CN113037461B - Multi-candidate anonymous electronic voting method based on homomorphic encryption - Google Patents

Multi-candidate anonymous electronic voting method based on homomorphic encryption Download PDF

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CN113037461B
CN113037461B CN202110240530.3A CN202110240530A CN113037461B CN 113037461 B CN113037461 B CN 113037461B CN 202110240530 A CN202110240530 A CN 202110240530A CN 113037461 B CN113037461 B CN 113037461B
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CN113037461A (en
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马文平
李�杰
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Xidian University
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    • 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
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L63/00—Network architectures or network communication protocols for network security
    • H04L63/04—Network architectures or network communication protocols for network security for providing a confidential data exchange among entities communicating through data packet networks
    • H04L63/0428—Network architectures or network communication protocols for network security for providing a confidential data exchange among entities communicating through data packet networks wherein the data content is protected, e.g. by encrypting or encapsulating the payload
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L63/00—Network architectures or network communication protocols for network security
    • H04L63/08—Network architectures or network communication protocols for network security for authentication of entities
    • H04L63/083—Network architectures or network communication protocols for network security for authentication of entities using passwords
    • 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
    • H04L9/0819—Key transport or distribution, i.e. key establishment techniques where one party creates or otherwise obtains a secret value, and securely transfers it to the other(s)
    • H04L9/083—Key transport or distribution, i.e. key establishment techniques where one party creates or otherwise obtains a secret value, and securely transfers it to the other(s) involving central third party, e.g. key distribution center [KDC] or trusted third party [TTP]
    • 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
    • H04L9/0861—Generation of secret information including derivation or calculation of cryptographic keys or passwords
    • H04L9/0869—Generation of secret information including derivation or calculation of cryptographic keys or passwords involving random numbers or seeds
    • 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
    • H04L9/3236—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 using cryptographic hash functions
    • H04L9/3239—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 using cryptographic hash functions involving non-keyed hash functions, e.g. modification detection codes [MDCs], MD5, SHA or RIPEMD
    • 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
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L2209/00—Additional information or applications relating to cryptographic mechanisms or cryptographic arrangements for secret or secure communication H04L9/00
    • H04L2209/46—Secure multiparty computation, e.g. millionaire problem
    • H04L2209/463—Electronic voting

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Abstract

The invention discloses a multiple candidate anonymous electronic voting method based on fully homomorphic encryption, which mainly solves the problems that in the prior art, the cascade depth of homomorphic operation is small, so that votes are counted wrongly when the number of voters is fixed, the voting is invalid, and the voting efficiency is influenced. The implementation scheme is as follows: the trusted third party verifies the identity of the voter through an NTRU-Prime algorithm, and distributes a fully homomorphic algorithm public key to the voter after verification; all voters construct a specific vote matrix, call the GSW-FHE algorithm to encrypt the vote matrix, and send the ciphertext matrix to a cloud service provider for homomorphic addition calculation to obtain a voting result ciphertext matrix; and the credible third party calls a decryption algorithm to decrypt the voting result ciphertext matrix to obtain a voting result plaintext matrix, and the rationality of the plaintext matrix is verified to obtain voting results of all candidates. The invention ensures the cascade depth of homomorphic operation, improves the voting efficiency and can be used for a multi-candidate anonymous electronic voting system in a cloud environment.

Description

Multi-candidate anonymous electronic voting method based on homomorphic encryption
Technical Field
The invention belongs to the field of cloud computing and information security, and particularly relates to a multi-candidate anonymous electronic voting method which can be used for an anonymous electronic voting system based on a trusted third party in a cloud environment.
Background
With the rapid development of computers and cloud computing technologies, electronic voting is gradually becoming a hot spot for research in the cryptology field as a large application of cloud computing. Compared with the traditional voting mode, the electronic voting method has the advantages that the voting is fast and accurate, the expenditure such as manpower is saved, the good anonymity is realized, and the justness, the safety and the high efficiency of the election are ensured to the greatest extent.
The fully homomorphic encryption FHE is a brand new encryption form, allows the calculation of messages under the condition of ciphertext, has high theoretical and research values under the current cloud environment, and can be widely applied to electronic voting, ciphertext retrieval, safe multi-party calculation and cloud calculation analysis.
