CN112383394A - Novel incremental signature method based on ideal lattice - Google Patents

Novel incremental signature method based on ideal lattice Download PDF

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CN112383394A
CN112383394A CN202011319580.2A CN202011319580A CN112383394A CN 112383394 A CN112383394 A CN 112383394A CN 202011319580 A CN202011319580 A CN 202011319580A CN 112383394 A CN112383394 A CN 112383394A
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朱锦德
吴开贵
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Chongqing University
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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
    • H04L9/08—Key distribution or management, e.g. generation, sharing or updating, of cryptographic keys or passwords
    • H04L9/0816—Key establishment, i.e. cryptographic processes or cryptographic protocols whereby a shared secret becomes available to two or more parties, for subsequent use
    • H04L9/0852—Quantum cryptography
    • 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
    • H04L9/32—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols including means for verifying the identity or authority of a user of the system or for message authentication, e.g. authorization, entity authentication, data integrity or data verification, non-repudiation, key authentication or verification of credentials
    • H04L9/3247—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols including means for verifying the identity or authority of a user of the system or for message authentication, e.g. authorization, entity authentication, data integrity or data verification, non-repudiation, key authentication or verification of credentials involving digital signatures

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Abstract

The invention discloses a novel incremental signature method based on ideal lattices, and particularly relates to the technical field of lattice cryptography. A novel incremental signature method based on ideal lattices comprises the following steps: initializing system parameters and generating a public key and a private key; whether a signature generated by a standard signature scheme exists; acquiring an original signature sigma and a corresponding message under a standard signature, generating a corresponding new signature sigma' for the new message mu through an increment scheme, executing a verification algorithm Verify (PK, mu and sigma) for the message and the generated signature, and judging whether the output is 1. The technical scheme of the invention solves the problem that the traditional computer can not resist the attack of the quantum computer, and can be used for high-efficiency encrypted signature in the existing computer.

