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Reversed genetic algorithms for generation of bijective s-boxes with good cryptographic properties

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Abstract

Often the S-boxes are the only nonlinear components in a block cipher and as such play an important role in ensuring its resistance to cryptanalysis. Cryptographic properties and constructions of S-boxes have been studied for many years. The most common techniques for constructing S-boxes are: algebraic constructions, pseudo-random generation and a variety of heuristic approaches. Among the latter are the genetic algorithms. In this paper, a genetic algorithm working in a reversed way is proposed. Using the algorithm we can rapidly and repeatedly generate a large number of strong bijective S-boxes of each dimension from (8 × 8) to (16 × 16), which have sub-optimal properties close to the ones of S-boxes based on finite field inversion, but have more complex algebraic structure and possess no linear redundancy.

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Notes

  1. “m of 255” in Table 1 means that m out of 255 S-box component Boolean functions belong to distinct extended affine equivalence classes

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Author information

Authors and Affiliations

  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Georgi Ivanov & Nikolay Nikolov

  2. Department ESAT/COSIC and iMinds, KU Leuven, Leuven, Belgium

    Svetla Nikova

Authors
  1. Georgi Ivanov
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  2. Nikolay Nikolov
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  3. Svetla Nikova
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Corresponding author

Correspondence to Svetla Nikova.

Additional information

The research is done as a part of the project “Finite geometries, coding theory and cryptography” between the Research Foundation - Flanders (FWO) and the Bulgarian Academy of Sciences. This research has been supported by the ICT COST Action IC1306 “Cryptography for Secure Digital Interaction”.

Appendices

Appendix A:

1.1 A.1 Genetic Algorithm 1 pseudo code

STEP 1:

(Initializing step) - defining the algorithm parameters

  • Define an integer n, representing the dimensions (n × n) of the bijective S-box.

  • Define an integer T, representing the number of S-boxes in the parent pool (P P).

  • Define an even integer N t h r ≤N i n v , representing the nonlinearity threshold value.

  • Generate a number of T S-boxes of dimensions (n × n) and put them into the (P P). Some based on the inversion in the finite field G F(2n), while the other obtained in result of the application of affine transformations to the outputs of the former.

  • Create an empty offspring pool (O P) of size T.

  • Set the parents indexes t and r to be 1.

STEP 2:

(Breeding step)

figure a
STEP 3:

(Mutation step) C h 1 = m o d e l i n g(C h 1) and C h 2 = m o d e l i n g(C h 2) go to Step 4

STEP 4:

(Fitness step)

figure b
STEP 5:

(Solution pool)The number of all T desired S-boxes, having nonlinearity N = N t h r , are disposed in the offspring pool OP.

1.2 A.2 Genetic Algorithm 2 pseudo code

STEP 1:

(Initializing step) - defining the algorithm parameters

  • Define an integer n, representing the dimensions (n × n) of the bijective S-box.

  • Define an integer T, representing the number of S-boxes in the parent pool (P P).

  • Define an even integer N t h r ≤N i n v , representing the nonlinearity threshold value.

  • Generate a number of T S-boxes of dimensions (n × n) and put them into the (P P). Some based on the inversion in the finite field G F(2n), while the other obtained in result of the application of affine transformations to the outputs of the former.

  • Create an empty offspring pool (O P) of size T.

  • Set the counter cnt value to be 0.

  • Set the parents indexes t and r to be 1.

STEP 2:

(Breeding step)

figure c
STEP 3:

(Mutation step)C h 1 = m o d e l i n g(C h 1) and C h 2 = m o d e l i n g(C h 2) go to Step 4

STEP 4:

(Fitness step)

figure d
STEP 5:

(Solution pool)The number of all T desired S-boxes, having nonlinearity N = N t h r , are disposed in the offspring pool OP.

1.3 A.3 Genetic Algorithm 3 pseudo code

STEP 1:

(Initializing step) - defining the algorithm parameters

  • Define an integer n, representing the dimensions (n × n) of the bijective S-box.

  • Define an integer T, representing the number of S-boxes in the parent pool (P P).

  • Define an even integer N t h r ≤N i n v , representing the nonlinearity threshold value.

