CN114124351B - Rapid calculation method of nonlinear invariant - Google Patents
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Abstract
The invention provides a quick calculation method of a nonlinear invariant, which is based on a computer program to execute a calculation process and comprises the following steps: step 1, setting algebraic times of an effective nonlinear invariant; step 2, calculating the Boolean expression of each component function according to the truth table of the Boolean function; step 3, calculating positive integers with the weight of all the hamming weights not exceeding algebraic times, and acquiring a coefficient matrix according to the positive integers; step 4, introducing an auxiliary matrix to expand the coefficient matrix to obtain an expansion matrix, and performing Gaussian elimination on the expansion matrix; step 5, judging row vectors of the matrix after the cells are eliminated, and converting corresponding partial row vectors into Boolean expressions according to judging results; and 6, all the non-zero linear combinations of the calculated Boolean expression are output as non-linear invariant sub-if the algebraic number of the non-zero linear combinations is the set algebraic number. The scheme provided by the invention avoids the waste of computing resources and storage resources, and simultaneously greatly improves the computing efficiency.
Description
Technical Field
The invention relates to the field of passwords, in particular to a quick calculation method of a nonlinear invariant.
Background
The design technology and the analysis technology of the symmetric cipher are two branches in the field of symmetric ciphers, which complement each other and are promoted together. A secure symmetric cryptographic algorithm often requires repeated iterations of algorithm design, analysis, improvement optimization, and ultimately is able to resist all known attacks. The nonlinear invariant analysis is a novel attack method proposed in 2016, and a discriminator about the plaintext is constructed by calculating nonlinear invariant with the algebraic number of S-box of 2, so that the discrimination attack with the success probability of 1 is realized. Since the existence of a weak key needs to be ensured in the nonlinear invariant analysis, which requires that the algorithm key arrangement algorithm is simple and the round constant is set sparse, the nonlinear invariant analysis is mainly used for analyzing the lightweight block cipher algorithm. Currently, nonlinear invariant analysis has successfully analyzed Midori algorithm, screen algorithm and iScreen algorithm, and the precondition for attack is that nonlinear invariant with the algebraic number of S-boxes of 2 is calculated first.
Considering S-boxes as boolean functions of n-bit input n-bit output, then for arbitrary input x= (X) n-1 ,x n-2 ,…,x 0 ) With output y= (Y) n-1 ,y n-2 ,…,y 0 ) Y=s (X) is satisfied. If a nonlinear boolean function g is present, so that for any input X, the formula g (S (X))=g (X) +c is constant, where c e F 2 ，F 2 Representing a binary field set consisting of 0 and 1, the boolean function g is called the nonlinear invariant of the S-box.
Essentially nonlinear invariants are an n-ary Boolean function, and any one of the n-ary Boolean functions f can be written in algebraic formal form, i.e.
Wherein a is u ∈F 2 Called single-item X u Coefficient of (a), single item X u The specific form of (2) is as follows: when u=0, X u =1; otherwise, converting the positive integer u into an n-bit binary number u n-1 u n-2 ...u 0 ，u 0 Called least significant bits, u n-1 Called the most significant bit, at this time
Extracting all coefficient components 2 n Dimension vector->
The coefficient vector of f, called the boolean function. The number of the non-zero single-term of the coefficient in the algebraic normal type of f is called the term number of f, and the maximum value of the algebraic times of all the non-zero single-term of the coefficient is called the algebraic times of the Boolean function f. The hamming weight of a non-negative integer is defined as the number of non-zero bits in its binary representation.
Calculating the nonlinear invariant with the algebraic number of S-boxes of 2 is the key to successfully implement nonlinear invariant analysis. The traditional calculation method mainly comprises three steps: calculating a Boolean expression of the component function, calculating a coefficient matrix, and eliminating Gaussian elements of the matrix. Because algebraic times cannot be specified, the conventional calculation method needs to calculate X for all non-negative integers k when calculating coefficient matrixes k +Y k Count up to 2 n And as the hamming weight of k increases, Y k The computational complexity of (a) increases dramatically. When the Gaussian matrix is eliminated, the size of the matrix is 2 n Line 2 n+1 Columns.