Description

Novel incremental signature method based on ideal lattice
Technical Field
The invention relates to the technical field of lattice cryptography, in particular to a novel incremental signature method based on ideal lattices.
Background
With the rapid development of quantum computers, quantum algorithms are a corresponding breakthrough. Under a quantum computing model, a cryptosystem assumed by classical number theory (such as large integer decomposition, discrete logarithm problem on a computational finite field/elliptic curve and the like) has a quantum algorithm of polynomial time (PPT), in other words, the classical number theory cryptosystem is greatly impacted and can possibly become tears of the old era. Therefore, the password capable of resisting quantum computer attack- "post quantum" or "anti quantum" password is produced at the same time, and lattice passwords are a class of concerned public key cryptosystems resisting quantum computer attack.
Lattice cryptography has developed significantly over the past few decades, and lattice digital signatures have become one of the hot spots of academic interest in recent years, because of the significant efficiency gains achieved with cryptographic schemes using lattices with algebraic structures, and the power and diversity of the lattice classical difficulty problem, the error learning problem (LWE) and the small integer solution problem (SIS). Theoretically, any one-way function can be used for constructing the digital signature, so that a large number of digital signature schemes based on the lattice difficulty problem are derived directly on the basis of the pioneering work of Ajtai. However, the general structure brings serious inefficiency, and the parameters of the lattice code scheme usually comprise several parameters
Figure BDA0002792427790000011
A dimension vector. If a general digital signature construct is used, it requires many one-way functions, which results in the generation of a need when using algebraic lattices
Figure BDA0002792427790000012
Huge digital signatures of storage spaces. Therefore, finding a practical construction of a "short" signature based on the lattice difficulty problem, such as a lattice signature consisting of a single lattice vector, has been a very critical problem since the advent of lattice cryptography.
An ideal lattice is a class of lattices that consists of an ideal of lattices. For convenience of introducing the ideal lattice, we cut in from two common lattice difficulties, namely the LWE problem and the SIS problem. An example of the LWE problem is defined by a random n × m integer matrix a and a vector b, where b ═ aTs + e mod q, where s is a secret vector and e is a small noise vector, the difficult problem is to find this vector s. As a "dual" problem for LWE, an example of a SIS problem is defined by the same random matrix a, which requires finding a short vector x such that Ax is 0 mod q. Ring-LWE and Ring-SIS are "Ring" up versions of LWE and SIS, respectively, specific examples of LWE and SIS, respectively, defined for certain structured matrices A, which are detailedThe introduction of which is as follows:
in an ideal lattice or so-called "ring setup", the above matrix a needs some additional algebraic structure. A common example is to interpret each column of A as a coefficient of a polynomial p (x) of degree (n-1), and require that xp (x) mod (x) be included in some columns of A as welln+1)). In this case, the matrix multiplication of a is equivalent to polynomial multiplication. Thus, we can consider each vector v as a ring
Figure BDA0002792427790000021
And each n × n sub-matrix of A is regarded as
Figure BDA0002792427790000022
A ring element ofi。
Due to the algebraic structure of ideal lattices, the cryptosystem based on ideal lattices (security based on Ring-LWE or Ring-SIS difficulty assumptions) is more efficient than that in general lattices, as shown in:
1. since each nxn sub-matrix is now represented as a ring element, the size of some of the parameters that would otherwise be matrices is reduced by a factor n;
2.
Figure BDA0002792427790000023
the multiplication of the ring elements can be implemented in hardware by FFT (fast fourier transform).
Thus, the use of ideal lattices is an important approach to the construction of highly efficient cryptographic schemes on lattices.
The incremental signature scheme is a good method for quickly generating the signature. Suppose we need to generate signatures for many messages, usually we should sign them separately. That is, if the signature time of one message is T, then the total signature time of k messages should be k × T, which means that the efficiency is obviously very low in some practical scenarios, such as large-scale software authentication and large data verification, where there are many messages and each message is long, and it obviously takes time to generate a signature. Note that the messages in the above scenario are generally similar. For example, software may be updated frequently to correct errors with the goal of improving quality, so there is often a lot of overlap between the previous software and the updated software. Therefore, to speed up the signature generation process, using a delta signature scheme is a more preferred option with only small differences between messages.
Bellare et al first proposed the concept of an incremental cryptosystem. In order to reflect the unique advantages of the method in pioneering work, the method also provides an incremental signature scheme, and the main ideas are as follows: when the signer obtains the signature σ for the message μ, then the signer does not need to generate the signature σ for the updated message μ' of μ again. Since it takes some time for him to extract σ from (μ, σ, μ'), which ideally is proportional to the total message modification of μ; thus, the delta signature scheme takes less time than the generic signature scheme for k similar messages.
Incremental cryptographic techniques have gained widespread attention in the cryptographic community in recent years, driven by real-world applications. Based on the pioneering work of Bellare et al, many researchers have proposed many new incremental cryptographic schemes, and the traditional difficult assumptions ensure the security of the schemes, which will be threatened by quantum computers.
Disclosure of Invention
The invention provides a novel incremental signature method based on an ideal lattice, aiming at the problem of how to set an efficient cryptographic signature to resist quantum computer attack in a traditional computer.
In order to achieve the purpose, the technical scheme of the invention is as follows: a novel incremental signature method based on ideal lattices comprises the following steps:
s1: setup (n), system parameter setting: setting a security parameter n, giving two fixed public large integer parameters p, q and an integer b, and then generating other parameters according to the following operations:
s1-1: selecting vectors
Figure BDA0002792427790000031
I is 0, 1,., l;
s1-2: performing the trap-gate-on-Ring Generation Algorithm, ringGenTrap (A', H, s)R)；
S1-3: generating public key PK ═ (a, C)0，C1，...，Cl) And the private key SK ═ R;
s2: sign (PK, SK, μ), standard signature generation: for a given ciphertext mu ═ v1，v2，...，vl]∈{0，1}d×lUsing the public key PK and the private key R, a signature is generated according to the following steps:
s2-1: selecting
Figure BDA0002792427790000032
S2-2: computing
Figure BDA0002792427790000033
S2-3: computing
Figure BDA00027924277900000312
S2-4: executing ring pre-image sampling algorithm ringSampleD (A, R, qb.u, s) to obtain
Figure BDA0002792427790000034
S2-5: generating signatures
Figure BDA0002792427790000035
S3: IncSig (PK, SK, mu, sigma, mu') incremental signature generates a given message mu e {0, 1}d×lThe corresponding signature μ e {0, 1}d×lAnd a modified message mu '═ v'1，v′2，...，v′l]∈{0，1}d×lMu' and mu satisfy the same condition except for column j; generating a delta signature using the public key PK and the private key R by:
