Nonlinear Properties of Rijndael S-boxes Represented by the Many-Valued Logic Functions Artem Sokolov [0000-0003-0283-7229] and Djiofack Temgoua Vanіssa Noel [0000-0003-1651-0702] Odessa National Polytechnic University, Odessa, Ukraine radiosquid@gmail.com Abstract. S-boxes of the Nyberg construction are one of the most important cryptographic primitives, which are used in the AES cryptographic algorithm and largely determines its effectiveness. Numerous researches have confirmed the high cryptographic quality of their component Boolean functions. Neverthe- less, the cryptanalyst is not constrained in the methods used and can also use the mathematical apparatus of the functions of many-valued logic for cryptanalysis. This work is devoted to the research of the nonlinear properties of S-boxes of the Nyberg construction, presented in the form of component 4-functions and 16-functions. The paper proposes a method for calculating the nonlinearity val- ue of 16-functions, for which the formula of the recursive construction of hexa- decimal Vilenkin-Chrestenson matrices of arbitrary order is discovered. The performed researches made it possible to establish that the nonlinearity values of component 4-functions and 16-functions of S-boxes of the Nyberg construc- tion is not stable and depends on the type of irreducible polynomial used to construct them. In the paper we present the irreducible polynomial for which the nonlinearity values of component 4-functions and 16-functions is evenly high. At the same time, it was established that the same polynomial also pro- vides the uniform minimization of the correlation coefficients between output and input vectors of the S-box. The specified polynomial can be recommended for the practical use. Keywords: cryptography, logic, function, Nyberg construction, nonlinearity, S-box. 1 Introduction and problem statement Block symmetric cryptographic algorithms are a very important component of mod- ern information security systems. The main component of block symmetric crypto- graphic algorithms, on which the overall quality of the cryptographic transform de- pends, is a cryptographic S-box. Today, there are many constructive methods for the synthesis of high-quality S-boxes. As one of the most effective methods for S-boxes design the Nyberg construction can be mentioned [1]. S-boxes of this construction are used in the Rijndael cryptographic algorithm, which is approved as the AES encryp- tion standard [2]. Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons Li- cense Attribution 4.0 International (CC BY 4.0). CybHyg-2019: International Workshop on Cyber Hygiene, Kyiv, Ukraine, November 30, 2019. S-boxes of the Nyberg construction are determined by using a mapping in the form of multiplicatively inverse elements of the Galois field GF (2k ) y  x1 modd[f (z), p], y, x  GF (2k ) , (1) which is in general combined with an affine transform b  A  y  a, a, b  GF (2k ) , (2) where as f ( z ) the standard AES irreducible over the field GF (2) polynomial is used f ( z )  z8  z 4  z 3  z  1 ; A is the nonsingular affine transform matrix; a is the shift vector; p  2 is the characteristic of the extended Galois field, 01  0 is taken a-priory; a, b, x, y are the elements of the extended Galois field, that can be considered as decimal numbers, or binary vectors, or polynomials of degree k  1 . A detailed research of the cryptographic properties of Nyberg construction S-boxes of length N  256 was performed in [3], where it was established that the crypto- graphic quality of the S-box depends on the type of irreducible polynomial