CN114553408B - Galois ring-based threshold linear encryption and decryption method for RS code - Google Patents

Galois ring-based threshold linear encryption and decryption method for RS code Download PDF

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CN114553408B
CN114553408B CN202210165966.5A CN202210165966A CN114553408B CN 114553408 B CN114553408 B CN 114553408B CN 202210165966 A CN202210165966 A CN 202210165966A CN 114553408 B CN114553408 B CN 114553408B
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陈博涵
邢朝平
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Shanghai Jiaotong 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
    • H04L9/085—Secret sharing or secret splitting, e.g. threshold schemes
    • 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
    • H04L9/3026—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 polynomials generation, e.g. generation of irreducible polynomials

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Abstract

A threshold linear encryption decryption method of RS code based on Galois ring includes initializing and selecting Galois ring, making data preparation for generation of RS code, judging user to carry out interactive calculation, directly calculating locally when addition exists only, otherwise obtaining double secret sharing to carry out calculation. The invention designs a threshold linear secret sharing method based on the RS code by introducing a linear secret sharing method (LSSS) and a Reed-Solomon (RS) code, exerts the advantages of LSSS secret information protection and RS code polynomial information transmission, and has high information transmission efficiency and safety. The invention is applied to the communication between users on MPC, and the realization of the addition and multiplication of the users during the data exchange is completed through the coding protocol.

Description

Galois ring-based threshold linear encryption and decryption method for RS code
Technical Field
The invention relates to a technology in the field of information security, in particular to a Galois-ring-based threshold linear encryption and decryption method for RS codes.
Background
Multiparty secure computing (MPC) refers to a method of how multiple parties securely compute the same contract function without a trusted third party. The technology can safely communicate among users on the premise of protecting private data of the users. Assume that there are n participants P 1 ,P 2 ,…,P n Each participant P i All have a private data x i . Participants need to guarantee their own private data x i Calculating f (x) without leakage 1 ,x 2 ,…,x n ). Two properties need to be met in MPC: privacy, each participant does not know the private data of other participants except the private data of the participant; correctness, the result of the function calculation is unique and correct.
Most of the existing MPC-based passwords and encoding protocols areAnd->The two large domains are performed on the ringThe above application is lacking.
Disclosure of Invention
Aiming at the defects existing in the prior art, the invention provides a Galois ring-based threshold linear encryption and decryption method for RS codes, and a threshold linear secret sharing method based on the RS codes is designed by introducing a linear secret sharing method (LSSS) and a Reed-Solomon (RS) code, so that the advantages of LSSS secret information protection and RS code polynomial information transmission are exerted, and the efficient information transmission efficiency and safety are realized. The invention is applied to the communication between users on MPC, and the realization of the addition and multiplication of the users during the data exchange is completed through the coding protocol.
The invention is realized by the following technical methods:
the invention relates to a Galois ring-based threshold linear encryption and decryption method for RS codes, which comprises the following steps:
step 1, initializing and selecting Galois rings, which specifically comprises the following steps:
1.1 selecting prime number p and exponent s to construct a ringSum field->
1.2 polynomial h (Y) =a is constructed by selecting the polynomial highest degree r 0 +a 1 Y+···+a r Y r ， Wherein a is r ＝β r =1, other a i Belonging to the set {0,1, …, p s Data within-1 }, beta i Data belonging to the set {0,1, …, p-1}, i.e. the polynomial with the highest degree of the first one being r; obtaining a new polynomial h' (Y) =beta after all coefficient modulo p of h (Y) 0 +β 1 Y+···+β r Y r ，/>h' (Y) is the first primitive polynomial of irreducible degree with highest degree r.
The polynomial h' (Y) satisfies primitive polynomial property, namelyAnd for any prime number p i |p r -1 have->The order of h' (Y) is p r -1。
1.3 h' (Y) satisfiesLet h (Y) =h' (Y), then +.>Is the primitive polynomial with the first one and the highest term degree of r and the order of p r -1. For h (Y), there is a class of root +.>Satisfy->Then->Is Galois ring with upper order p r -non-zero elements of 1.
1.4 construction Galois Ring And set->The elements in set T are all on Galois rings.
Step 2, preparing data for generating RS codes, which specifically comprises the following steps:
2.1 randomly selecting n+1 different elements from the set T to form the setWherein n.ltoreq.p r -1. Randomly selecting any two elements in the set T for subtraction, and carrying out n+1 times to obtain +.> Each element in v is a unit element on the Galois ring.