s3-1: selecting
Figure BDA0002792427790000036
S3-2: computing
Figure BDA00027924277900000313
S3-3: performing the on-Ring proto-image sampling algorithm ringSampleD (A, R, qb u', s) to obtain
Figure BDA0002792427790000037
S3-4: generating incremental signatures
Figure BDA0002792427790000038
S4: verify (PK, μ, σ), complete the verification. According to the public key PK, the message mu-v1，v2，...，vl]∈{0，1}d×lAnd signatures
Figure BDA0002792427790000039
Output 1 if the following conditions are met:
Figure BDA00027924277900000310
e ═ qb · u (mod pq) and
Figure BDA00027924277900000311
compared with the prior art, the beneficial effect of this scheme:
compared with the existing incremental signature scheme, the scheme is constructed based on the difficult problem of ideal lattices, parameters in the scheme are all obtained from a ring, and the algebraic characteristics of the ideal lattices are fully utilized, so that the sizes of the public and private keys and the generated signature are almost reduced by O (n). At the same time, due to the ring
Figure BDA00027924277900000415
The multiplication between the elements in (1) can be accelerated in hardware using FFT (fast Fourier transform), so the present invention is in memory complexity toAnd the calculation efficiency is obviously superior to that of the existing scheme, and the method has great use and practical value.
Drawings
Fig. 1 is a flow chart of a novel incremental signature method based on ideal lattices in the invention.
Detailed Description
The present invention will be described in further detail below by way of specific embodiments:
examples
The following description will be made with reference to the following examples:
Figure BDA0002792427790000041
represents a ring of modulo q, where q ≧ 2.
Figure BDA0002792427790000042
Representing a set of n x m matrices, in which the entries are
Figure BDA0002792427790000043
In (1). s1(A)＝maxuAu | | represents the maximum singular value of the matrix a, where u represents the identity matrix and | | · | | represents
Figure BDA00027924277900000413
A paradigm. When S refers to a collection, x ← S denotes uniform random retrieval of x from the collection S. When in use
Figure BDA00027924277900000417
When referring to a distribution of the plurality of clusters,
Figure BDA00027924277900000416
represents x according to distribution
Figure BDA00027924277900000418
And (6) obtaining.
Figure BDA0002792427790000044
And
Figure BDA0002792427790000045
respectively representing polynomials with coefficients being integers and real numbers,
Figure BDA00027924277900000414
indicating ring
Figure BDA0002792427790000046
And assume that all operations in the scheme are done on the ring.
Figure BDA0002792427790000047
Is a polynomial ring of modulo f (x) and has a coefficient range of
Figure BDA0002792427790000048
The above is obtained.
Figure BDA0002792427790000049
Refers to a ring polynomial of length m.
Figure BDA00027924277900000410
Represents an element
Figure BDA00027924277900000411
As a ring polynomial vector
Figure BDA00027924277900000412
The coefficient of (a).
Incremental signature definition: an incremental signature scheme consists of four Probabilistic Polynomial Time (PPT) algorithms, namely the system Setup algorithm Setup, the standard signature algorithm Sign, the incremental signature algorithm incusig, and the verification algorithm Verify.
Setup (n): given a security parameter n, the algorithm outputs a system security parameter PP and a private key SK.
Sign (PP, SK, μ): given the public parameter PP, the private key SK and the message μ, the algorithm outputs a signature σ for μ.
Incusig (PP, SK, μ, σ, μ'): given the public parameter PP, the private key SK, a set of message-signatures (μ, σ), and a new message μ ', the algorithm outputs a delta signature σ ' for μ '.
Vetify (PP, μ, σ): given the common parameters PP, one message mu and one signature sigma, the algorithm outputs 1 with an overwhelming probability if sigma is a valid signature of the message mu, and 0 otherwise.
Correctness: we require that the delta signature must satisfy the following correctness: for all n and messages μ, μ ', if (PP, SK) ═ setup (n), σ ═ Sign (PP, SK, μ) and σ ' ═ incusig (PP, SK, μ, σ, μ '), then Verify (PP, μ, σ) ═ 1 and Verify (PP, μ ', σ ') -1 must hold with an overwhelming probability.
As shown in fig. 1, a novel incremental signature method based on ideal lattices includes the following steps:
s1: setup (n), system parameter setting. Setting the safety parameter n to be more than or equal to 4 and the power of 2. Given two fixed large integer common parameters p and q, satisfying p.q ═ 3tAnd one integer b satisfies qb ═ 1(mod p). Determining a ring from the above parameters
Figure BDA0002792427790000051
Wherein phi2n(x)＝Xn+1 is a cyclotomic polynomial of order n. Other parameters are then generated according to the following operations:
s1-1: selecting vectors
Figure BDA0002792427790000052
i＝0，1，...，l；
S1-2: performing the trap-gate-on-Ring Generation Algorithm, ringGenTrap (A', H, s)R) Wherein
Figure BDA0002792427790000053
Mark matrix
Figure BDA00027924277900000516
Obtaining a matrix
Figure BDA0002792427790000054
And a matrix with a marking matrix H
Figure BDA0002792427790000055
G-trap door
Figure BDA0002792427790000056
S1-3: generating public key PK ═ (a, C)0，C1，...，Cl) And the private key SK ═ R.
S2: sign (PK, SK, μ), standard signature generation: for a given ciphertext mu ═ v1，v2，...，vl]∈{0，1}d×lUsing the public key PK and the private key R, a signature is generated according to the following steps:
s2-1: selecting
Figure BDA0002792427790000057
R obtained here satisfies
Figure BDA0002792427790000058
S2-2: computing
Figure BDA0002792427790000059
S2-3: computing
Figure BDA00027924277900000518
S2-4: executing ring pre-image sampling algorithm ringSampleD (A, R, qb.u, s) to obtain
Figure BDA00027924277900000510
S2-5: generating signatures
Figure BDA00027924277900000511
S3: incusig (PK, SK, μ, σ, μ'), incremental signature generation: given a message μ e {0, 1}d×lThe corresponding signature μ e {0, 1}d×lAnd a modified message mu '═ v'1，v′2，...，v′l]∈{0，1}d×lMu' andmu satisfies the same condition except for the j-th column. Generating a delta signature using the public key PK and the private key R by:
s3-1: selecting
Figure BDA00027924277900000512
R' obtained here satisfies
Figure BDA00027924277900000513
S3-2: computing
Figure BDA00027924277900000517
S3-3: performing an on-Ring proto-image sampling algorithm, ringSampleD (A, R, qb u', s) acquisition
Figure BDA00027924277900000514
S3-4: obtaining delta signatures
Figure BDA00027924277900000515
S4: verify (PK, μ, σ), signature verification. According to the public key PK, the message mu-v1，v2，...，vl]∈{0，1}d×lAnd signatures
Figure BDA0002792427790000061
Output 1 if the following conditions are met:
Figure BDA0002792427790000062
e ═ qb · u (mod pq) and
Figure BDA0002792427790000063
the table for comparing the efficiency of the signature method with the efficiency of the general incremental signature method is shown in the following table 1:
table 1:
Figure BDA0002792427790000064
note:
Figure BDA0002792427790000065
m＝w+t。
the foregoing are merely examples of the present invention and common general knowledge of known specific structures and/or features of the schemes has not been described herein in any greater detail. It should be noted that, for those skilled in the art, without departing from the structure of the present invention, several changes and modifications can be made, which should also be regarded as the protection scope of the present invention, and these will not affect the effect of the implementation of the present invention and the practicability of the patent. The scope of the claims of the present application shall be determined by the contents of the claims, and the description of the embodiments and the like in the specification shall be used to explain the contents of the claims.

