CN111339577B - Construction method of S box with excellent DPA resistance - Google Patents

Construction method of S box with excellent DPA resistance Download PDF

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CN111339577B
CN111339577B CN202010088063.2A CN202010088063A CN111339577B CN 111339577 B CN111339577 B CN 111339577B CN 202010088063 A CN202010088063 A CN 202010088063A CN 111339577 B CN111339577 B CN 111339577B
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CN111339577A (en
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徐友乐
王启春
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Nanjing Normal University
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F21/00—Security arrangements for protecting computers, components thereof, programs or data against unauthorised activity
    • G06F21/70—Protecting specific internal or peripheral components, in which the protection of a component leads to protection of the entire computer
    • G06F21/71—Protecting specific internal or peripheral components, in which the protection of a component leads to protection of the entire computer to assure secure computing or processing of information
    • G06F21/75—Protecting specific internal or peripheral components, in which the protection of a component leads to protection of the entire computer to assure secure computing or processing of information by inhibiting the analysis of circuitry or operation
    • G06F21/755—Protecting specific internal or peripheral components, in which the protection of a component leads to protection of the entire computer to assure secure computing or processing of information by inhibiting the analysis of circuitry or operation with measures against power attack
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F7/00—Methods or arrangements for processing data by operating upon the order or content of the data handled
    • G06F7/60—Methods or arrangements for performing computations using a digital non-denominational number representation, i.e. number representation without radix; Computing devices using combinations of denominational and non-denominational quantity representations, e.g. using difunction pulse trains, STEELE computers, phase computers
    • G06F7/72—Methods or arrangements for performing computations using a digital non-denominational number representation, i.e. number representation without radix; Computing devices using combinations of denominational and non-denominational quantity representations, e.g. using difunction pulse trains, STEELE computers, phase computers using residue arithmetic
    • G06F7/727—Modulo N arithmetic, with N being either (2**n)-1,2**n or (2**n)+1, e.g. mod 3, mod 4 or mod 5
    • 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
    • H04L9/002—Countermeasures against attacks on cryptographic mechanisms
    • H04L9/003—Countermeasures against attacks on cryptographic mechanisms for power analysis, e.g. differential power analysis [DPA] or simple power analysis [SPA]
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2207/00—Indexing scheme relating to methods or arrangements for processing data by operating upon the order or content of the data handled
    • G06F2207/72—Indexing scheme relating to groups G06F7/72 - G06F7/729
    • G06F2207/7219—Countermeasures against side channel or fault attacks

Abstract

The invention discloses a construction method of an S box with excellent DPA resistance, which relates to the technical field of information security, can obtain the S box with excellent DPA resistance, can also be used for improving the DPA resistance of the existing S box, and simultaneously can retain other important cryptology properties such as nonlinearity, algebraic times, difference consistency, absolute value indexes and the like. By contrast, the S box obtained by the method has the DPA resistance obviously superior to that of the existing practical S box, and other various important cryptology properties are the same, so that the S box has stronger safety and practicability.