The elements in the set T can form a unit element set { mu } 0 +μ 1 p+···+μ s-1 p s-1 ，μ i ∈T，μ 0 ≠0}
2.2 randomly generated integers k satisfy 0.ltoreq.k.ltoreq.n-1, and the Galois ring randomly generates polynomials f (x) E GR (p) s ,r)[x] <k The RS code for f (x) can be expressed as (v) 0 f(α 0 ),v 1 f(α 1 ),…,v n f(α n ) The RS code is expressed as
Step 3, judging that the user performs interactive calculation, directly performing local calculation when only addition exists, otherwise, obtaining double secret sharing to perform calculation, and specifically comprising the following steps:
3.1 totally n usersSelecting RS code with length of n+1On the above, codeword information (x, x 1 ,x 2 ,…,x n ) And (y, y) 1 ,y 2 ,…,y n ) Where x and y are secret data, x i And y i Is the holding data of each user i, will (x 1 ,…,x n ) And (y) 1 ,…,y n ) Respectively denoted as [ x ]] t And [ y ]] t
3.2 when only the addition calculation of x and y is needed, user i only needs to hold the data x i And y i Locally calculating to obtain x i +y i As long as t+1 users share their own data x i +y i Then all users can have t+1 [ x+y ]] t And thereby reconstruct x+y.
3.3 when the multiplication of x and y is required, the user first needs to obtain a duplicate secret share ([ z)] t ,[z] 2t ). Selecting t+1 users and RS codes in n+1And->Respectively generating codewords (c) i ,[c i ] t ) And (c) i ,[c i ] 2t ) Secret data z=c 0 +...+c t It is apparent that this data is not revealed to the user. [ z ]] t ＝([c 0 ] t ,…,[c t ] t )＝(z,z 1 ,z 2 ,…,z n ) And [ z ]] 2t ＝([c 0 ] 2t ,…,[c t ] 2t )＝(z,z 1 `,z 2 `,…,z n ' constitutes a duplicate secret sharing ([ z ]] t ,[z] 2t )。
3.4 per user i local calculations (x i y i )`＝x i *y i And e i `＝(x i y i )`-z i Then as long as 2t+1 users share their own data e i All users can have 2t+1 [ e ]] 2t Thereby reconstructing e.
3.5 randomly generating a set of codewords according to e (e, [ e ]] t ) Disclosing, each user locally calculates t+1 x i y i ＝e i +z i I.e. by t+1 [ xy ]] t And reconstructing xy therefrom.
Technical effects
The invention is in the ringThe upper run is applied to the current multiparty secure computing field through complete application, and the upper run is in the field +.>And->The method can be applied to practical application, and the integrity of the MPC on the application scene on the data application is perfectly improved.
Drawings
FIG. 1 is a flow chart of an embodiment.
Detailed Description
As shown in fig. 1, this embodiment relates to a threshold linear encryption and decryption method for RS codes based on Galois rings, which includes the following steps:
step 1) negotiating to generate related parameters of Galois ring and RS code, which comprises the following steps:
1.1 Prime number p=2, exponent s is random integer of arbitrary bit, highest term number r is random integer of arbitrary bit, participating user n=p r -1, when RS codeThe highest degree of the polynomial on the selected Galois ring is 2t=n-2=p r -3, RS code->The polynomial highest degree on the selected Galois ring is t= (p) r -3)/2。
1.2 Generating the highest term p r First-order polynomial of-1At->Factorization of the same to obtain +.>For irreducible polynomials where the highest term degree r is satisfied, a test is performed which must satisfy the requirement for any prime number p i |p r -1 have->The selected primitive polynomial of degree r, which is first, irreducible, and highest, of polynomial factor, is p r -1, let the polynomial factor be +.>H (Y) above. h (Y) has a group of roots->Satisfy->Then->Can represent the upper order p of Galois ring r -non-zero elements of 1.
1.3 Constructing Galois rings as Sequentially constructing a set->Co p r Elements, each element on a Galois ring; structure set->Co p r Element(s), wherein->Others->Is a random non-zero element in the set T. The set T and the set α are equal, except that the order of the data is not necessarily the same; structure set-> Co p r Elements, each element->The reconstruction of the RS code can be performed, and the requirement that the element in v is a unit element on the Galois ring is also met; structure-> Wherein set α and set v are all user-fixed, f (x) ∈GR [ x ]] <k Then it is randomly generated and f (x) takes the highest term number k-1.