Claims (1)

1. A novel incremental signature method based on ideal lattices is characterized in that: the method comprises the following steps:
s1: setup (n), system parameter setting: setting a security parameter n, giving two fixed public large integer parameters p, q and an integer b, and then generating other parameters according to the following operations:
s1-1: selecting vectors
Figure FDA0002792427780000011
I is 0, 1,., l;
s1-2: performing the trap-gate-on-Ring Generation Algorithm, ringGenTrap (A', H, s)R)；
S1-3: generating public key PK ═ (a, C)0，C1，...，Cl) And the private key SK ═ R;
s2: sign (PK, SK, μ), standard signature generation: for a given ciphertext mu ═ v1，v2，...，vl]∈{0，1}d×lUsing the public key PK and the private key R, a signature is generated according to the following steps:
s2-1: selecting
Figure FDA0002792427780000012
S2-2: computing
Figure FDA0002792427780000013
S2-3: computing
Figure FDA0002792427780000014
S2-4: executing ring pre-image sampling algorithm ringSampleD (A, R, qb.u, s) to obtain
Figure FDA0002792427780000015
S2-5: generating signatures
Figure FDA0002792427780000016
S3: IncSig (PK, SK, mu, sigma, mu') incremental signature generates a given message mu e {0, 1}d×lThe corresponding signature μ e {0, 1}d×lAnd a modified message mu '═ v'1，v′2，...，v′l]∈{0，1}d×lMu' and mu satisfy the same condition except for column j; generating a delta signature using the public key PK and the private key R by:
s3-1: selecting
Figure FDA0002792427780000017
S3-2: computing
Figure FDA0002792427780000018
S3-3: performing the on-Ring proto-image sampling algorithm ringSampleD (A, R, qb u', s) to obtain
Figure FDA0002792427780000019
S3-4: generating deltasSignature
Figure FDA00027924277800000110
S4: verify (PK, μ, σ), complete the verification. According to the public key PK, the message mu-v1，v2，...，vl]∈{0，1}d×lAnd signatures
Figure FDA00027924277800000111
Output 1 if the following conditions are met:
Figure FDA00027924277800000112
and is
Figure FDA00027924277800000113
e ═ qb · u (mod pq) and
Figure FDA00027924277800000114
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