-12-23 | 2019-09-09 | 한국전자통신연구원 | Apparatus and methdo for giving the compressed encryption functionality to integer-based homomorphic encryption schemes |
| CN104283669B (en) * | 2014-08-25 | 2017-07-18 | 东南大学 | Re-encryption depth optimization method in full homomorphic cryptography |
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN103916248A (en) * | 2014-04-10 | 2014-07-09 | 东南大学 | Fully homomorphic encryption public key space compression method |
| CN111585743A (en) * | 2020-04-28 | 2020-08-25 | 西安电子科技大学 | Homomorphic encryption public key compression method for many-to-one on integer |
Non-Patent Citations (1)
| Title |
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| Fully homomorphic encryption over the integers with shorter public keys;Jean-sebastien coron;《P.Rogaway(Ed.):CRYPTO 2011,LNCS 6841》;20110818;正文第4-7页 * |
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