CN106453285B - A kind of verification method and device that secret data is shared - Google Patents

A kind of verification method and device that secret data is shared Download PDF

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Publication number
CN106453285B
CN106453285B CN201610856701.4A CN201610856701A CN106453285B CN 106453285 B CN106453285 B CN 106453285B CN 201610856701 A CN201610856701 A CN 201610856701A CN 106453285 B CN106453285 B CN 106453285B
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vector
sub
preset quantity
secret
share
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CN106453285A (en
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吕春利
张博
曾子川
申长虹
刘瑜
马钦
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China Agricultural University
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China Agricultural University
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    • 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
    • H04L63/06—Network architectures or network communication protocols for network security for supporting key management in a packet data network
    • H04L63/062—Network architectures or network communication protocols for network security for supporting key management in a packet data network for key distribution, e.g. centrally by trusted party
    • 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
    • H04L63/06—Network architectures or network communication protocols for network security for supporting key management in a packet data network
    • H04L63/067—Network architectures or network communication protocols for network security for supporting key management in a packet data network using one-time keys
    • 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
    • H04L9/08—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

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  • Engineering & Computer Science (AREA)
  • Computer Security & Cryptography (AREA)
  • Computer Networks & Wireless Communication (AREA)
  • Signal Processing (AREA)
  • Computer Hardware Design (AREA)
  • Computing Systems (AREA)
  • General Engineering & Computer Science (AREA)
  • Management, Administration, Business Operations System, And Electronic Commerce (AREA)
  • Storage Device Security (AREA)

Abstract

The invention discloses verification methods and device that a kind of secret data is shared, which comprises secret data is expressed as to the secret vector of the first preset quantity block number；The random string vector of the second preset quantity is selected, and determines the shared multinomial of secret vector according to the random string vector；Identity is calculated, the sub- share vector of the third preset quantity of secret vector is calculated according to the shared multinomial and identity；Calculate shared promise to undertakeThe sub- share vector is verified according to the shared promise, determines the correctness of the sub- share vector.The present invention determines the shared multinomial of secret data by random string vector, and the sub- share vector of secret vector is obtained according to shared polynomial computation, and sub- share vector is verified by verifying formula, calculation amount and the traffic are all smaller, and computational efficiency is higher.

Description

A kind of verification method and device that secret data is shared
Technical field
The present invention relates to field of information security technology, and in particular to a kind of verification method and device that secret data is shared.
Background technique
Privacy sharing is the important means of information security and data confidentiality, it is protected in the safety of important information and secret data Deposit, transmit and legal utilization in play key effect.Wherein the scheme of Shamir is the classics in secret sharing scheme, base This thought is that shared secret S is divided into n share secret to be distributed to different participants, meets and is only more than or equal to t service Device joint could reconstruct shared secret, and any information of the secret cannot be arbitrarily obtained less than t server.But Shamir scheme needs to assume two premises: first is that secret distributor is always honest；Second is that the preserver of n share has The status of complete equality and right, and there is identical safety and reliability.The two are assumed in reality often It is difficult to be met, to will lead to privacy sharing, there are safety problems in practical application scene, it may be verified that privacy sharing What concept exactly generated in this context.Since the privacy sharing that can verify that is the base of the secret sharing scheme of active safety Fault-tolerant important tool in plinth and practical application, so the secret sharing scheme that many work can verify that in research again.
One of classical Verified secret sharing scheme is the scheme of Feldman.Feldman scheme is based on Shamir Scheme and dispersed accumulation, to being all condition safety in secret confidentiality and verifiability, and it is simpler and clear. But the secret sharing scheme based on Shamir needs to be calculated in the finite field that a rank is prime number p, and in the side VSS In case, need to find the cyclic group that a rank is prime number q, and q is the prime factor of p-1, will increase promise size, scheme in this way It can be not good enough.The problem is studied in the prior art, but all there are shortcomings: each shared secret need to be expected It calculates, and the certification of sub-secret needs each side to cooperate online, so that calculation amount and the traffic are all very big；Or safety is to be based on The intractability of discrete logarithm, the in order to prevent fraud between participant need to be implemented an interactive indentification protocol, and calculation amount is non- Chang great；Or use multi-secret sharing scheme scheme, but that there are distributors is computationally intensive for the program, it is inefficient the disadvantages of.