Cloud computing is an inexpensive distributed computing power that provides scalability over a network. Currently, cloud computing services have been advanced into various industries, such as cloud storage services, data mining services, and the like, and the outsourced computing services are the most widely used. The outsourcing computation refers to a strategy for serving the interior or the individual of the enterprise by utilizing cloud computing resources. For resource-constrained users, it is difficult for them to complete the computational tasks for extremely demanding resource configurations in an efficient time. In this case, cloud computing provides another option for these users.
There are many types of outsourcing services, and providing users with large mathematical computation services is also one of the key points in research. Since the introduction of matrix theory, matrix operation has become an important branch in mathematical development, and matrix operation plays an important role in processing the relationship between large finite dimensional space form and quantity in the fields of scientific engineering, genetic engineering and the like, and large matrix operation often needs huge computing power. In order to solve this problem, researchers have conducted many studies on the problem.
In 2009, Gentry proposed an ideal lattice-based homomorphic scheme, and the architecture of the scheme becomes the basis of more and more homomorphic encryption schemes, such as DGHV10, BV11a, BV11b, BGV12 and GSW13, which all use the architecture proposed by Gentry. In 2015, Yasuda et al proposed a new encryption scheme for secure solution of multiple inner products that could be applied to secure hamming distance and pattern matching computational studies, but this method was only effective for very small matrices.
In 2016, Hiromasa proposes a first Matrix-based fully homomorphic encryption technology Matrix GSW-FHE based on a GSW-FHE scheme, the scheme supports homomorphic operation of Matrix addition and multiplication, expands the original GSW-FHE scheme and optimizes the bootstrap process of the FHE scheme. However, in the scheme, the space dimension of the ciphertext is too large, and the communication complexity and the calculation cost are increased.
In 2018, Wang et al optimized the scheme for Hiromasa, and although the dimension of the ciphertext matrix was reduced from (n + r) × logq to (n + r) × r, under the same parameters, the cascade depth of homomorphic operation was reduced, which resulted in errors in timing votes when the number of voters was fixed, and the votes were discarded, which affected the shortage of votes.
Disclosure of Invention
The invention aims to provide a multi-candidate anonymous electronic voting method based on fully homomorphic encryption aiming at the defects of the prior art, so that homomorphic addition of a GSW-FHE fully homomorphic encryption scheme is expanded to a matrix range, and the voting efficiency is improved.
In order to achieve the purpose, the technical scheme of the invention comprises the following steps:
(1) voter (P) participating in a vote1,P2,…,Pn) An identity verification request is sent to a trusted third party PKI, and the trusted third party verifies the identity of the voter: if the identity is valid, distributing a public key A of the homomorphic encryption algorithm to the voter, otherwise, not distributing;
(2) optional voter P participating in voting and obtaining public key AiVote matrix M for public key AiEncrypting to obtain a ciphertext matrix Ci：
Ci＝BitDecomp(BitDecomp-1(Mi+BitDecomp(R·A)))，
Wherein, R is a matrix generated by uniform sampling, elements of the matrix are integers and belong to [0,1 ];
(3) cipher text matrix CiTransmitting to an untrusted cloud service provider, and calling a server end of a GSW-FHE algorithm by the cloud service provider to all the ciphertext matrixes CiPerforming homomorphic addition calculation to obtain sum matrix C of all cipher text matrixes, and transmitting the sum matrix C back to the trusted second partyA three-way PKI;
(4) a trusted third party calls a decryption algorithm of the GSW-FHE algorithm, the sum matrix C is decrypted through a private key V of the algorithm, namely a plaintext matrix M is obtained through a decryption formula CV (MV + small), wherein small is an error vector small enough;
(5) the trusted third party PKI verifies whether the plaintext matrix M is correct and reasonable:
if the voting is reasonable, the voting is finished,
if not, the voting is cancelled and the voting is carried out again.
Compared with the prior art, the invention has the following advantages:
1. the GSW-FHE is naturally expanded into the matrix operation, excessive conversion is not needed to be carried out on the plaintext, the cascade depth of the original algorithm is not reduced, and the voting efficiency is improved.
2. The invention uses GSW-FHE full homomorphic encryption algorithm, so that a trusted third party can only obtain the final voting result and can not obtain the voting of a voter, thereby realizing anonymity.
3. The identity authentication algorithm NTRU-Prime algorithm and the homomorphic encryption algorithm GSW-FHE algorithm adopted by the invention both adopt the structure of the homomorphic scheme based on ideal lattices provided by Gentry, so that quantum password attack can be effectively resisted, and the safety of the voting process is ensured.