Description

Construction method of S box with excellent DPA resistance
Technical Field
The invention relates to the technical field of information security, in particular to a construction method of an S box with excellent DPA resistance.
Background
The S-box is the only nonlinear component in the block cipher, and the cryptographic property directly influences the security of the related cipher system. Therefore, the S-box plays a very important role in symmetric cryptography. The defense against various attack means is started from constructing an S box with good cryptology properties, such as nonlinearity, difference consistency, algebraic times, absolute value indexes and the like. A sufficiently safe S-box should have a high degree of non-linearity, not low algebraic degree, low differential consistency, low absolute value index, etc.
In 1996, the side channel attack proposed by Kocher posed a serious threat to the security of physical cryptographic devices. Among the types of side channel attacks, Differential Power Analysis (DPA) is one of the most effective attack means. Compared with the traditional attack means such as linear attack, algebraic attack, differential attack and the like, the attack means has very strong attack efficiency, and can extract the key information only by needing little physical information, thereby successfully attacking an encryption system. Especially the widely used AES, which uses S-boxes with very poor resistance to DPA attacks. Therefore, many researchers have been exploring in recent years how to construct an S-box with excellent DPA resistance, thereby increasing DPA resistance of the relevant cryptographic device at a small cost. The transparent order is a cryptographic property that has been proven in practice to be an effective measure of resistance to S-box DPA. Recent studies have focused on the S-box resulting in DPA resistance, which has been greatly improved over AES. However, the S-boxes obtained by these techniques have important cryptographic properties such as non-linearity, difference consistency, absolute value index, etc. which cannot satisfy the requirements of security and practicability.
Disclosure of Invention
The technical problem to be solved by the invention is to overcome the defects of the prior art and provide a construction method of an S box with excellent DPA resistance, and meanwhile, the important cryptology properties of nonlinearity, difference consistency, algebraic times, absolute value indexes and the like also meet the requirements of safety and practicability.
The invention adopts the following technical scheme for solving the technical problems:
according to the invention, the construction method of the S box with excellent DPA resistance comprises the following steps:
step 1, giving an n-bit input and balanced S box F, wherein the S box F is used as an initial input of an iterative construction process, and the S box F is put into a set SF; the iterative construction process is step 2-step 7;
step 2, generating a candidate pool according to the input truth table of the S box F, wherein the candidate pool comprises all bit positions capable of being reversed;
step 3, b elements are taken out from the candidate pool, and the b elements are b bit positions; then, carrying out bit inversion operation on the b bit positions of the S box to generate a new S box;
the b elements taken out need to meet the following requirements: 1) b 2k, k 1,2, …,2n-1(ii) a 2) The same component function attributed to the S-box; 3) the Hamming weight of the vector composed of the bits corresponding to the b elements is 2b-1；
Step 4, repeating the process of step 3 until the elements in the candidate pool are emptied, and generating all new S boxes associated with the F;
step 5, calculating the value of the loss function of each newly generated S box, and putting the S box and the corresponding value of the loss function into a list L;
step 6, judging whether the list L is empty, if so, increasing the value of b, and meanwhile, b meets the requirements 1), 2) and 3) in the step 3, and executing the step 3-5; if L is not empty, selecting an S box with the minimum loss function value from the list L;
and 7, judging whether the S box selected in the step 6 exists in the set SF:
if yes, deleting the S box and the corresponding loss function value from the list L, and executing the step 6;
if not, judging whether the S box meets the preset safety requirement or not: if yes, outputting the S box, and terminating the iterative construction process; if not, putting the S box into the set SF, and taking the S box as an input S box of the next iteration construction process, namely taking the S box as the input S box in the step 2; and steps 2-7 are performed.
As a further optimization of the construction method of an S-box with excellent DPA resistance according to the present invention, the loss function in step 5 is:
Figure BDA0002382743370000021
where n represents the S-box as n input bits,
Figure BDA0002382743370000022
is an n-dimensional vector space over a binary domain,
Figure BDA0002382743370000023
is a non-zero n-dimensional vector space, W, over a binary domainF(c, b) is Walsh spectrum value of S-box relative to vector c, b, beta is N-dimensional vector on binary domain, c, alpha is non-zero N-dimensional vector on binary domain, R is constant, sigma (alpha, beta) is difference spectrum value of S-box relative to vector alpha, beta, N4(σ (α, β)) is a function for determining whether σ (α, β) is greater than 4, τFIs a transparent step of the S-box.
As a further optimization of the construction method of an S-box with excellent DPA resistance according to the present invention, the preset safety requirements in step 7 are: the nonlinearity of the S-box is 112, the algebraic degree is greater than 3, the difference consistency is 4, the absolute value index is 32, and the transparency level is less than 6.9160.
As a further optimization scheme of the construction method of the S box with excellent DPA resistance, R is set to be 2-5.
Compared with the prior art, the invention adopting the technical scheme has the following technical effects:
(1) the method can generate the S box with excellent DPA resistance, very high nonlinearity, low difference consistency, low absolute value index and high algebraic times, and compared with the prior art research scheme, the S box obtained by construction has stronger safety and practicability;
(2) the invention can improve the DPA resistance of the existing S box, and simultaneously reserve other important cryptology properties such as nonlinearity, difference consistency, algebraic times, absolute value indexes and the like; the S box of the invention has strong enough capability of resisting other attacks, so that the S box of the invention can have higher security to meet the practical requirement.
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FIG. 1 is a process of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in detail with reference to the accompanying drawings and specific embodiments.