The traditional calculation method can calculate all nonlinear invariants of the S box, and then the nonlinear invariants with algebraic frequency of 2 are screened out. However, the number of non-linear invariants tends to be exponential, for example, the number of non-linear invariants of an 8-bit S-box of the iScaram algorithm reaches 2 136 -1, whereas the number of algebraic non-linear invariants of number 2 often represents only a small fraction. This renders the computed non-linear invariants mostly ineffective, resulting in serious waste of computing resources and storage resources, and quite low computing efficiency.
Disclosure of Invention
Aiming at the problems existing in the prior art, the rapid calculation method of the nonlinear invariant is provided, the nonlinear invariant with the algebraic number of S-box being 2 can be rapidly calculated, the waste of calculation resources and storage resources is avoided, and the calculation efficiency is greatly improved.
The technical scheme adopted by the invention is as follows: a method for fast computation of non-linear invariants, based on a computer program, performing a computation process, comprising:
step 2, boolean function F of n-bit input and n-bit output to be calculated satisfies (y n-1 ,y n-2 ,...,y 0 )＝F(x n-1 ,x n-2 ,...,x 0 ) Calculating the Boolean expression of each component function according to the truth table of the Boolean function F;
step 3, obtaining positive integers with the weight of all the hamming weights not exceeding algebraic times, and obtaining a coefficient matrix M according to the positive integers F ；
Step 4, introducing an auxiliary matrix E F To coefficient matrix M F Expansion to obtain an expansion matrix [ M ] F ,E F ]For the expansion matrix [ M ] F ,E F ]Gaussian elimination is carried out to obtain a matrix [ M ] 1 ,M 2 ]；
Step 5, traversing the matrix M 1 If the row vector in (a) takes the value of [0, …,0]Or [1,0, …,0 ]]Then M is 2 The line vector at the same position in the row is regarded as a coefficient vector and is converted into a Boolean expression;
and 6, all the non-zero linear combinations of the calculated Boolean expression are output as non-linear invariant sub-if the algebraic number of the non-zero linear combinations is the set algebraic number.
Further, in the step 3, the positive integer is k i Satisfies the following conditions
Wherein,,
d is the set algebraic number.
Further, the acquisition coefficient matrix M F The method comprises the following steps:
step 3.1 according to positive integer k i Calculation of
Step 3.2, extracting coefficient vector according to the calculation result of step 3.1
Construction of coefficient matrix from coefficient vector
Further, the specific process of the step 3.1 is as follows:
step 3.1.1, let k be i Binary number k converted into n bits i,n-1 k i,n-2 ...k i,0 ；
Step 3.1.2, calculating
Step 3.1.3, substituting the Boolean expression of each component function obtained in the step 2 into the calculation formula of the step 3.1.2 respectively, and eliminating all variables y j And simplifying and outputting a calculation result.
Further, in the step 3.2, the coefficient vector
Wherein->
Is either 0 or 1, j=0, 1,..2 n -1；
When single item X j Occurs at
In the simplified Boolean expression, < >>
Otherwise, go (L)>
Wherein the single item X j The method comprises the following steps: converting j to n-bit binary number j n-1 j n-2 …j 0 ，/>
When j=0, X j ＝1。
Further, the auxiliary matrix E F From 2 n Dimension unit vector
The composition specifically comprises:
wherein 2 is n Dimension unit vector
Refers to the kth i The +1 element having a value of 1, i.e. a unit vector
Wherein->
The specific values are as follows: when j=k i When (I)>
Otherwise, go (L)>
Further, M in the step S4 1 And M 2 Is that
Line 2 n A matrix of columns.
Compared with the prior art, the beneficial effects of adopting the technical scheme are as follows: the method provided by the invention saves the computing resource and the storage resource to a great extent, effectively reduces the complexity of computation and screening, and remarkably improves the computing efficiency.
Drawings
FIG. 1 is a flow chart of a fast calculation of a nonlinear invariant proposed by the present invention.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
As shown in fig. 1, the present embodiment provides a fast calculation method of a nonlinear invariant, and a computer executes a calculation process, which specifically includes:
s1, limiting algebraic times d of effective nonlinear invariants;
s2, a Boolean function F of n-bit output is input to the selected n bits to satisfy (y n-1 ,y n-2 ,...,y 0 )＝F(x n-1 ,x n-2 ,...,x 0 ) Calculating the Boolean expression of each component function according to the truth table;
s3, obtaining positive integers k with the hamming weight not exceeding d i ，
Satisfy the following requirements
Separately calculate->
Extracting coefficient vector->
Obtaining coefficient matrix M F The method comprises the steps of carrying out a first treatment on the surface of the The hamming weight of a positive integer refers to the number of non-zero bits in its corresponding binary number.