used. The number of irreducible polynomials is defined as 1 Wk     d   p(k / d ) , k d (3) d k where d are the divisors of the k ,  (d ) is the Mobius function, the notation d k means that d divides k . Moreover, to determine the cryptographic quality of S-boxes, the generally accept- ed approach is to represent the S-box using the mathematical apparatus of Boolean algebra: the original S-box is decomposed into component Boolean functions, to each of which a generally accepted set of criteria for cryptographic quality is used. In this set of criteria, the criterion of high nonlinearity is adopted as the most important crite- rion [4]. Nevertheless, when describing cryptographic algorithms, a cryptanalyst is not con- strained in the facilities used, in particular, the mathematical apparatus of functions of many-valued logic can be used [5]. Today in the literature there are no researches of the nonlinear properties of S-boxes of the Nyberg construction, presented using the functions of many-valued logic. The purpose of this work is to research the nonlinear properties of the component many-valued logic functions of Nyberg construction S-boxes of length N  256 based on the full set of irreducible polynomials. 2 Possible representations of AES S-boxes by functions of many-valued logic We introduce the definition of the S-box which is necessary for further research and consider the possible forms of its representation. Definition 1. S-box is a substitution of the form  y0 0 N 1 , y N 1 (4) where the first row is a sequence of numbers from 0 to N  1 , the second row is a sequence { yi } consisting of elements of the first row, rearranged according to the law specified by the designers of the S-box. The second row of the substitution (4) is called as the coding Q-sequence and denoted Q  { yi } , i  0,1,..., N  1 . Each coding Q-sequence can be unambiguously represented in the form of k  log q N its component q-functions, where q belongs to the set of such values that the length N of the S-box can be represented in the form N  q k . Obviously, the Nyberg S-boxes used in the AES cryptoalgorithm can be uniquely represented using component Boolean functions (2-functions), using component 4- functions, and also using component 16-functions. Moreover, each of these functions completely determines the structure and cryptographic quality of the S-box in the sense of the corresponding logic. For example, consider the S-box of the Nyberg construction (1) based on a poly- nomial f ( z)  28310  z8  z 4  z3  z  1 used in the AES cipher in the form of its coding Q-sequence Q={0 1 141 246 203 82 123 209 232 79 41 192 176 225 229 199 116 180 170 75 153 43 96 95 88 63 253 204 255 64 238 178 58 110 90 241 85 77 168 201 193 10 152 21 48 68 162 194 44 69 146 108 243 57 102 66 242 53 32 111 119 187 89 25 29 254 55 103 45 49 245 105 167 100 171 19 84 37 233 9 237 92 5 202 76 36 135 191 24 62 34 240 81 236 97 23 22 94 175 211 73 166 54 67 244 71 145 223 51 147 33 59 121 183 151 133 16 181 186 60 182 112 208 6 161 250 129 130 131 126 127 128 150 115 (5) 190 86 155 158 149 217 247 2 185 164 222 106 50 109 216 138 132 114 42 20 159 136 249 220 137 154 251 124 46 195 143 184 101 72 38 200 18 74 206 231 210 98 12 224 31 239 17 117 120 113 165 142 118 61 189 188 134 87 11 40 47 163 218 212 228 15 169 39 83 4 27 252 172 230 122 7 174 99 197 219 226 234 148 139 196 213 157 248 144 107 177 13 214 235 198 14 207 173 8 78 215 227 93 80 30 179 91 35 56 52 104 70 3 140 221 156 125 160 205 26 65 28}. We consider the possible representations of the S-box (5) using the functions of many-valued