  • Define an even integer δ t h r ≥δ i n v representing the δ-uniformity threshold value.

  • Generate a number of T S-boxes of dimensions (n × n) and put them into the (P P). Some based on the inversion in the finite field G F(2n), while the other obtained in result of the application of affine transformations to the outputs of the former.

  • Create an empty offspring pool (O P) of size T.

  • Set the counter cnt value to be 0.

  • Set the parents indexes t and r to be 1.

STEP 2:

(Breeding step)

figure e
STEP 3:

(Mutation step)C h 1 = m o d e l i n g(C h 1) and C h 2 = m o d e l i n g(C h 2) go to Step 4

STEP 4:

(Fitness step)

figure f
STEP 5:

(Solution pool)The number of all T desired S-boxes, having N = N t h r and δ = δ t h r (and at least one component Boolean function of nonlinearity greater than N i n v - for the case of the advanced version advGA3 only), are disposed in the offspring pool OP.

Appendix B: S-boxes generated with GA1

1.1 B.1 S-box No 1 (N S =104,d e g(S)=7,A C(S) m a x =64,δ = 6)

$$\begin{array}{@{}rcl@{}} &&\{0x52\ 0x53\ 0xDF\ 0xA4\ 0x99\ 0x00\ 0x29\ 0x83\ 0xBA\ 0x1D\ 0x7B\ 0x92\ 0xE2\ 0xB3\ 0xB7\ 0x95\\ &&0x26\ 0xE6\ 0xF8\ 0x19\ 0xCB\ 0x79\ 0x32\ 0x0D\ 0x0A\ 0x6D\ 0xAF\ 0x9E\ 0xAD\ 0x12\ 0xBC\ 0xE0\\ &&0x68\ 0x3C\ 0x08\ 0xA3\ 0x07\ 0x1F\ 0xFA\ 0x9B\ 0x93\ 0x58\ 0xCA\ 