S4, introducing an auxiliary matrix E F To coefficient matrix M F Expansion to obtain an expansion matrix [ M ] F ,E F ]For the expansion matrix [ M ] F ,E F ]Gaussian elimination is carried out to obtain a matrix [ M ] 1 ,M 2 ]The method comprises the steps of carrying out a first treatment on the surface of the Wherein M is 1 And M 2 Du Shi
Line 2 n A matrix of columns.
S5, traversing the matrix M 1 If the row vector in (a) takes a value of [0, …,0 ]]Or [1,0, …,0 ]]Then M is 2 The line vector at the same position in the row is regarded as a coefficient vector and is converted into a Boolean expression;
s6, calculating all nonzero linear combinations of the Boolean expressions, and outputting the nonlinearly invariant sub-output if the algebraic degree is d.
In particular in the step S3
The calculation process of (1) is as follows:
s31, let k i Binary number k converted into n bits i,n-1 k i,n-2 ...k i,0 ；
S32, calculating
k i,j Is used to determine whether a certain input variable or output variable is present in the product term.
S33, substituting the Boolean expression of each component function obtained in the step S2 into the calculation formula of the step S32, and simplifying and outputting the Boolean expression.
Specifically, the boolean expression of each component is calculated in step S2, and the form is as follows: y is 0 ＝f 0 (x 0 ,x 1 ,...,x n-1 ),y 1 ＝f 1 (x 0 ,x 1 ,...,x n-1 ),...,y n-1 ＝f n-1 (x 0 ,x 1 ,...,x n-1 ) Substituting these n-in 1-out Boolean function expressions into the formula y of step S32 j Thereby eliminating the output variable y 0 ,y 1 ,...,y n-1 Finally, the method is simplified to obtain
With respect to input variable x 0 ,x 1 ,...,x n-1 Is a boolean function of (c).
After the simplified Boolean expression is obtained, a coefficient vector can be calculated; the coefficient vector
Wherein->
Is either 0 or 1, j=0, 1,..2 n -1；
The coefficient vector value method comprises the following steps: when single item X j Occurs at
In the simplified boolean expression,
otherwise, go (L)>
Wherein the single formula X j Is as follows: converting j to n-bit binary number j n-1 j n-2 ...j 0 ，
In particular, when j=0, X j ＝1。
Finally, the obtained coefficient vectors are arranged to obtain a coefficient matrix M F In particular to
In this embodiment, the auxiliary matrix E introduced in S4 F From 2 n Dimension unit vector
The constitution is specifically that
Wherein 2 is n Dimension unit vector
Refers to the kth i The +1 element having a value of 1, i.e. a unit vector
Wherein->
The specific values of (2) are as follows: when j=k i When (I)>
Otherwise, go (L)>
In this embodiment, a fast calculation method of a nonlinear invariant with algebraic frequency of 2 is also provided for verification.
The algebraic number d=2 of the effective non-linear invariant is defined, taking the 4-bit S-box of the lightweight cryptographic algorithm Midori64 algorithm as an example, S e F, and the non-linear invariant with algebraic number 2 is calculated by using a fast calculation method. The truth table for the S-box is shown in table 1.