Drawings
FIG. 1 is a schematic flow diagram of the present invention.
Detailed Description
Referring to fig. 1, the multiple candidate anonymous electronic voting method based on the homomorphic encryption algorithm in this example is performed in a cloud environment, and the implementation steps are as follows:
the method comprises the following steps: and the trusted third party PKI verifies the identity of the voter.
Existing digital signature algorithms for verifying identity include RSA-based digital signature algorithms, NTRU-based digital signature algorithms, and this example employs, but is not limited to, NTRU-Prime-based digital signature algorithms. The concrete implementation is as follows:
1.1) trusted third party PKI invokes key generation in the NTRU-Prime algorithmAn algorithm randomly generates two reversible polynomials g and f, calculates the inverse g of the polynomial g-1Inverse f of sum polynomial f-1Let the public key pkiG/(3. f), private key ski＝(f,g-1) Form a public and private key pair (sk)i,pki) The public and private key pair is the voter PiRegistered identity information;
1.2) voter PiGenerating a random number x, generating a hash value mess:
existing algorithms for generating the hash value include SM3 algorithm, MD5 algorithm, SHA-256 algorithm, but the present embodiment adopts but is not limited to SHA-256 hash algorithm, and takes random number x as input of SHA-256 hash algorithm, and outputs hash value mess;
1.3) voter PiCalling encryption algorithm in NTRU-Prime algorithm, using public key pkiEncrypting the hash value mess and the random number x to obtain a ciphertext c ═ pki(mess x), which is a concatenation of a hash value mess and a random number x, and transmits the ciphertext c to a trusted third party PKI;
1.4) the trusted third party PKI calls a decryption algorithm in the NTRU-Prime algorithm and uses the private key skiThe ciphertext c is decrypted by first calculating the polynomial e ═ 3 · f · c) mod3, and then calculating (mess | | | x) ═ e · g-1Obtaining a hash value mess and a random number x, where mod3 is modulo 3 for each coefficient of the polynomial;
1.5) trusted third party PKI verification voter PiThe identity of (c):
the trusted third party PKI calls an SHA-256 algorithm, calculates the hash value mess 'of the random number x, and compares the calculation result mess' with the hash value mess:
if mess' is mess, the voter PiThe identity is valid, otherwise, the identity is invalid.
Step two: the trusted third party PKI distributes the public key to the voter.
2.1) initialization parameters: selecting a modulus q, a number of candidates n, an error distribution delta, and a
Figure BDA0002962044750000046
2.2) generating a private key:
uniformly sampling to generate an n.n-dimensional private key blinding matrix T, and making a cascade matrix
Figure BDA0002962044750000041
Wherein, InIs an n-n dimensional identity matrix,
Figure BDA0002962044750000042
is an identity matrix InAnd vertical concatenation of matrix-T;
each column vector of the matrix S is to be concatenated
Figure BDA0002962044750000043
The transformation is:
Figure BDA0002962044750000044
obtaining a private key matrix V ← Powersof2(S), wherein biIs a column vector
Figure BDA0002962044750000045
I ═ 1,2, …,2 n;
and 2.3) generating a public key, namely uniformly sampling to generate an n & n-dimensional public key blinding matrix B, generating an n & n-dimensional error matrix E according to the error distribution delta to obtain a public key matrix A [ B & T + E | | | B ], and distributing the public key matrix A to the voter by a trusted third party PKI, wherein [ B & T + E | | | | B ] is the cascade of the matrix B & T + E and the public key blinding matrix B.
Step three: and the voter encrypts the vote matrix.