0x47\ 0x62\ 0x16\ 0xF0\ 0x90\\ &&0x7E\ 0x17\ 0xC0\ 0x3E\ 0xA1\ 0x6B\ 0x34\ 0x10\ 0xA0\ 0x67\ 0x72\ 0x3D\ 0x25\ 0xE9\ 0x0B\ 0x4B \\ &&0x4F\ 0xAC\ 0x65\ 0x35\ 0x7F\ 0x63\ 0xA7\ 0x3B\ 0xF5\ 0x36\ 0xF9\ 0x41\ 0x06\ 0x77\ 0xBB\ 0x5B\\ &&0xBF\ 0x0E\ 0x57\ 0x98\ 0x1E\ 0x76\ 0xD5\ 0xED\ 0x4A\ 0x6C\ 0x70\ 0xA2\ 0x03\ 0xBE\ 0x33\ 0x45\\ &&0x44\ 0x0C\ 0xFD\ 0x81\ 0x1B\ 0xF4\ 0x64\ 0x11\ 0xA6\ 0x15\ 0xC3\ 0x8D\ 0x61\ 0xC1\ 0x73\ 0x69\\ &&0x2B\ 0xE5\ 0xC5\ 0xD7\ 0x42\ 0xE7\ 0xE8\ 0x6E\ 0xE4\ 0x22\ 0x82\ 0x54\ 0xF3\ 0xA8\ 0xD3\ 0xD0\\ &&0xD1\ 0x2C\ 0x2D\ 0xD2\ 0xC4\ 0x21\ 0xEC\ 0x04\ 0xC9\ 0xCC\ 0xC7\ 0x8B\ 0xA5\ 0x50\ 0xEB\ 0xF6\\ &&0x8C\ 0x38\ 0x60\ 0x3F\ 0x8A\ 0xD8\ 0xD6\ 0x20\ 0x78\ 0x46\ 0xCD\ 0xDA\ 0xAB\ 0x8E\ 0xDB\ 0xC8\\ &&0xA9\ 0x2E\ 0x7C\ 0x91\ 0xDD\ 0xEA\ 0x37\ 0x1A\ 0x74\ 0x9A\ 0x40\ 0x18\ 0x9C\ 0xB5\ 0x80\ 0x30\\ &&0x5E\ 0xB2\ 0x4D\ 0xBD\ 0x43\ 0x27\ 0x2A\ 0x23\ 0xF7\ 0xDC\ 0x24\ 0x6F\ 0xEF\ 0xEE\ 0xD4\ 0x05\\ &&0x59\ 0x7A\ 0x7D\ 0xF1\ 0x88\ 0x86\ 0xB6\ 0x5D\ 0xFB\ 0x75\ 0x01\ 0x56\ 0x49\ 0xAE\ 0xFE\ 0xB4\\ &&0x28\ 0x55\ 0xFC\ 0x31\ 0x97\ 0x89\ 0xB0\ 0xB8\ 0xC6\ 0xD9\ 0x96\ 0x87\ 0xCF\ 0xAA\ 0xC2\ 0x39\\ &&0xE3\ 0x5F\ 0x84\ 0xB9\ 0x94\ 0x5C\ 0x9D\ 0xFF\ 0x5A\ 0x1C\ 0x85\ 0xB1\ 0x0F\ 0x02\ 0x4C\ 0xF2\\ &&0x4E\ 0x71\ 0x48\ 0x9F\ 0x14\ 0xCE\ 0x09\ 0x51\ 0x66\ 0xE1\ 0x6A\ 0x3A\ 0xDE\ 0x13\ 0x8F\ 0x2F\} \end{array} $$