Summary of the invention
It is inefficient since that there are calculation amount and the traffics is all very big for the indentification protocol of existing secret data sharing method The problems such as, the present invention proposes a kind of verification method and device that secret data is shared.
In a first aspect, the present invention proposes a kind of verification method that secret data is shared, comprising:
Secret data is divided into the isometric secret vector for being expressed as the first preset quantity block number；
The random string vector of the second preset quantity is selected, and secret data is determined according to the random string vector Shared multinomial；
Identity is calculated, and the third of secret vector is obtained according to the identity and the shared polynomial computation The sub- share vector of preset quantity；
Calculate shared promise to undertake
The sub- share vector is verified according to the shared promise, determines the correctness of the sub- share vector； Wherein, i, j are nonnegative integer, (0≤i≤p-1,0≤j≤t-1)；P is positive integer, and p-1 is first preset quantity；T is Positive integer, t-1 are second preset quantity；ri,jFor the component of the random string vector,It is one of cyclic group Generate member；Q be the circulation order of a group, and for can disclose, sufficiently large prime number.
Preferably, further includes:
According to reconstruct expression formula, secret reconstruct is carried out to the sub- share vector of at least the 4th preset quantity, is obtained described secret Ciphertext data.
Preferably, the shared polynomial f (x) are as follows:
Wherein, x expression is described shares polynomial independent variable,Respectively t-1 random character String vector, t are positive integer, and t-1 indicates second preset quantity；Indicate the secret data.
Preferably, the sub- share vector of the third preset quantityAre as follows:
Wherein, m is positive integer；P is positive integer, and p-1 indicates first preset quantity；y0, m,…,yP-2, mRespectively institute The daughter element of sub- share vector is stated, n indicates third preset quantity.
Preferably, the reconstruct expression formulaAre as follows:
Wherein,BottomBlockRowof () expression takes matrix in block formMost A line piecemeal row vector afterwards；Indicate the sub- share vector of at least the 4th preset quantity；λ0, λ1... ... λt-1For positive integer, the subscript of the optional t sub- shares from sub- share is indicated；T is positive integer, and t indicates the 4th present count Amount.
Second aspect, the present invention also propose a kind of verifying device that secret data is shared, comprising:
Data division module, for by secret data be divided into it is isometric be expressed as the first preset quantity block number it is secret to Amount；
Multinomial determining module, for selecting the random string vector of the second preset quantity, and according to the random words Symbol string vector determines the shared multinomial of secret data；
Vector calculation module, for calculating identity, and according to the identity and according to the shared multinomial The sub- share vector of the third preset quantity of secret vector is calculated；
Computing module is promised to undertake, for calculating shared promise
Vector authentication module determines the son for verifying according to the shared promise to the sub- share vector The correctness of share vector；
Wherein, i, j are nonnegative integer, (0≤i≤p-1,0≤j≤t-1)；P is positive integer, and p-1 is described first default Quantity；T is positive integer, and t-1 is second preset quantity；ri,jFor the component of the random string vector,For circulation The generation member of group；Q be the circulation order of a group, and for can disclose, sufficiently large prime number.
Preferably, further includes:
Secret reconstructed module, for being carried out to the sub- share vector of at least the 4th preset quantity secret according to reconstruct expression formula Close reconstruct obtains the secret data.
Preferably, the shared polynomial f (x) in the multinomial determining module are as follows:
Wherein, x expression is described shares polynomial independent variable,Respectively t-1 random character String vector, t are positive integer, and t-1 indicates second preset quantity；Indicate the secret data.
Preferably, the sub- share vector of third preset quantity described in the vector calculation moduleAre as follows:
Wherein, m is positive integer；P is positive integer, and p-1 indicates first preset quantity；y0,m,…,yp-2,mRespectively institute The daughter element of sub- share vector is stated, n indicates third preset quantity.