Table 1S box truth table
The calculation process is as follows:
s1, inputting 4 bits to the selected 4 bits and outputting an S box, wherein (y) is satisfied 3 ,y 2 ,y 1 ,y 0 )＝S(x 3 ,x 2 ,x 1 ,x 0 ) The boolean expression for each component function is calculated according to the truth table, with the following results:
y 0 ＝x 1 +x 0 x 2 +x 0 x 1 x 2 +x 0 x 3 +x 0 x 1 x 3 +x 1 x 2 x 3 ；
y 1 ＝x 0 +x 2 +x 0 x 2 +x 0 x 3 +x 2 x 3 ；
y 2 ＝1+x 0 +x 0 x 1 x 2 +x 3 +x 0 x 3 +x 0 x 1 x 3 +x 1 x 2 x 3 ；
y 3 ＝1+x 0 x 1 +x 1 x 3 +x 0 x 1 x 3 +x 2 x 3 +x 1 x 2 x 3 ；
s2, positive integer k with weight not exceeding 2 for all Hamming i I=0, 1, 9; in the present embodiment, k 0 ＝1，k 1 ＝2，k 2 ＝3，k 3 ＝4，k 4 ＝5，k 5 ＝6，k 6 ＝8，k 7 ＝9，k 8 ＝10，k 9 =12, the specific description is as follows:
4-bit S-boxes, according to positive integer k i Value range 0 of (2)<k i <16, the hamming weight is not more than 2, and 1, 2, 3, 4, 5, 6, 8, 9, 10 and 12 are all 10 positive integers. Wherein the number of binary non-zero bits corresponding to 1, 2, 4 and 8 is 1, the hamming weights thereof are 1,3, 5, 6, 9, 10 and 12, the number of binary non-zero bits corresponding to 2, and the hamming weights thereof are 2. Thus, according to the value rule k 0 <k 1 <…<k 9 It can be seen that k 0 ＝1，k 1 ＝2，k 2 ＝3，k 3 ＝4，k 4 ＝5，k 5 ＝6，k 6 ＝8，k 7 ＝9，k 8 ＝10，k 9 ＝12。
Separately calculate
Extracting coefficient vector->
Obtaining coefficient matrix M F The results were as follows:
s3, introducing an auxiliary matrix E F Obtain an expansion matrix [ M ] F ,E F ]Wherein
For the expansion matrix [ M ] F ,E F ]Gaussian elimination is carried out to obtain a matrix [ M ] 1 ,M 2 ]The results were as follows:
s4, traversing the matrix M 1 The row vector in (a) takes the value of [0, …,0 ]]Is 4 in total, M 2 The co-located row vectors in (a) are considered as coefficient vectors, and are converted to boolean expressions, with the following results:
x 0 +x 3 +x 0 x 3 +x 2 x 3 ；
x 2 +x 0 x 1 +x 1 x 2 +x 2 x 3 ；
x 0 +x 1 +x 2 +x 2 x 3 ；
x 0 x 2 +x 0 x 3 ；
s5, calculating all non-zero linear combinations of the Boolean expressions, wherein the total number of the non-zero linear combinations is 15, and the algebraic times are all 2, so that 15 non-linear invariant sub-groups with the algebraic times of 2 of the S box can be obtained, and the method specifically comprises the following steps:
g 0 ＝x 0 +x 3 +x 0 x 3 +x 2 x 3 ；
g 1 ＝x 2 +x 0 x 1 +x 1 x 2 +x 2 x 3 ；
g 2 ＝x 0 +x 2 +x 3 +x 0 x 3 +x 0 x 1 +x 1 x 2 ；
g 3 ＝x 0 +x 1 +x 2 +x 2 x 3 ；
g 4 ＝x 1 +x 2 +x 3 +x 0 x 3 ；
g 5 ＝x 0 +x 1 +x 0 x 1 +x 1 x 2 ；
g 6 ＝x 1 +x 3 +x 0 x 3 +x 0 x 1 +x 1 x 2 +x 2 x 3 ；
g 7 ＝x 0 x 2 +x 0 x 3 ；
g 8 ＝x 0 +x 3 +x 2 x 3 +x 0 x 2 ；
g 9 ＝x 2 +x 0 x 1 +x 1 x 2 +x 2 x 3 +x 0 x 2 +x 0 x 3 ；
g 10 ＝x 0 +x 2 +x 3 +x 0 x 1 +x 1 x 2 +x 0 x 2 ；
g 11 ＝x 0 +x 1 +x 2 +x 2 x 3 +x 0 x 2 +x 0 x 3 ；
g 12 ＝x 1 +x 2 +x 3 +x 0 x 2 ；
g 13 ＝x 0 +x 1 +x 0 x 1 +x 1 x 2 +x 0 x 2 +x 0 x 3 ；
g 14 ＝x 1 +x 3 +x 0 x 1 +x 1 x 2 +x 2 x 3 +x 0 x 2 。
in the embodiment, the method provided by the invention can rapidly calculate the nonlinear invariant with the algebraic number of S-box of 2; by defining the algebraic number d=2 of the effective nonlinear invariant, X is calculated by considering only the positive integer k of 1 by Hamming weight and 2 by Hamming weight k +Y k The number of times of calculation of (2) n The next most significant reduction to
Second, at the same time avoid Y corresponding to Gao Hanming weight k k Is calculated by (a), the coefficient matrix M is greatly reduced F The dimension of the expansion matrix is also 2 n Line 2 n+1 Column is reduced to +.>
Line 2 n+1 And the complexity of the Gaussian elimination of the matrix is reduced.