FIG. 1 is a process of the present invention, which is specifically embodied as follows:
step 1, an n-bit input and balanced S box F is given as an initial input of an iterative construction process and is put into a set SF. The iterative construction process is step 2-7:
step 2, according to the truth table of the S-box F as input, adding all the bit positions of the S-box into a candidate pool PF, wherein the pool contains all the possible reversal bit positions, and the total number is n x 2nAnd (4) each element.
And 3, taking out b elements, namely bit positions, from the candidate pool. The b elements taken out need to meet the requirements: 1) b 2k, (k 1,2, …, 2)n-1). 2) The same component function attributed to the S-box. 3) The Hamming weight of the vector composed of the bits corresponding to the b elements is 2b-1. The bit reversal operation is then performed at these b positions of the S-box, generating a new S-box F'.
Step 4, repeat the process of step 3 until all elements in the candidate pool are emptied, generating all S-boxes F' associated with F.
And 5, calculating the value CF of the loss function of each newly generated S box F ', and putting the S box F' and the corresponding loss function value CF into the list L. Where the loss function used for the calculation is:
Figure BDA0002382743370000031
where n represents the number of input bits of S-box F,
Figure BDA0002382743370000032
is an n-dimensional vector space over a binary domain,
Figure BDA0002382743370000033
the vector space is a non-zero n-dimensional vector space on a binary domain, and R is a constant and is set to be 2-5.
WF(c, b) is the Walsh spectrum value of the S-box with respect to vector c, b, defined as:
Figure BDA0002382743370000034
wherein x is (x)1,x2,…,xn),b＝(b1,b2,…,bn) Is an n-dimensional vector on a binary domain, "·" is an inner product operation, i.e. x · b ═ x1b1⊕x2b2⊕…⊕xnbn. ≧ is addition on the binary field, i.e., modulo-2 addition.
N4(σ (α, β)) is a judgment function that judges whether the S-box has a difference spectrum value σ (α, β) greater than 4 with respect to the n-dimensional vector α, β on the binary domain, and if greater than 4, returns 1, otherwise returns 0. Wherein σ (α, β) is defined as
Figure BDA0002382743370000041
Where # represents the number of collection elements.
τFTransparent order for S-box F, defined as:
Figure BDA0002382743370000042
herein, the
Figure BDA0002382743370000043
Is the cross-correlation coefficient of S-box F, defined as
Figure BDA0002382743370000044
Where max is the maximum value, and β ═ β (β)0,β1,…,βn) Is an n-dimensional vector over a binary field, and F ═ F1,f2,…,fn),fi(x),fj(x) As a function of the component of S-box F, betai,βjIs the bit information of i and j positions in the vector beta, wherein 0 is less than or equal to i, j, and less than or equal to n.
For a given S-box F, the loss function value designed above is calculated, and the smaller the calculated value is, the higher the nonlinearity, the lower the difference consistency, the smaller the absolute value index, the lower the transparency level and the higher the DPA resistance are possessed in the cryptographic property of the S-box. Therefore, the S box can resist attacks such as DPA attack, linear attack, algebraic attack, differential attack and the like, and the safety and the practicability of relevant equipment are improved.
And 6, judging whether the list L is empty, if so, increasing the value of b, and simultaneously b meets the requirements 1), 2) and 3) in the step 3). Step 3 in the starting iterative construction process is reproduced. If L is not empty, selecting an S box in the list L, wherein the S box needs to meet the requirement of having the minimum loss function value;
and 7, judging whether the S box selected in the step 6 exists in the set SF or not. If yes, deleting the S box and the corresponding loss function value from the list L, and repeating the step 6; if not, whether the S box meets the safety requirement designed in advance is judged. The method comprises the following specific steps:
and calculating whether the properties of the S box, such as nonlinearity, transparent order, difference consistency, algebraic times, absolute value index and the like, meet the requirement that the nonlinearity is equal to 112, the transparent order is less than 6.9160, the algebraic times is greater than 3, the difference consistency is equal to 4, and the absolute value index is equal to 32. The specific calculation process is as follows:
degree of non-linearity
Figure BDA0002382743370000045
Differential coherency
Figure BDA0002382743370000046
Index of absolute value
Figure BDA0002382743370000047
Where r (a, b) is its autocorrelation function defined as:
Figure BDA0002382743370000051
if yes, outputting the S box, and terminating the iterative construction process. If not, the S box is put into the set SF and is used as an input S box of the next iteration construction process, and the iteration construction process is repeated, namely the steps 2-7.
The cryptographic properties of the S-box of the AES of the present invention are shown in Table 1
TABLE 1
Degree of non-linearity Degree of algebra Differential coherency Index of absolute value Transparent stage
The method of the invention 112 7 4 32 6.8930
AES S box 112 7 4 32 6.9160
The method of the invention constructs the S box as follows: [65,158,207,138,121,103,233,69,72,60,88,51,244, 215,34,37,164,28,30,78,129,172,77,153,56,122,95,235,17,251,18,200,82,71, 14,66,180,15,86,39,232,64,203,214,166,97,204,202,156,234,189,11,197,175, 218,245,24,136,196,253,9,128,100,239,41,165,35,93,114,7,127,33,8,90,160, 135,171,22,19,188,116,254,32,89,249,229,38,107,10,83,73,176,230,67,101, 26,206,45,117,194,62,94,212,5,146,226,205,87,109,108,250,237,140,185,68, 184,80,98,223,126,167,4,25,192,178,221,119,70,61,20,243,210,145,177,174, 75,57,157,3,154,112,191,247,144,143,132,104,173,208,151,195,149,213,23, 139,181,48,137,1,222,74,186,228,255,16,65,44,120,252,179,242,76,21,147, 134,53,91,133,124,169,36,155,216,219,115,113,161,190,79,50,187,141,123, 231,99,150,58,52,225,238,31,209,47,42,46,106,96,2,148,201,131,241,102, 152,43,13,182,248,54,183,227,125,159,118,246,198,105,220,162,85,92,193, 40,6,49,27,240,111,199,63,236,211,170,130,12,55,224,142,217,84,110,29, 168,59,16,163]
The above embodiments are only for illustrating that the present invention aims to improve DPA resistance of S-boxes, and do not limit the present invention to take only the above parameter values and parameter forms, and it is obvious to those skilled in the art that the present invention can be changed in various forms without departing from the spirit of the present invention, so the protection scope of the present invention is not limited to the above embodiments. Any modifications, equivalents, improvements and the like which come within the spirit of the invention are intended to be included within the scope of the claims.