1.2 B.2 S-box No 2 (N S =106,d e g(S)=6,A C(S) m a x =56,δ = 6)

$$\begin{array}{@{}rcl@{}} &&\{0x50\ 0x51\ 0x93\ 0xD2\ 0xF2\ 0x2E\ 0x11\ 0x0A\ 0x01\ 0x66\ 0x6F\ 0xFC\ 0xB3\ 0x38\ 0x7D\ 0x7A\\ &&0xBB\ 0xCB\ 0x4B\ 0x65\ 0x8C\ 0x4E\ 0x06\ 0xF5\ 0xE2\ 0x24\ 0x64\ 0x42\ 0x85\ 0x34\ 0x45\ 0x8D \\ &&0xE6\ 0x1B\ 0xDE\ 0xAB\ 0x9E\ 0xB9\ 0x89\ 0xF1\ 0x3E\ 0x8B\ 0x5F\ 0x7C\ 0x7B\ 0x5E\ 0xC1\ 0xA1 \\ &&0x09\ 0x87\ 0x6A\ 0xA4\ 0x4A\ 0x43\ 0x59\ 0x00\ 0xF9\ 0x33\ 0x62\ 0xA5\ 0x99\ 0x9C\ 0xFD\ 0x5A \\ &&0x0B\ 0x56\ 0xB6\ 0xA7\ 0x17\ 0xEF\ 0xEE\ 0x14\ 0x37\ 0x2B\ 0xE7\ 0x71\ 0xFF\ 0x03\ 0xC3\ 0xAF \\ &&0x67\ 0x58\ 0xFE\ 0x1D\ 0x94\ 0x81\ 0x46\ 0xF4\ 0x86\ 0x60\ 0x57\ 0x10\ 0xDB\ 0xCD\ 0xEB\ 0xDC \\ &&0xBF\ 0xD1\ 0xF8\ 0x69\ 0x4D\ 0x84\ 0x2A\ 0x18\ 0x5D\ 0xB2\ 0x9A\ 0xE0\ 0x97\ 0x8E\ 0x78\ 0x8A \\ &&0xC7\ 0x82\ 0xA2\ 0xD4\ 0x49\ 0xE3\ 0xE9\ 0xD7\ 0xF7\ 0xB4\ 0x36\ 0x19\ 0xC5\ 0xC9\ 0x55\ 0xF3 \\ &&0xBE\ 0x31\ 0x53\ 0x92\ 0x23\ 0xA3\ 0xE8\ 0x27\ 0xB0\ 0xA8\ 0xCC\ 0x0C\ 0x0F\ 0xEA\ 0x72\ 0xAA \\ &&0xA0\ 0x7E\ 0xAE\ 0x1E\ 0xC8\ 0x2C\ 0x83\ 0x20\ 0xC4\ 0x2D\ 0xBA\ 0x41\ 0xDA\ 0x0D\ 0xEC\ 0xBC \\ &&0x88\ 0x77\ 0x54\ 0x2F\ 0x07\ 0x47\ 0xB5\ 0x28\ 0x32\ 0x68\ 0xFB\ 0xFA\ 0x5B\ 0x6E\ 0x02\ 0x1C\\ &&0x3B\ 0x9B\ 0x48\ 0x25\ 0x90\ 0xAD\ 0x70\ 0x1A\ 0xD6\ 0x26\ 0xDD\ 0x0E\ 0xCE\ 0xBD\ 0x16\ 0x15 \end{array} $$
$$\begin{array}{@{}rcl@{}} &&0xE4\ 0xAC\ 0xD3\ 0x52\ 0x04\ 0x80\ 0x8F\ 0x3C\ 0x9D\ 0x6C\ 0x3A\ 0xE1\ 0x6D\ 0x98\ 0x74\ 0xB8 \\ &&0x95\ 0x05\ 0x21\ 0xC6\ 0x35\ 0x4C\ 0x08\ 0x61\ 0xF0\ 0x76\ 0x3F\ 0x79\ 0x44\ 0x4F\ 0x3D\ 0x96 \\ &&0xD8\ 0xA9\ 0x39\ 0x5C\ 0x29\ 0xF6\ 0x12\ 0xA6\ 0x9F\ 0x75\ 0xCA\ 0x40\ 0xCF\ 0xED\ 0xD0\ 0x30 \\ &&0xC0\ 0x7F\ 0x22\ 0xD5\ 0x63\ 0x6B\ 0xB7\ 0x13\ 0xC2\ 0xDF\ 0xE5\ 0x73\ 0xB1\ 0x91\ 0x1F\ 0xD9 \} \end{array} $$