When the data on the Galois ring is needed, only one data is generated randomlyThe elements of the above-mentioned material are,its mould->The latter elements are all on the Galois ring, which provides for the subsequent generation of polynomials on the Galois ring.
Step 2) when the x and y addition operation is to be carried out, setting the RS code with the length of n+1 from the third partyAcquiring codeword information (x, x 1 ,x 2 ,…,x n ) And (y, y) 1 ,y 2 ,…,y n ) Wherein x and y are as non-public data, x i And y i Distributed to each user i, who locally calculates x i +y i Selecting fixed t+1 honest users to disclose own data x i +y i All users can pass through t+1 [ x+y ]] t To reconstruct x+y.
The reconstruction means: for RS codesCodeword (x+y, [ x+y)] t ) There is a Galois-ring polynomial q (x) with the highest degree of t, i.e., the polynomial has t+1 coefficients to be solved, and the polynomial coefficients can be solved by lagrangian interpolation or matrix operation to obtain q (x), where x+y=v 0 q(α 0 )。
Step 3) when an x and y multiplication is to be performed, then a double secret sharing ([ z) is generated] t ,[z] 2t ): t+1 users are selected from n users, and each participant randomly selects Galois ring polynomial d i (x) And l i (x) Wherein d is i (x) The highest term number of the highest term is t, l i (x) The highest term number of the highest term is 2t, and d i (x) And l i (x) The lowest order coefficients of (2) are equal. Generating codewords from two polynomials (c i ,[c i ] t ) And (c) i ,[c i ] 2t ) Will ([ c) i ] t ,[c i ] 2t ) All of uses are disclosedUser calculation z] t ＝([c 0 ] t ,…,[c t ] t )＝(z,z 1 ,z 2 ,…,z n ) And [ z ]] 2t ＝([c 0 ] 2t ,…,[c t ] 2t )＝(z,z 1 `,z 2 `,…,z n '), thereby ensuring that z does not leak.
Step 4) after obtaining the duplicate secret sharing, each user i calculates (x i y i )`＝x i *y i And e i `＝(x i y i )`-z i Then 2t+1 honest users are selected to share the owned data ei', so that all users can pass through 2t+1 [ e ]] 2t E is reconstructed from the data in (c).
Step 5) when the third party randomly generates a set of codewords (e, [ e ] by obtaining a common e] t ) Will [ e ]] t Disclosed are methods and apparatus for controlling the flow of liquid. Local calculation of t+1 x per user i y i ＝e i +z i Then all users can have t+1 xy] t And reconstructing xy therefrom.
Through specific practical experiments, the polynomials are expressed in the form of vectors, such as 1+Y+Y 4 Denoted as [1 1 0 0 1]]。
The experimental parameters obtained by negotiation are as follows: prime number p=2, exponent s=3, degree r=4, number of users n=15, primitive polynomial h (Y) = [1 1 0 0 1], when 2t=13, t=6.
Set t= ([ 0)],[1],[0 1],[0 0 1],[0 0 0 1],[7 7],[0 7 7],[0 0 7 7],[1 1 0 7],[1 2 1],[0 1 2 1],[7 7 1 2],[6 5 7 1],[7 5 5 7],[1 0 5 5],[3 4 0 5]) Totally 16 are concernedIs a polynomial of (a).
Set α= ([ 0)],[7 7],[7 7 1 2],[6 5 7 1],[7 5 5 7],[3 4 0 5],[0 7 7],[1 0 5 5],[0 1],[0 0 7 7],[0 1 2 1],[0 0 0 1],[1 1 0 7],[1 2 1],[0 0 1],[1]) Totally 16 are concernedIs a polynomial of (a).
Set v= ([ 1)],[1],[1],[1],[1],[1],[1],[1],[1],[1],[1],[1],[1],[1],[1],[1]) Totally 16 are concernedIs a polynomial of (a).