3.1) voter PiConstructing ballot matrix MiExpressed as follows:
Figure BDA0002962044750000051
wherein, 0(n·l)(n·l)Is a 0 matrix of (n · l) (n · l) dimensions,
g is a l.l dimensional matrix representing the voter PiFor the voting result of a candidate, if the voter PiVoting to the candidate, then
Figure BDA0002962044750000052
If not, then,
Figure BDA0002962044750000053
Figure BDA0002962044750000054
gnis a voter PiVoting results for the nth candidate;
3.2) voters P participating in the voteiCalculating vote matrix M by using public key matrix AiCiphertext matrix C ofi：
3.2.1) voter PiUniformly sampling to generate an encrypted blinding matrix R, and dividing each row vector of the matrix R.A
Figure BDA0002962044750000055
The transformation is:
Figure BDA0002962044750000056
obtaining a plaintext blinding matrix BitDecomp (R.A), wherein (a)i,0,…,ai,l-1) Representing row vectors
Figure BDA0002962044750000057
The ith element of (a)iThe bit order is from least significant bit to most significant bit;
3.2.2) vote matrix MiAdding the plaintext blinded matrix BitDecomp (R.A) to obtain a transformation basis matrix Mi+BitDecomp(R·A)；
3.2.3) transforming the basis matrix MiEach row vector of + BitDecomp (R.A)
Figure BDA0002962044750000058
The following transformations are made:
Figure BDA0002962044750000059
wherein c ist·l+jIs a line vector
Figure BDA00029620447500000510
Is the t.l + j element of (2 nl) is a row vector
Figure BDA00029620447500000511
To obtain a plaintext transformation matrix BitDecomp-1(Mi+BitDecomp(R·A))；
3.2.4) matrix BitDecomp-1(MiThe BitDecomp (R.A)) is transformed into a ciphertext matrix
Ci＝BitDecomp(BitDecomp-1(Mi+ BitDecomp (R · a))) and the ciphertext matrix CiAnd transmitting to the cloud service provider.
Step four: the cloud service provider computes the sum matrix of all the ciphertext matrices.
The cloud service provider obtains the ciphertext matrix (C) of all voters1,…,Ci,…,Ch) Calculating the sum matrix C ═ C of all the ciphertext matrices1+…+Ci+…+ChAnd h is the total number of voters, and the sum matrix C is finally transmitted back to the trusted third party.
Step five: and the trusted third party decrypts the sum matrix C and verifies the rationality of the voting result.
5.1) the trusted third party calls the decryption algorithm of the GSW-FHE algorithm and utilizes the private key matrix V to sum the matrix C
And (3) decryption:
5.1.1) multiplying the ciphertext matrix C with the r-th column of the private key matrix V to obtain a ciphertext vector
Figure BDA0002962044750000061
Wherein r is 1,2, … h,
Figure BDA0002962044750000062
is a vector of coefficients that is a function of,
Figure BDA0002962044750000063
mris the total votes for the r-th candidate,
Figure BDA0002962044750000064
is an error vector;
5.1.2) mixing of mrExpressed in its binary form (m)r,0,mr,1,…,mr,l-2) The bit order is from least significant bit to most significant bit, i.e. mr＝mr,0+21mr,1+…+2l-2mr,l-2；
5.1.3) vector based on ciphertext
Figure BDA0002962044750000065
L-2 th element x ofr,l-2The following formula is satisfied:
xr,l-2＝(mr,0+21mr,1+…+2l-2mr,l-2)·2l-2+el-2and e is al-2Under the condition of < q/8, m is obtained by solutionrThe least significant bit of the binary form of (a):
Figure BDA0002962044750000066
5.1.4) by mr,0Iteratively calculating mrOther binary digit of
Figure BDA0002962044750000067
Wherein p is 1,2, …, l-2;
5.1.5) mixing of mrBinary (m) ofr,0,mr,1,…,mr,l-2) Reducing to decimal form to obtain the total ticket number m of the candidater＝mr,0+21mr,1+…+2l-2mr,l-2；
5.2) the trusted third party PKI verifies whether the elements of the plaintext matrix M are reasonable:
if the elements of the plaintext matrix M simultaneously satisfy the following three conditions, the plaintext matrix M is reasonable, and the voting is finished:
when w is u, M(w,u)＜＝n w,u＝0,1,…,nl，
When w ≠ u, M(w,u)＝0 w,u＝0,1,…,nl，
M(k·l,k·l)＝M(k·l+1,k·l+1)＝…＝M(k·l+l-1,k·l+l-1) k＝0,1,…,n-1，
Wherein M is(w,u)Is the element of the w-th row and u-th column of the plaintext matrix M, nl is the number of rows of the plaintext matrix M, M(k·l,k·l)Is an element of the k · l th row and the k · l th column of the plaintext matrix M;
if the elements of the plaintext matrix M do not meet the three conditions simultaneously, the plaintext matrix M is unreasonable, the voting is cancelled, and the voting is carried out again.
While the invention has been particularly shown and described with reference to exemplary embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention.