The invention is not limited to the specific embodiments described above. The invention extends to any novel one, or any novel combination, of the features disclosed in this specification, as well as to any novel one, or any novel combination, of the steps of the method or process disclosed. It is intended that insubstantial changes or modifications from the invention as described herein be covered by the claims below, as viewed by a person skilled in the art, without departing from the true spirit of the invention.
All of the features disclosed in this specification, or all of the steps in a method or process disclosed, may be combined in any combination, except for mutually exclusive features and/or steps.
Any feature disclosed in this specification may be replaced by alternative features serving the same or equivalent purpose, unless expressly stated otherwise. That is, each feature is one example only of a generic series of equivalent or similar features, unless expressly stated otherwise.
Claims (1)
1. A method for fast computation of non-linear invariants, which performs a computation process based on a computer program, comprising:
step 1, setting algebraic times of an effective nonlinear invariant;
step 2, boolean function F of n-bit input and n-bit output to be calculated satisfies (y n-1 ,y n-2 ,...,y 0 )＝F(x n-1 ,x n-2 ,...,x 0 ) Calculating the Boolean expression of each component function according to the truth table of the Boolean function F;
step 3, obtaining positive integers with the weight of all the hamming weights not exceeding algebraic times, and obtaining a coefficient matrix M according to the positive integers F ；
Step 4, introducing an auxiliary matrix E F To coefficient matrix M F Expansion to obtain an expansion matrix [ M ] F ,E F ]For the expansion matrix [ M ] F ,E F ]Gaussian elimination is carried out to obtain a matrix [ M ] 1 ,M 2 ]；
Step 5, traversing the matrix M 1 If the row vector in (a) takes the value of [0, …,0]Or [1,0, …,0 ]]Then M is 2 The line vector at the same position in the row is regarded as a coefficient vector and is converted into a Boolean expression;
step 6, all the non-zero linear combinations of the calculated Boolean expression are output as non-linear invariant sub-if the algebraic number of the non-zero linear combinations is the set algebraic number;
in the step 3, the positive integer is k i Satisfies the following conditions
Wherein,,
d is the set algebraic times;
the acquisition coefficient matrix M F Is the step of (a)The method comprises the following steps:
step 3.1 according to positive integer k i Calculation of
Step 3.2, extracting coefficient vector according to the calculation result of step 3.1
Construction of coefficient matrix from coefficient vector
The specific process of the step 3.1 is as follows:
step 3.1.1, let k be i Binary number k converted into n bits i,n-1 k i,n-2 ...k i,0 ；
Step 3.1.2, calculating
Step 3.1.3, substituting the Boolean expression of each component function obtained in the step 2 into the calculation formula obtained in the step 3.1.2 respectively, and eliminating all variables y j Simplifying and outputting a calculation result;
in the step 3.2, the coefficient vector
Wherein->
Is 0 or 1, j=0, 1.,. 2 n -1；
When single item X j Occurs at
In the simplified Boolean expression, < >>
Otherwise, go (L)>
Wherein the single item X j The method comprises the following steps: converting j to n-bit binary number j n-1 j n-2 ...j 0 ，/>
When j=0, X j ＝1；
The auxiliary matrix E F From 2 n Dimension unit vector
The composition specifically comprises:
wherein 2 is n Dimension unit vector
Refers to the kth i The +1 elements take a value of 1, i.e. +.>
Wherein->
The specific values are as follows: when j=k i When (I)>
Otherwise->
M in the step 4 1 And M 2 Is that
Line 2 n A matrix of columns.
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