Preferably, expression formula is reconstructed described in the secret reconstructed moduleAre as follows:
Wherein,BottomBlockRowof () expression takes matrix in block formMost A line piecemeal row vector afterwards；Indicate the sub- share of at least third preset quantity of the 4th preset quantity to Amount；λ0, λ1... ... λt-1For positive integer, the subscript of the optional t sub- shares from sub- share is indicated；T is positive integer, described in t expression 4th preset quantity.
As shown from the above technical solution, the present invention determines the shared multinomial of secret data by random string vector, And the sub- share vector of secret vector is obtained according to shared polynomial computation, and verify to sub- share vector by verifying formula, Calculation amount and the traffic are all smaller, and computational efficiency is higher.
Detailed description of the invention
In order to more clearly explain the embodiment of the invention or the technical proposal in the existing technology, to embodiment or will show below There is attached drawing needed in technical description to be briefly described, it should be apparent that, the accompanying drawings in the following description is only this Some embodiments of invention for those of ordinary skill in the art without creative efforts, can be with Other attached drawings are obtained according to these figures.
Fig. 1 is the flow diagram for the verification method that a kind of secret data that one embodiment of the invention provides is shared；
Fig. 2 is renewable (t, n) threshold secret sharing scheme flow chart for the calculating safety that one embodiment of the invention provides；
Fig. 3 is (t, n) threshold secret sharing scheme illustraton of model that one embodiment of the invention provides；
Fig. 4 is the geometric description figure for upper (t, n) threshold secret sharing process of group that one embodiment of the invention provides；
Fig. 5 is the structural schematic diagram for the verifying device that a kind of secret data that one embodiment of the invention provides is shared.
Specific embodiment
With reference to the accompanying drawing, the specific embodiment of invention is further described.Following embodiment is only used for more clear Illustrate to Chu technical solution of the present invention, and not intended to limit the protection scope of the present invention.
The flow diagram for the verification method that a kind of secret data that Fig. 1 shows one embodiment of the invention offer is shared, Include:
S1, secret data is divided into the isometric secret vector for being expressed as the first preset quantity block number；
S2, the random string vector for selecting the second preset quantity, and secret is determined according to the random string vector The shared multinomial of data；
S3, identity is calculated, and secret vector is obtained according to the identity and the shared polynomial computation The sub- share vector of third preset quantity；
S4, shared promise to undertake is calculated
S5, the sub- share vector is verified according to the shared promise, determines the correct of the sub- share vector Property；
Wherein, i, j are nonnegative integer, (0≤i≤p-1,0≤j≤t-1)；P is positive integer, and p-1 is described first default Quantity；T is positive integer, and t-1 is second preset quantity；ri,jFor the component of the random string vector,For circulation The generation member of group；Q be the circulation order of a group, and for can disclose, sufficiently large prime number.
The present embodiment determines the shared multinomial of secret data by random string vector, and according to shared multinomial meter Calculation obtains the sub- share vector of secret vector, and is verified by verifying formula to sub- share vector, calculation amount and the traffic all compared with It is small, and computational efficiency is higher.
Further, the method also includes:
S6, according to reconstruct expression formula, secret reconstruct is carried out to the sub- share vector of at least the 4th preset quantity, is obtained described Secret data.
Secret reconstruct is carried out by reconstruct expression formula, it being capable of quick correctly Restore Secret data.
Specifically, the shared polynomial f (x) are as follows:
Wherein, x expression is described shares polynomial independent variable,Respectively t-1 random character String vector, t are positive integer, and t-1 indicates second preset quantity；Indicate the secret data.
By using multinomial is shared shown in formula one, calculation amount can be reduced, improves computational efficiency.
Further, the sub- share vector of the third preset quantityAre as follows:
Wherein, m is positive integer；P is positive integer, and p-1 indicates first preset quantity；y0,m,…,yp-2,mRespectively institute The daughter element of sub- share vector is stated, n indicates third preset quantity.
It, being capable of the traffic between less banker and member by using sub- share vector shown in formula two.