Claims (6)

1. A multiple candidate anonymous electronic voting method based on homomorphic encryption is characterized by comprising the following steps:
(1) voter (P) participating in voting1,P2,…,Ph) An identity verification request is sent to a trusted third party PKI, and the trusted third party verifies the identity of the voter: if the identity is valid, distributing a public key A of the homomorphic encryption algorithm to the voter, otherwise, not distributing;
(2) optional voter P participating in voting and obtaining public key AiVote matrix M for public key AiEncrypting to obtain a ciphertext matrix Ci：
Ci＝BitDecomp(BitDecomp-1(Mi+BitDecomp(R·A)))，
Wherein, R is a matrix generated by uniform sampling, elements of the matrix are integers and belong to [0,1 ];
(3) voter PiCipher text matrix CiTransmitting to an untrusted cloud service provider, and calling a server end of a GSW-FHE algorithm by the cloud service provider to all the ciphertext matrixes CiPerforming homomorphic addition calculation to obtain a sum matrix C of all the ciphertext matrixes, and transmitting the sum matrix C back to the trusted third party PKI;
(4) a trusted third party calls a decryption algorithm of the GSW-FHE algorithm, the sum matrix C is decrypted through a private key V of the algorithm, namely a plaintext matrix M is obtained through a decryption formula CV (MV + small), wherein small is an error vector small enough;
(5) the trusted third party PKI verifies whether the plaintext matrix M is correct and reasonable:
if the voting is reasonable, the voting is finished,
if not, the voting is cancelled and the voting is carried out again.
2. The method of claim 1, wherein the trusted third party in (1) verifies the identity of the voter by:
(1a) calling NTRU-Prime algorithm to each registered voter P by trusted third party PKIiGenerating identity information, i.e. public and private key pair (sk), in advancei,pki) Voter PiA random number x is randomly generated and passes through the private key sk in the identity informationiGenerating a hash value mess, and transmitting the random number x and the hash value mess to a trusted third party PKI;
(1b) trusted third party PKI through voter PiThe public key pk in the identity information ofiCalling an NTRU-Prime algorithm, calculating a hash value mess 'of the random number x, and comparing the calculated result mess' with the hash value mess: if the mess' is the mess, the identity is valid, otherwise, the identity is invalid.
3. The method of claim 1, wherein the public key A of the fully homomorphic encryption algorithm in (1) is expressed as follows:
A＝[B·T+E||B]，
b and T are two different n.n dimensional matrixes generated by uniform sampling, matrix elements of the matrixes belong to [0, q), and q represents a modulus; e is an n-n dimensional error matrix generated by Gaussian sampling, wherein n is the number of candidates.
4. The method of claim 1, wherein the vote matrix M selected in (2) isiExpressed as follows:
Figure FDA0003565262590000021
wherein, InIs an identity matrix of n.n,
Figure FDA0003565262590000022
q represents a modulus; 0(n·l)(n·l)Is a 0 matrix of (n · l) (n · l) dimensions,
g is a l.l dimensional matrix representing the voter PiFor the voting result of a candidate, if the voter PiVoting to the candidate, then
Figure FDA0003565262590000023
If not, then,
Figure FDA0003565262590000024
Figure FDA0003565262590000025
gnis a voter PiVoting results for the nth candidate.
5. The method of claim 1, wherein the sum matrix C of all the ciphertext matrices in (3) is expressed as follows:
Figure FDA0003565262590000026
wherein h is the number of people selected.
6. The method according to claim 1, wherein the trusted third party PKI in (6) verifies the plaintext matrix M that whether the elements of the plaintext matrix M are reasonable is verified by the trusted third party PKI:
if the elements of the plaintext matrix M satisfy the following three conditions at the same time, the plaintext matrix M is reasonable:
when w is u, M(w,u)＜＝n w,u＝0,1,…,nl，
When w ≠ u, M(w,u)＝0 w,u＝0,1,…,nl，
M(k·l,k·l)＝M(k·l+1,k·l+1)＝…＝M(k·l+l-1,k·l+l-1) k＝0,1,…,n-1，
Wherein M is(w,u)Is the element of the w-th row and u-th column of the plaintext matrix M, nl is the number of rows of the plaintext matrix M, M(k·l,k·l)Is the element of the k · l th row and column of the plaintext matrix M, n is the number of candidates, l is the voter PiDimension of voting result matrix for some candidate;
if the elements of the plaintext matrix M do not satisfy the above three conditions at the same time, the plaintext matrix M is not reasonable.
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