1.3 B.3 S-box No 3 (N S =108,d e g(S)=6,A C(S) m a x =48,δ = 6)

$$\begin{array}{@{}rcl@{}} &&\{0x97\ 0x96\ 0x1A\ 0x26\ 0x1B\ 0x82\ 0xAB\ 0x01\ 0x38\ 0x9F\ 0xF9\ 0x10\ 0x60\ 0x31\ 0x35\ 0x17\\ &&0xA4\ 0x64\ 0x7A\ 0x9B\ 0x49\ 0xFB\ 0xB0\ 0x8F\ 0x88\ 0xEF\ 0x2D\ 0x1C\ 0x2F\ 0x90\ 0x3E\ 0x62\\ &&0xEA\ 0xBE\ 0x8A\ 0x21\ 0x85\ 0x9D\ 0x78\ 0x19\ 0x11\ 0xDA\ 0x48\ 0xC5\ 0xE0\ 0x94\ 0x72\ 0x12\\ &&0xFC\ 0x95\ 0x42\ 0xBC\ 0x23\ 0xE9\ 0xB6\ 0x92\ 0x22\ 0xE5\ 0xF0\ 0xBF\ 0xA7\ 0x6B\ 0x89\ 0xC9\\ &&0xCD\ 0x2E\ 0xE7\ 0xB7\ 0xFD\ 0xE1\ 0x25\ 0xB9\ 0x77\ 0xB4\ 0x7B\ 0xC3\ 0x84\ 0xF5\ 0x39\ 0xD9\\ &&0x3D\ 0x8C\ 0xD5\ 0xD1\ 0x9C\ 0xF4\ 0x57\ 0x6F\ 0xC8\ 0xEE\ 0xF2\ 0x20\ 0x81\ 0x3C\ 0xB1\ 0xC7\\ &&0xC6\ 0x8E\ 0x7F\ 0x03\ 0x99\ 0x76\ 0xE6\ 0x93\ 0x24\ 0x5D\ 0x41\ 0x0F\ 0xE3\ 0x43\ 0xF1\ 0xEB\\ &&0xA9\ 0x67\ 0x47\ 0x55\ 0xC0\ 0x65\ 0x6A\ 0xEC\ 0x66\ 0xA0\ 0x00\ 0xD6\ 0x71\ 0x2A\ 0x51\ 0x52\\ &&0x53\ 0xAE\ 0xAF\ 0x50\ 0x46\ 0xA3\ 0x6E\ 0x86\ 0x4B\ 0x4E\ 0x45\ 0x09\ 0x27\ 0xD2\ 0x69\ 0x74\\ &&0x0E\ 0xBA\ 0xE2\ 0xBD\ 0x08\ 0x5A\ 0x54\ 0xA2\ 0xFA\ 0xC4\ 0x4F\ 0x58\ 0x29\ 0x0C\ 0x59\ 0x4A\\ &&0x2B\ 0xAC\ 0xFE\ 0x13\ 0x5F\ 0x68\ 0xB5\ 0x98\ 0xF6\ 0x18\ 0xC2\ 0x9A\ 0x1E\ 0x37\ 0x02\ 0xB2\\ &&0xDC\ 0x30\ 0xCF\ 0x3F\ 0xC1\ 0xA5\ 0xA8\ 0xA1\ 0x75\ 0x5E\ 0xA6\ 0xED\ 0x6D\ 0x6C\ 0x56\ 0x87\\ &&0xDB\ 0xF8\ 0xFF\ 0x73\ 0x0A\ 0x04\ 0x34\ 0xDF\ 0x79\ 0xF7\ 0x83\ 0xD4\ 0xCB\ 0x2C\ 0x7C\ 0x36\\ &&0xAA\ 0xD7\ 0x7E\ 0xB3\ 0x15\ 0x0B\ 0x32\ 0x3A\ 0x44\ 0x5B\ 0x14\ 0x05\ 0x4D\ 0x28\ 0x40\ 0xBB\\ &&0x61\ 0xDD\ 0x06\ 0x3B\ 0x16\ 0xDE\ 0x1F\ 0x7D\ 0xD8\ 0x9E\ 0x07\ 0x33\ 0x8D\ 0x80\ 0xCE\ 0x63\\ &&0x8B\ 0xF3\ 0xE8\ 0xE4\ 0xB8\ 0xD0\ 0xD3\ 0x5C\ 0x0D\ 0x4C\ 0xAD\ 0x70\ 0x1D\ 0xCA\ 0x91\ 0xCC\} \end{array} $$