Further, the reconstruct expression formulaAre as follows:
Wherein,BottomBlockRowof () expression takes matrix in block formMost A line piecemeal row vector afterwards；Indicate the sub- share vector of at least the 4th preset quantity；λ0, λ1... ... λt-1For positive integer, the subscript of the optional t sub- shares from sub- share is indicated；T is positive integer, and t indicates the 4th present count Amount.
Expression formula is reconstructed by using shown in formula three, calculation amount can be reduced.
The shared verification method of secret data provided in this embodiment similar to Feldman scheme, group's Upper threshold is secret Shared to combine with dispersed accumulation, the confidentiality and verifiability for secret information can achieve the safety on calculating Property, and improve computational efficiency, reduce promise size, so that privacy sharing method is more suitable for the data storage of big data quantity On.
For example, it is secret to respectively illustrate renewable (t, n) thresholding provided in this embodiment for calculating safety by Fig. 2 and Fig. 3 Close secret sharing flow chart and (t, n) threshold secret sharing scheme illustraton of model；Assuming that there is banker D need to be in n participant U= {U1,…UnBetween shared secret S only when t or the t above participant join together that shared secret could be restored it is a to be less than t Any combination of participant is all unable to get about secret any information.Concrete scheme is made of 4 sub- agreements: system is initial Change, secret distribution protocol, the indentification protocol of sub- share vector and secret reconstruct agreement.
A1, system initialization
A11, banker define and announce a sufficiently large prime number q, define one that the cyclic group G, g that a rank is q are G Generate member；
A12, a finitely Abelian group is defined Rank is integer0 generates member for it；
A13, it looks for a prime number p (>=n), andRe-define a vector space [0,…0]TTo generate member, scalar operation result thereon is all fallen withinOn.
A2, secret distribution protocol
A21, it is divided into p-1 block for secret S is isometric, is denoted asChoose t-2 random string vectorWherein
A22, banker choose privacy sharing multinomial:
It enablesSub- share vector can be calculated by formula two
Wherein,For participant PmIdentity, and is defined as:0≤m≤p-1
For example, as p=5,
Indicate that above-mentioned privacy sharing process is as follows with matrix form:
Wherein,
For example, with k=3, for n=4, p=5, by above-mentionedAnd Hn×tDefinition H can be obtained4×3It is as follows:
According to the expression matrix of privacy sharing, i.e. formula four, to calculate sub- share vectorFor Example, has:
Therefore, according to the expression matrix of privacy sharing, the geometric description of available scheme, as shown in Figure 4.It is retouched from geometry It states as can be seen that calculating sub- share vectorMathematic(al) representation are as follows:
Wherein 0≤i≤p-2,0≤j≤t-1, ri,t-1=si, rp-1,j=0.
A23, banker secret distribute sub- share vectorTo Um(0≤m≤n-1)；
A24, banker make following promise using discrete logarithm, that is, calculate(0≤i≤p-1,0≤j≤ T-1), wherein ri,t-1=si, rp-1,j=0, and it is broadcast to Um(0≤m≤n-1)。
The indentification protocol of A3, sub- share vector
UmReceive sub- share vectorAnd Ai,jAfterwards, sub- share vector can be verified by formula six Correctness:
Wherein, 0≤i≤p-2.
Illustrate the correctness of formula six below:
According to A in the formula five and step A24 of the geometric description of privacy sharingi,jCalculating formula can obtain:
A4, secret reconstruct agreement
As at least t member Uλ(λ ∈ B, and | B | >=t) their sub- share vector is providedAfterwards, using formula (4) into The secret reconstruct of row (here by taking t sub- share vectors as an example):
Wherein,
Vk×kIt is according to t shadow secret vectorFrom Hn×tIt is middle to take out what corresponding t row vector formed Square matrix, wherein
Therefore secret can be reconstructed
Integer mould is added group by the present embodimentOn efficient secret sharing scheme combined with dispersed accumulation, With below the utility model has the advantages that
Based on efficient threshold secret sharing scheme on a population, shared and reduction efficiency can achieve theoretically highest；
Secret vector generating process can be obtained by visual in image geometric expression, thus to can verify that Threshold Secret is total It enjoys scheme and calculating basis is provided；
Confidentiality and verifiability for secret information may be implemented that safety (condition safety) can be calculated；
It promises to undertake having a size of everyOnly need | q | the promise of bitsCompared with the promise ruler of Feldman scheme It is very little small；
Compared to Feldman scheme each element need 1 exponent arithmetic andSecondary multiplying, we The computational efficiency of case is 1 exponent arithmetic and t-1 multiplying, so this scheme is applied to the privacy sharing of big data quantity When, computational efficiency is higher.