logic, bringing as an example the first of the corresponding component q-functions. So, the S-box (5) can be represented as 8 component Boolean functions, the first of which is given as an example Fbin1  {0110101101100111000111010110100000011101100100000 10011000101111110111111101101111010001100001011001110010 11111111111010000001010101001001011101000010000001010101 (6) 00110100000010000111101100110011011000111101000010111000 101100111010011001110011100001010101010}. The S-box (5) can also be represented as four component 4-functions Ffouri , i  1, 2,..., 4 , the first of which has the form Ffour1  {0112323103100113002313030310302222211101120100220 120312221033311123311113033011110120033022010132233122303 133313133101202002121232302322321132102221020220301012302 (7) 330102022232200331101122110233033200313303002232313220301 100311232231023310233300023010101210}. And also, the S-box (5) can be represented as two component 16-functions, the first of which we give as an example Fhex1  {01D6B2B18F90015744AB9B0F8FDCF0E2AEA15D891A85 0422C52C3962250F7B99DE77D15974B34599DC5AC47F8E201C17 6EF39663471F331B977505AC60061A123EF063E6BE597294EA2D8 (8) A42A4F89C9ABCE3F858682AE722C0FF15815E6DDC67B8F3A44F 9734BCC6A7E35B2A4B45D80B1D6B6EFD8E73D0E3B384863CDC D0DA1C}. 3 Method for determining the nonlinearity value of many- valued logic functions The most important characteristic of the cryptographic quality of S-boxes is its non- linearity distance. The binary case is classical, in which the nonlinearity distance is defined as the minimum Hamming distance between Boolean function f and all codewords of an affine code [6] 2N f  min(dist ( f , A j )), j  0,1,..., 2k 1 1 . (9) Definition 2. For an arbitrary positive integer k , an affine code A( N , k ) of length N  2k is defined as the set of all rows of those Boolean functions whose algebraic   degree of nonlinearity does not exceed 1, that is A( N , k )  A f | f  Fk ,deg f  1 [7]. In turn, the nonlinearity distance of the entire S-box is determined by the worst from its component Boolean functions, i.e. as 2 NS  min{2 NFi }, i  1, 2,..., k2 . (10) Moreover, since the set of codewords of the non-inverse part of the affine code co- incides with the rows of the Walsh-Hadamard matrix, the nonlinearity distance of the component Boolean functions can also be found in the domain of the Walsh- Hadamard transform coefficients in accordance with the following formula 1 2 N f  2k 1  maxk W f (v) , (11) 2 vZ 2 where W f (v)  f  AN is the vector of coefficients of the Walsh-Hadamard transform of the component Boolean function f , AN is the Walsh-Hadamard matrix, which is constructed in accordance with the following recurrence rule A k A2k  A2k 1   2  A2k  , (12)  A2k where A1  1 . Applying formulas (9) or (11) to the first component Boolean function (6) of the S- box (5), it is easy to verify that its nonlinearity distance is equal to N Fbin1  112 , while the nonlinearity distances of all component Boolean functions of the S-box (5) are equal to NFbini  {112,112,112,112,112,112,112,112}, i  1, 2,...,8 . Nevertheless, formulas (9) and (11) are not applicable for the estimation of the nonlinearity value of functions of many-valued logic, in particular, for the 4-functions and 16-functions that we are researching. Estimation of the nonlinearity value of 4-functions is an important problem, which was solved in [8, 11]. The proposed method for estimating the nonlinearity value of 4- functions is based on finding the coefficients of the Vilenkin-Chrestenson transform  f  f V of the investigated 4-function f , where the investigated 4-function f and