When a user wants to perform a simple multiplication operation, such as 2*3, a double secret share ([ z) is generated] t ,[z] 2t )。
[z] t ＝([3 6 4 1],[1 6 0 2],[4 0 0 6],[0 7 7 6],[1 4 3 2],[6 2 5 4],[7 4 0 7],[1 2 2 6],[1 6 2 2],[4 5 1 1],[0 6 2 4],[7 2 7 2],[6 4],[5 7 6 5],[5 3 7 6])。
[z] 2t ＝([1 1 2 1],[1 6 6 2],[6 4 2 4],[7 7 3 5],[3 6 0 7],[7 0 3 1],[5 2 6 1],[7 1 4 4],[1 2 1 2],[6 3 6 3],[6 6 6 5],[7 0 0 6],[5 0 7 3],[2 5 4 1],[1 7 2 5])。
From the RS code of length 16The codeword information is obtained (x=2, x 1 ,x 2 ,…,x n ) And (y=3, y 1 ,y 2 ,…,y n ). Where user i holds x i And y i 。
(x＝2,x 1 ,x 2 ,…,xn)＝([2],[0 2 5 7],[6 7 6 7],[4 2 4 4],[0 7 2 4],[0 1 0 5],[7 0 5 6],[2 4 6 3],[5 6 0 1],[3 3 5 4],[4 1 6 1],[5 3 7 3],[4 5 0 6],[2 4 1 5],[0 7 7 5],[4 2 2 3])。
(y＝3,y 1 ,y 2 ,…,y n )＝([3],[2 2 1 3],[3 2 0 2],[6 1],[2 0 6 5],[3 2 2 5],[2 7 6 5],[6 2 4 2],[1 6 7 6],[6 5 3 6],[5 3 1 6],[5 6 7 1],[7 3 7 1],[1 7 1],[2 4 0 6],[0 4 1])。
User i performs (x) i y i )`＝x i *y i The arithmetic of (1) is [ xy ]] 2t ＝([7 5 3 5],[6 1 6 7],[4 4 2 4],[1 5 6 6],[1 2 7],[6 4 1 4],[6 6 6 4],[3 3 2 3],[5 2 4 4],[5 5 2 2],[3 3 1],[5 6 3 3],[6 1 2],[2 6 2],[2 7 1 2]) These data are private to the individual user.
User i performse i `＝(x i y i )`-z i The operation of [ e ]] 2t ＝([6 4 1 4],[5 3 0 5],[6],[2 6 3 1],[6 4 7 1],[7 4 6 3],[1 4 0 3],[4 2 6 7],[4 0 3 2],[7 2 4 7],[5 5 3 3],[6 6 3 5],[1 1 3 5],[0 1 6 7],[1 0 7 5]) These data are private to the individual user.
Select e of 14 users disclosing themselves i Then the codeword (e, [ e ]] 2t ) The polynomial q (x) on the corresponding Galois ring has a highest degree of 13.
q(x)＝[[1 6 4 4][3 6 6][7 7 1 2][7 0 2 2][6 7 1 6][3 0 3 3][7 7 6][7 1 1 1][7 3 7 6][2 7 2 2][7 3 5 3][3 7 5 6][6 7 1 1][7 3 3 1]]。
The user can calculate e=v 0 q(α 0 )＝[1 6 4 4]Then randomly generating codeword (e, [ e ]] t )。
(e,[e] t )＝([1 6 4 4],[0 7 6 4],[4 1 4],[2 4 4 3],[3 3 3 1],[1 1 3 6],[0 5 7],[4 5 5 2],[3 7 4 5],[5 5 7 7],[0 0 2 7],[7 7 4 2],[4 5 6],[2 5 2 1],[2 4 6 4],[0 3 3 6])。
At this time, each user has [ e ]] t And [ z ]] t Then [ xy ] can be calculated by itself] t 。
[xy] t ＝([3 5 2 5],[5 7 4 2],[6 4 4 1],[3 2 2 7],[2 5 6],[6 7 4 4],[3 1 5 1],[4 1 6 3],[6 3 1 1],[4 5 3],[7 5 6 6],[3 7 5 2],[0 1 2 1],[7 3 4 1],[5 6 2 4])。
Finally each user reconstruct [ xy ] by oneself] t Resulting in x y = 6.
Galois ring based Reed-Solomon code length on MPC is at most 2 r And the secret sharing size is log|gr|=rs, where s is fixed. When rs is a fixed value, the number of participants is also fixed. Therefore, the present invention expects that s, r corresponding to Galois ring based codes are fixed and the code length is as long as possible. The invention supplements the current situation of insufficient application of MPC in the field of rings.