Appendix C: S-boxes generated with GA2

1.1 C.1 S-box No 1 (N S =106,d e g(S)=6,A C(S) m a x =48,δ = 6)

$$\begin{array}{@{}rcl@{}} &&\{0x82\ 0xD3\ 0x21\ 0x1F\ 0x95\ 0xDC\ 0x4E\ 0x86\ 0x5A\ 0x68\ 0x8D\ 0x47\ 0xC4\ 0x31\ 0xA0\ 0x5E\\ &&0xCE\ 0x20\ 0xD7\ 0xB6\ 0x56\ 0x2C\ 0x33\ 0x05\ 0x81\ 0x8F\ 0x08\ 0x32\ 0xB3\ 0x8A\ 0xCC\ 0x58\\ &&0x84\ 0x22\ 0xF3\ 0x5C\ 0x7B\ 0x1E\ 0xB8\ 0x6C\ 0xC8\ 0x71\ 0xF5\ 0x6F\ 0x09\ 0x04\ 0x12\ 0xC5\\ &&0x50\ 0xD4\ 0x57\ 0x0A\ 0xE7\ 0x78\ 0xFA\ 0x4D\ 0x49\ 0xB4\ 0xA6\ 0x97\ 0x85\ 0x3E\ 0xCF\ 0x0E\\ &&0xA1\ 0x10\ 0xF2\ 0x3C\ 0x69\ 0x17\ 0xCD\ 0x00\ 0x2D\ 0x0B\ 0xA2\ 0xDB\ 0xBF\ 0x67\ 0xD5\ 0x2F\\ &&0x87\ 0x19\ 0x28\ 0xFB\ 0x6A\ 0xB1\ 0x27\ 0xB5\ 0x14\ 0x8C\ 0xE1\ 0xD9\ 0xEA\ 0x9C\ 0x72\ 0x9F\\ &&0xCB\ 0xEE\ 0x89\ 0xA9\ 0x3B\ 0x83\ 0xE6\ 0x2A\ 0x63\ 0x93\ 0xDF\ 0xC1\ 0x9E\ 0x41\ 0x36\ 0xC9\\ &&0x34\ 0xF4\ 0xB9\ 0x38\ 0xB0\ 0x4B\ 0x5B\ 0x16\ 0x52\ 0xBE\ 0xFC\ 0x98\ 0x77\ 0x92\ 0xE4\ 0xAB\\ &&0x40\ 0x73\ 0xEB\ 0x42\ 0x9A\ 0x9B\ 0xFD\ 0x64\ 0x24\ 0xA7\ 0x1B\ 0xDD\ 0x76\ 0x62\ 0xE3\ 0xEC\\ &&0x06\ 0x80\ 0x15\ 0x46\ 0xB2\ 0x02\ 0x7D\ 0xA8\ 0x4F\ 0x18\ 0x23\ 0x3F\ 0x7A\ 0x3A\ 0x07\ 0x8E\\ &&0x53\ 0x66\ 0x1C\ 0xED\ 0xF7\ 0xD0\ 0x6D\ 0x39\ 0xD6\ 0x0C\ 0x48\ 0x26\ 0x03\ 0x8B\ 0x4A\ 0x6B\\ &&0xE9\ 0xF0\ 0xA5\ 0xC7\ 0x60\ 0xE5\ 0x7C\ 0x88\ 0x96\ 0x25\ 0xAD\ 0xC6\ 0xDA\ 0x55\ 0x5F\ 0x11 \end{array} $$
$$\begin{array}{@{}rcl@{}} &&0x75\ 0xC0\ 0x94\ 0x1A\ 0x54\ 0xA3\ 0x44\ 0xE0\ 0x0D\ 0xB7\ 0x51\ 0x2E\ 0x90\ 0xE2\ 0xF6\ 0xBA\\ &&0xBD\ 0x0F\ 0x59\ 0x01\ 0x7F\ 0xEF\ 0x70\ 0x37\ 0xAC\ 0xA4\ 0x30\ 0x13\ 0xF8\ 0xFE\ 0x74\ 0xDE\\ &&0xF9\ 0xC2\ 0x99\ 0x65\ 0x4C\ 0x29\ 0xFF\ 0xC3\ 0xBB\ 0xD1\ 0x35\ 0x6E\ 0x3D\ 0x5D\ 0xE8\ 0xAE\\ &&0xCA\ 0x7E\ 0xBC\ 0x1D\ 0x9D\ 0x43\ 0xAF\ 0xF1\ 0x2B\ 0x79\ 0xAA\ 0x61\ 0x91\ 0xD2\ 0x45\ 0xD8\} \end{array} $$