Claims (3)

1. A method of constructing an S-cassette with excellent DPA resistance, comprising the steps of:
step 1, giving an n-bit input and balanced S box F, wherein the S box F is used as an initial input of an iterative construction process, and the S box F is put into a set SF; the iterative construction process is step 2-step 7;
step 2, generating a candidate pool according to the input truth table of the S box F, wherein the candidate pool comprises all bit positions capable of being reversed;
step 3, b elements are taken out from the candidate pool, and the b elements are b bit positions; then, carrying out bit inversion operation on the b bit positions of the S box to generate a new S box;
the b elements taken out need to meet the following requirements: 1) b 2k, k 1,2, …,2n-1(ii) a 2) The same component function attributed to the S-box; 3) the Hamming weight of the vector composed of the bits corresponding to the b elements is 2b-1；
Step 4, repeating the process of step 3 until the elements in the candidate pool are emptied, and generating all new S boxes associated with the F;
step 5, calculating the value of the loss function of each newly generated S box, and putting the S box and the corresponding value of the loss function into a list L;
step 6, judging whether the list L is empty, if so, increasing the value of b, and simultaneously b meets the requirements 1), 2) and 3) in the step 3, and executing the step 3 to the step 5; if L is not empty, selecting an S box with the minimum loss function value from the list L;
and 7, judging whether the S box selected in the step 6 exists in the set SF:
if yes, deleting the S box and the corresponding loss function value from the list L, and executing the step 6;
if not, judging whether the S box meets the preset safety requirement: if yes, outputting the S box, and terminating the iterative construction process; if not, putting the S box into the set SF, and taking the S box as an input S box of the next iteration construction process, namely taking the S box as the input S box in the step 2; and executing the step 2 to the step 7;
the loss function in step 5 is:
Figure FDA0003470425490000011
where n represents the S-box as n input bits,
Figure FDA0003470425490000012
is an n-dimensional vector space over a binary domain,
Figure FDA0003470425490000013
is a non-zero n-dimensional vector space, W, over a binary domainF(c, b) Walsh spectrum values of S-box with respect to vector c, b, β are N-dimensional vectors in binary domain, c, α are non-zero N-dimensional vectors in binary domain, R is a constant, σ (α, β) is a differential spectrum value of S-box with respect to vector α, β, N4(σ (α, β)) is a function for determining whether σ (α, β) is greater than 4, τFIs a transparent step of the S-box.
2. The method of claim 1, wherein the safety requirements predetermined in step 7 are: the nonlinearity of the S-box is 112, the algebraic degree is greater than 3, the difference consistency is 4, the absolute value index is 32, and the transparency level is less than 6.9160.
3. The method of claim 1, wherein R is set to 2-5.
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