Fig. 5 shows a kind of structural schematic diagram described device of the shared verifying device of secret data provided in this embodiment Including data division module 11, multinomial determining module 12, vector calculation module 13 and vector authentication module 14；Wherein,
Data division module 11 be used to for secret data being divided into it is isometric be expressed as the first preset quantity block number it is secret to Amount；
Multinomial determining module 12 is used to select the random string vector of the second preset quantity, and according to the random words Symbol string vector determines the shared multinomial of secret data；
Vector calculation module 13 is used to calculate identity, and according to the identity and the shared polynomial computation Obtain the sub- share vector of the third preset quantity of secret vector；
Computing module 14 is promised to undertake, for calculating shared promise
Vector authentication module 15 is used to verify the sub- share vector according to the shared promise, determines the son The correctness of share vector；
Wherein, i, j are nonnegative integer, (0≤i≤p-1,0≤j≤t-1)；P is positive integer, and p-1 is described first default Quantity；T is positive integer, and t-1 is second preset quantity；ri,jFor the component of the random string vector,For circulation The generation member of group；Q be the circulation order of a group, and for can disclose, sufficiently large prime number.
Specifically, secret data is divided into the isometric secret for being expressed as the first preset quantity block number by data division module 11 Vector；Multinomial determining module 12 select the second preset quantity random string vector, and according to the random string to Measure the shared multinomial for determining secret data；Vector calculation module 13 obtains secret vector according to the shared polynomial computation Sub- share vector；Promise to undertake that computing module 14 calculates shared promise to undertakeVector authentication module 15 is according to described shared Promise verifies the sub- share vector, determines the correctness of the sub- share vector.
The present embodiment determines the shared multinomial of secret data by random string vector, and according to shared multinomial meter Calculation obtains the sub- share vector of secret vector, and is verified by verifying formula to sub- share vector, calculation amount and the traffic all compared with It is small, and computational efficiency is higher.
Further, described device further include:
Secret reconstructed module 16, for being carried out to the sub- share vector of at least the 4th preset quantity according to reconstruct expression formula Secret reconstruct, obtains the secret data.
Specifically, the shared polynomial f (x) in the multinomial determining module are as follows:
Wherein, x expression is described shares polynomial independent variable,Respectively t-1 random character String vector, t are positive integer, and t-1 indicates second preset quantity；Indicate the secret data.
Further, the sub- share vector of third preset quantity described in the vector calculation moduleAre as follows:
Wherein, m is positive integer；P is positive integer, and p-1 indicates first preset quantity；y0,m,…,yp-2,mRespectively institute That states sub- share vector divides vector.
Further, expression formula is reconstructed described in the secret reconstructed moduleAre as follows:
Wherein,BottomBlockRowof () expression takes matrix in block formMost A line piecemeal row vector afterwards；Indicate the sub- share vector of at least the 4th preset quantity；λ0, λ1... ... λt-1For positive integer, the subscript of the optional t sub- shares from sub- share is indicated；T is positive integer, and t indicates the 4th present count Amount.
The shared verifying device of secret data described in the present embodiment can be used for executing above method embodiment, principle Similar with technical effect, details are not described herein again.
In specification of the invention, numerous specific details are set forth.It is to be appreciated, however, that the embodiment of the present invention can be with It practices without these specific details.In some instances, well known method, structure and skill is not been shown in detail Art, so as not to obscure the understanding of this specification.