The foregoing embodiments may be partially modified in numerous ways by those skilled in the art without departing from the principles and spirit of the invention, the scope of which is defined in the claims and not by the foregoing embodiments, and all such implementations are within the scope of the invention.

Claims (2)

1. A threshold linear encryption decryption method of RS code based on Galois ring is characterized in that through initializing and selecting Galois ring, after preparing data for generation of RS code, judging that user carries out interactive calculation, when only addition exists, local direct calculation is carried out, otherwise, double secret sharing is obtained for calculation;
the initializing and selecting Galois ring specifically includes:
1.1 selecting prime number p and exponent s to construct a ringSum field->
1.2 polynomial construction by selecting the highest degree of polynomial rWherein a is r ＝β r =1, other a i Belonging to the set 0,1, p s Data within-1 }, beta i Data within the set {0,1,., p-1}, i.e., a polynomial with the highest degree of the first one being r; obtaining new polynomial after all coefficient modulus p of h (Y) h' (Y) is a primitive polynomial of first, irreducible, and highest degree r;
1.3 h' (Y) satisfiesLet h (Y) =h' (Y),then->Is the primitive polynomial with the first one and the highest term degree of r and the order of p r -1; for h (Y), there is a class of root +.>Satisfy->Then->Is Galois ring with upper order p r -a non-zero element of 1;
1.4 construction Galois Ring And set->The elements in the set T are all on the Galois ring;
the preparation of data for the generation of the RS code specifically comprises the following steps:
2.1 randomly selecting n+1 different elements from the set T to form the setWherein n.ltoreq.p r -1; randomly selecting any two elements in the set T for subtraction, and carrying out n+1 times to obtain +.> v each element is a unit element on the Galois ring;
2.2 randomly generated integers k satisfy 0.ltoreq.k.ltoreq.n-1, and the Galois ring randomly generates polynomials f (x) E GR (p) s ，r)[x] <k The RS code for f (x) can be expressed as (v) 0 f(α 0 )，v 1 f(α 1 )，...，v n f(α n ) The RS code is expressed as
The judging user performs interactive calculation, which specifically comprises the following steps:
3.1 totally n users, selecting RS code with length of n+1On the above, codeword information (x, x 1 ,x 2 ,...，x n ) And (y, y) 1 ,y 2 ,...，y n ) Where x and y are secret data, x i And y i Is the holding data of each user i, will (x 1 ,...，x n ) And (y) 1 ，...，y n ) Respectively denoted as [ x ]] t And [ y ]] t ；
3.2 when only the addition calculation of x and y is needed, user i only needs to hold the data x i And y i Locally calculating to obtain x i +y i As long as t+1 users share their own data x i +y i Then all users can have t+1 [ x+y ]] t Reconstructing x+y therefrom;
3.3 when the multiplication of x and y is required, the user first needs to obtain a duplicate secret share ([ z)] t ，[z] 2t ) The method comprises the steps of carrying out a first treatment on the surface of the Selecting t+1 users and RS codes in n+1And->Respectively generating codewords (c) i ，[c i ] t ) And (c) i ，[c i ] 2t ) Secret data z=c 0 +...+c t ；[z] t ＝([c 0 ] t ，...，[c t ] t )＝(z,z 1 ,z 2 ,...，z n ) And [ z ]] 2i ＝([c 0 ] 2t ，...，[c t ] 2t )＝(z，z 1 `，z 2 `，...，z n ' constitutes a duplicate secret sharing ([ z ]] t ，[z] 2t )；
3.4 per user i local calculations (x i y i )`＝x i *y i And e i `＝(x i y i )`-z i Then as long as 2t+1 users share their own data e i All users can have 2t+1 [ e ]] 2t Reconstructing e therefrom;
3.5 randomly generating a set of codewords according to e (e, [ e ]] t ) Disclosing, each user locally calculates t+1 x i y i ＝e i +z i I.e. by t+1 [ xy ]] t And reconstructing xy therefrom.
2. The method for decrypting the threshold linear encryption of the Galois ring-based RS code according to claim 1, wherein the polynomial h' (Y) satisfies primitive polynomial properties, namelyAnd for any prime number p i |p r -1 have-> The order of h' (Y) is p r -1; the elements in the set T can form a unit element set { mu } 0 +μ 1 p+…+μ s-1 p s-1 ，μ i ∈T，μ 0 ≠0}。
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