the Vilenkin-Chrestenson matrix V are presented in exponential form using the unique transformation  2 2 2 2  j 0 j 1 j 2 j 3   {0,1, 2,3}  e 4 e 4 e 4 e 4  . (13)     The Vilenkin-Chrestenson matrix is constructed according to the following recurrence rule V4k V4k V4k  V4k V V4k  1 V4k  2 V4k  3  V4k 1   4 k , (14) V4k V4k  2 V4k V4k  2    V4k V4k  3 V4k  2 V4k  1  where the summation is performed modulo 4, and 0 0 0 0 0 1 2 3  V4   . (15) 0 2 0 2   0 3 2 1 Based on the Vilenkin-Chrestenson transform coefficients, a generalized formula for estimation of the nonlinearity value of q-valued logic functions is introduced in [8]  q k  max  f , q  2;    qN f   (16) 1  2k 1  max W f , q  2.  2   Same to the binary case, the nonlinearity value of the S-box is determined by its worst component q-function, respectively, qNS  min{qN Fi }, i  1, 2,..., kq . Using expression (14), it is easy to construct the Vilenkin-Chrestenson matrix over the alphabet {0,1, 2,3} of order N  256 , with help of which we can find the coeffi- cients of the Vilenkin-Chrestenson transform of sequence (7). Further, using expres- sion (16), it is not difficult to determine that the nonlinearity value of the 4-function (7) is equal to 4 N F1  219.5034 . The nonlinearity value of the remaining component 4-functions are equal to 4 NFfouri  {219.5034 221.9412 216.5538 219.2849} , i  1, 2,3, 4 . Accordingly, the nonlinearity value of the entire S-box (5) is equal to 4 N S  216.5538 . Note that, in contrast to the binary case, the nonlinearity values of the component 4-functions of S-boxes of the Nyberg construction are different for different component 4-functions [12-13]. Although the general formula for the nonlinearity value for an arbitrary q was in- troduced in [8], however, a specific mechanism for finding the nonlinearity value of 16-functions was not shown, and in order to evaluate the nonlinearity values of the component 16-functions of the Nyberg construction S-boxes, it is necessary to devel- op recurrence algorithm for constructing Vilenkin-Chrestenson matrices over the alphabet  0 1 2 3 4 5 6 7  j 2 0 j 2 1 j 2 2 j 2 3 2 j4 2 5 j 2 j 6 j 2 7 e 16 e 16 e 16 e 16 e 16 e 16 e 16 e 16 (17) 8 9 A B C D E F  2 2 2 2 2 2 2 2 j 8 j 9 j 10 j 11 j 12 j 13 j 14 j 15  . e 16 e 16 e 16 e 16 e 16 e 16 e 16 e 16  Obviously, the affine functions of a k  1 variable over the alphabet (17) have a general form i ( x1 )  a1 x1  a0 . Taking a0  0 , we construct 16 affine 16-functions 1 ,..., 16 1  0 0000000000000000 2  x1 0123456789ABCDEF 3  2 x1 02468ACE02468ACE 4  3 x1 0369CF258BE147AD 5  4 x1 048C048C048C048C 6  5 x1 05AF49E38D27C16B 7  6 x1 06C28E4A06C28E4A 8  7 x1 07E5C3A18F6D4B29 . (18) 9  8 x1 0808080808080808 10  9 x1 092B4D6F81A3C5E7 11  Ax1 0A4E82C60A4E82C6 12  Bx1 0B61C72D83E94FA5 13  Cx1 0C840C840C840C84 14  Dx1 0DA741EB852FC963 15  Ex1 0ECA86420ECA8642 16  Dx1 0FEDCBA987654321 The resulting set of the first 16 affine 16-functions (18) determines the Vilenkin- Chrestenson matrix of order N  16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0 1 2 3 4 5 6 7 8 9 A B C D E F  0 2 4 6 8 A C E 0 2 4 6 8 A C E  0 3 6 9 C F 2 5 8 B E 1 4 7 A D  0 4 8 C 0 4 8 C 0 4 8 C 0 4 8 C  0 5 A F 4 9 E 3 8 D 2 7 C 1 6 B  0 6 C 2 8 E 4 A 0 6 C 2 8 E 4 A  V16  0 7 E 5 C 3 A 1 8 F 6 D 4 B 2 9. (19) 0 8 0 8 0 8 0 808 0 8 0 8 0 8 0 9 2 B 4 D 6 F 8 1 A 3 C 5 E 7 0 A 4 E 8 2 C 6 0 A 4 E 8 2 C 6  0 B 6 1 C 7 2 D 8 3 E 9 4 F A 5  0 C 8 4 0 C 8 4 0 C 8 4 0 C 8 4  0 D A 7 4 1 E B 8 5 2 F C 9 6 3  0 E C A 8 6 4 2 0 E C A 8 6 4 2  0 F E D C B A 9 8 7 6 5 4 3 2 1  In view of the fact that for our purposes of researching