Claims (6)

1. a kind of shared verification method of secret data characterized by comprising
Secret data is divided into the isometric secret vector for being expressed as the first preset quantity block number；
The random string vector of the second preset quantity is selected, and being total to for secret data is determined according to the random string vector Enjoy multinomial；
Identity is calculated, and default according to the third that the identity and the shared polynomial computation obtain secret vector The sub- share vector of quantity；
Calculate shared promise to undertake
The sub- share vector is verified according to the shared promise, determines the correctness of the sub- share vector；
Wherein, i, j are nonnegative integer, (0≤i≤p-1,0≤j≤t-1)；P is positive integer, and p-1 is first preset quantity； T is positive integer, and t-1 is second preset quantity；ri,jFor the component of the random string vector,It is the one of cyclic group A generation member；Q is the circulation order of a group, and is ostensible prime number；
The shared polynomial f (x) are as follows:
Wherein, x expression is described shares polynomial independent variable,Respectively t-1 random string to Amount, t are positive integer, and t-1 indicates second preset quantity；Indicate the secret data；
The sub- share vector of the third preset quantityAre as follows:
Wherein, m is positive integer, and 0≤m≤n-1；y0,m,…,yp-2,mThe daughter element of the respectively described sub- share vector, n are indicated Third preset quantity.
2. the method according to claim 1, wherein further include:
According to reconstruct expression formula, secret reconstruct is carried out to the sub- share vector of at least the 4th preset quantity, obtains the secret number According to.
3. according to the method described in claim 2, it is characterized in that, the reconstruct expression formulaAre as follows:
Wherein,BottomBlockRowof () expression takes matrix in block formLast Row piecemeal row vector；Indicate the sub- share vector of at least the 4th preset quantity；λ0, λ1... ... λt-1For Positive integer indicates the subscript of the optional t sub- shares from sub- share；T indicates the 4th preset quantity；It indicatesp- 1 is first preset quantity,0≤m≤p-1,
4. a kind of shared verifying device of secret data characterized by comprising
Data division module, for secret data to be divided into the isometric secret vector for being expressed as the first preset quantity block number；
Multinomial determining module, for selecting the random string vector of the second preset quantity, and according to the random string Vector determines the shared multinomial of secret data；
Vector calculation module is obtained for calculating identity, and according to the identity and the shared polynomial computation The sub- share vector of the third preset quantity of secret vector；
Computing module is promised to undertake, for calculating shared promise
Vector authentication module determines the sub- share for verifying according to the shared promise to the sub- share vector The correctness of vector；
Wherein, i, j are nonnegative integer, (0≤i≤p-1,0≤j≤t-1)；P is positive integer, and p-1 is first preset quantity； T is positive integer, and t-1 is second preset quantity；ri,jFor the component of the random string vector,It is the one of cyclic group A generation member；Q is the circulation order of a group, and is ostensible prime number；
The shared polynomial f (x) are as follows:
Wherein, x expression is described shares polynomial independent variable,Respectively t-1 random string to Amount, t are positive integer, and t-1 indicates second preset quantity；Indicate the secret data；
The sub- share vector of the third preset quantityAre as follows:
Wherein, m is positive integer, and 0≤m≤n-1；y0,m,…,yp-2,mThe daughter element of the respectively described sub- share vector, n are indicated Third preset quantity.
5. device according to claim 4, which is characterized in that further include:
Secret reconstructed module, for carrying out secret weight to the sub- share vector of at least the 4th preset quantity according to reconstruct expression formula Structure obtains the secret data.
6. device according to claim 5, which is characterized in that reconstruct expression formula described in the secret reconstructed moduleAre as follows:
Wherein,BottomBlockRowof () expression takes matrix in block formLast Row piecemeal row vector；Indicate the sub- share vector of at least the 4th preset quantity；λ0, λ1... ... λt-1For Positive integer indicates the subscript of the optional t sub- shares from sub- share；T indicates the 4th preset quantity；It indicatesp- 1 is first preset quantity,0≤m≤p-1,
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