the nonlinearity values of component 16-functions of S-boxes of the Nyberg construction of length N  256 , we need a Vilenkin-Chrestenson matrix of order N  256 . Note that the previously used method [9] for constructing Vilenkin-Chrestenson matrices for arbitrary q is complex. It makes the task of developing of simple method for the synthesis of Vilen- kin-Chrestenson matrices over the alphabet (17) actual. Researches allowed us to derive a formula for the recurrence construction of Vilenkin-Chrestenson matrices of any given order N  16k V16i V16i V16i V16i V16i V16i V16i V16i V16i V16i  1 V16i  2 V16i  3 V16i  4 V16i  5 V16i  6 V16i  7 V i V i  2 V i  4 V i  6 V i  8 V i  10 V i  12 V i  14 V16 V16  3 V16  6 V16  9 V 16  12 V16  15 V16  2 V16  5  16i 16i 16i 16i 16i 16i 16i 16i V16i V16i  4 V16i  8 V16i  12 V16i V16i  4 V16i  8 V16i  12 V16i V16i  5 V16i  10 V16i  15 V16i  4 V16i  9 V16i  14 V16i  3 V16i V16i  6 V16i  12 V16i  2 V16i  8 V16i  14 V16i  4 V16i  10 V i V i  7 V i  14 V i  5 V i  12 V i  3 V i  10 V i  1 V16i   16 16 16 16 16 16 16 16 V V  8 V16i V16i  8 V16i V16i  8 V16i V16i  8  16i 16i V16i V16i  9 V16i  2 V16i  11 V16i  4 V16i  13 V16i  6 V16i  15 V16i V16i  10 V16i  4 V16i  14 V16i  8 V16i  2 V16i  12 V16i  6 V16i V16i  11 V16i  6 V16i  1 V16i  12 V16i  7 V16i  2 V16i  13 V i V i  12 V i  8 V i  4 V i V i  12 V i  8 V i  4 V16i V16i  13 V 16i  10 V16i  7 V 16 16 16 16  4 V16i  1 V16i  14 V16i  11  16 16 16 16 16i V16i V16i  14 V16i  12 V16i  10 V16i  8 V16i  6 V16i  4 V16i  2 V16i V16i  15 V16i  14 V16i  13 V16i  12 V16i  11 V16i  10 V16i  9 (20) V16i V16i V16i V16i V16i V16i V16i V16i  V16i  8 V16i  9 V16i  10 V16i  11 V16i  12 V16i  13 V16i  14 V16i  15  V16i V16i  2 V16i  4 V16i  6 V16i  8 V16i  10 V16i  12 V16i  14  V16i  8 V16i  11 V16i  14 V16i  1 V16i  4 V16i  7 V16i  10 V16i  13 V16i V16i  4 V16i  8 V16i  12 V16i V16i  4 V16i  8 V16i  12  V16i  8 V16i  13 V16i  2 V16i  7 V16i  12 V16i  1 V16i  6 V16i  11 V16i V16i  6 V16i  12 V16i  2 V16i  8 V16i  14 V16i  4 V16i  10  V16i  8 V16i  15 V16i  6 V16i  13 V16i  4 V16i  11 V16i  2 V16i  9  V16i  8  . V16i V16i  8 V16i V16i  8 V16i V16i  8 V16i  V16i  8 V16i  1 V16i  10 V16i  3 V16i  12 V16i  5 V16i  14 V16i  7  V16i V16i  10 V16i  4 V16i  14 V16i  8 V16i  2 V16i  12 V16i  6  V16i  8 V16i  3 V16i  14 V16i  9 V16i  4 V16i  15 V16i  10 V16i  5  V16i V16i  12 V16i  8 V16i  4 V16i V16i  12 V16i  8 V16i  4  V16i  8 V16i  5 V16i  2 V16i  15 V16i  12 V16i  9 V16i  6 V16i  3   V16i V16i  14 V16i  12 V16i  10 V16i  8 V16i  6 V16i  4 V16i  2  V16i  8 V16i  7 V16i  6 V16i  5 V16i  4 V16i  3 V16i  2 V16i  1  By constructing the Vilenkin-Chrestenson matrix with the help of (20), and also multiplying it by the component 16-function (8) of the S-box (5), it is easy to obtain the Vilenkin-Chrestenson transform coefficients of the 16-function (8). Further, ap- plying formula (16), we find that the nonlinearity value of the 16-function (8) is equal to 16 NFhex1  213.8184 . Moreover, the nonlinearity value of the second component 16-function is equal to 16 N Fhex2  212.4385 , and, accordingly, the nonlinearity value of the entire S-box is equal to 16 N S  212.4385 . 4 Research of the Nyberg construction S-boxes of length N=256 based on the full class of irreducible polynomials To compare nonlinearity values, it is convenient to use such perfect algebraic con- structions as bent-functions [10], which have the minimum possible value of the max- k imal Vilenkin-Chrestenson transform coefficient equal to q 2 , and, accordingly, the maximum nonlinearity value equal to k qN f  q k  q 2 . (21) Thus, in our case for q  4 and k  4 the maximum value of nonlinearity is equal to 4 N f  240 , while for q  16 and k  2 the maximum value of nonlinearity will also reach the value 16 N f  240 . Using the proposed method for estimating 2-nonlinearity, 4-nonlinearity and 16- nonlinearity values of S-boxes of Nyberg construction of length N  256 , it is not difficult to estimate the nonlinearity values for all S-boxes that can be built over a field GF (256) . These values are summarized in Table 1. Table 1. The values of nonlinearity for Nyberg construction S-boxes of length N=256. No. Irreducible polynomial 2NS 4NS 16NS 1 283 112 216.5538 212.4385 2 285 112 217.7901 208.0271 3 299 112 213.4794 215.6620 4 301 112 212.9187 211.2972 5 313 112 215.7508 213.2651 6 319 112 211.2786 213.6862 7 333 112 215.5031 215.3282 8 351 112 213.4794 219.9423 9 355 112 212.9187 216.2035 10 357 112 217.5292 219.6070 11 361 112 211.8186 215.3785 12 369 112 216.3011 212.2766 13 375 112 219.1218 213.1083 14 379 112 219.1218 200.4 15 391 112 216 214.2562 16 395 112 214.7689 220.3838 17 397 112 215.5031 217.4026 18 415 112 210.6569 202.9546 19 419 112 221.4746 215.3345 20 425 112 215.9500 213.2825 21 433 112 219.7785 217.0560 22 445 112 215.7508 218.4184 23 451 112 217.3736 218.5131 24 463 112 213.6208 211.3523 25 471 112 211.2786 218.9588 26 477 112 217.9474 209.9110 27 487 112 217.5292 207.0381 28 499 112 218.4234 219.6522 29 501 112 214.2388 217.5087 30 505 112 210.3930 212.3652 The data presented in Table 1 show that all Nyberg construction S-boxes of length N  256 based on the full set of irreducible polynomials have a nonlinearity distance of component Boolean functions 2 NS  112 . However, different polynomials provide different values of nonlinearity in the sense of 4-functions and 16-functions. So, an S- box based on a polynomial f19  41910 has the highest nonlinearity values of compo- nent 4-functions, while the S-box based on a polynomial f16  39510 has the highest nonlinearity values of component 16-functions. Moreover, both nonlinearity values of the component 4-functions and the nonlinearity values of the component 16-functions of S-box based on the polynomial f 28  49910 are optimal. Earlier in [3], it was found that the polynomial f 28  49910 (however, like polynomial f9  35510 ) also provides the most uniform minimization of the matrix of correlation coefficients. From our perspective, this S-box can be recommended for practical use in the AES crypto- graphic algorithm from the point of view of nonlinearity criteria for component func- tions of many-valued logic [14]. 5 Conclusions Let us to summarize the main results of the research: 1. The nonlinearity values of component 4-functions and 16-functions of S-boxes of Nyberg construction of length N  256 based on the full set of irreducible poly- nomials has been researched. It has been determined that the S-boxes of the Nyberg construction, which have the same nonlinearity distance of component Boolean functions, are at the same time characterized by different nonlinearity val- ues of 4-functions and 16-functions for various irreducible polynomials. It was found that the nonlinearity values of the component 4-fucntions and component 16-functions of the S-box based on the polynomial f 28  49910 are optimal, there- fore this polynomial can be recommended for practical use. 2. The method for researching the nonlinearity value of 4-functions was adapted to the case of 16-functions. This technique can be applied to S-boxes of other practi- cally valuable constructions. 3. 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