1.2 C.2 S-box No 2 (N S =110,d e g(S)=7,A C(S) m a x =40,δ = 6)

$$\begin{array}{@{}rcl@{}} &&\{0x1D\ 0x2B\ 0xD9\ 0x88\ 0xA0\ 0xE9\ 0x7B\ 0xB3\ 0x6F\ 0x5D\ 0xB8\ 0x72\ 0xF1\ 0x04\ 0x95\ 0x6B\\ &&0xFB\ 0x15\ 0xE2\ 0x83\ 0x63\ 0x19\ 0x06\ 0x30\ 0xB4\ 0xBA\ 0x3D\ 0x07\ 0x86\ 0xBF\ 0xF9\ 0x6D\\ &&0xB1\ 0x17\ 0xC6\ 0x69\ 0x4E\ 0xB7\ 0x8D\ 0x59\ 0xFD\ 0x44\ 0xC0\ 0x5A\ 0x3C\ 0x31\ 0x27\ 0xF0\\ &&0x65\ 0xE1\ 0x62\ 0x3F\ 0xD2\ 0x4D\ 0xCF\ 0x78\ 0x7C\ 0x81\ 0x93\ 0xA2\ 0xB0\ 0x0B\ 0xFA\ 0x3B\\ &&0x94\ 0x25\ 0xC7\ 0x09\ 0x5C\ 0x22\ 0xF8\ 0x35\ 0x18\ 0x3E\ 0x97\ 0xEE\ 0x8A\ 0x52\ 0xE0\ 0x1A\\ &&0xB2\ 0x2C\ 0xD5\ 0xCE\ 0x5F\ 0x84\ 0x12\ 0x80\ 0x21\ 0xB9\ 0xD4\ 0xEC\ 0xDF\ 0xA9\ 0x47\ 0xAA\\ &&0xFE\ 0xDB\ 0xBC\ 0x9C\ 0x0E\ 0xB6\ 0xD3\ 0x1F\ 0x56\ 0xA6\ 0xEA\ 0xF4\ 0x2A\ 0x74\ 0x03\ 0xFC\\ &&0x01\ 0xC1\ 0x8C\ 0x0D\ 0x85\ 0x7E\ 0x6E\ 0x23\ 0x67\ 0x8B\ 0xC9\ 0xAD\ 0x42\ 0xA7\ 0xD1\ 0x9E\\ &&0x75\ 0x46\ 0xDE\ 0x77\ 0xAF\ 0xAE\ 0xC8\ 0x51\ 0x11\ 0x92\ 0x2E\ 0xE8\ 0x43\ 0x57\ 0xD6\ 0xE6\\ &&0x33\ 0xB5\ 0x20\ 0x73\ 0x87\ 0x37\ 0x48\ 0x9D\ 0x7A\ 0x2D\ 0x16\ 0x0A\ 0x4F\ 0x0F\ 0x32\ 0xBB\\ &&0x66\ 0x53\ 0x29\ 0xD8\ 0xC2\ 0xE5\ 0x58\ 0x0C\ 0xE3\ 0x39\ 0x7D\ 0x13\ 0x36\ 0xBE\ 0x7F\ 0x5E\\ &&0xDC\ 0xC5\ 0x90\ 0xF2\ 0x55\ 0xD0\ 0x49\ 0xBD\ 0xA3\ 0x10\ 0x98\ 0xF3\ 0xEF\ 0x60\ 0x6A\ 0x24\\ &&0x40\ 0xF5\ 0xA1\ 0x2F\ 0x61\ 0x96\ 0x71\ 0xAB\ 0x38\ 0x82\ 0x64\ 0x1B\ 0xA5\ 0xD7\ 0xC3\ 0x8F\\ &&0x14\ 0x3A\ 0x6C\ 0x34\ 0x4A\ 0xDA\ 0x45\ 0x02\ 0x99\ 0x91\ 0x05\ 0x26\ 0xCD\ 0xCB\ 0x41\ 0xEB\\ &&0xCC\ 0xF7\ 0xAC\ 0x50\ 0x79\ 0x1C\ 0xCA\ 0xF6\ 0x8E\ 0xE4\ 0x00\ 0x5B\ 0x08\ 0x68\ 0xDD\ 0x9B\\ &&0xFF\ 0x4B\ 0x89\ 0x28\ 0xA8\ 0x76\ 0x9A\ 0xC4\ 0x1E\ 0x4C\ 0x9F\ 0x54\ 0xA4\ 0xE7\ 0x70\ 0xED\} \end{array} $$

1.3 C.3 S-box No 3 (N S =112,d e g(S)=7,A C(S) m a x =32,δ = 6)

$$\begin{array}{@{}rcl@{}} &&\{0x1E\ 0x2B\ 0xD9\ 0x88\ 0xA0\ 0xE9\ 0x7B\ 0xB3\ 0x6F\ 0x5D\ 0xB8\ 0x72\ 0xF1\ 0x04\ 0x95\ 0x6B\\ &&0xFB\ 0x15\ 0xE2\ 0x83\ 0x63\ 0x19\ 0x06\ 0x30\ 0xB4\ 0xBA\ 0x3D\ 0x07\ 0x86\ 0xBF\ 0xF9\ 0x6D\\ &&0xB1\ 0x17\ 0xC6\ 0x69\ 0x4E\ 0xB7\ 0x8D\ 0x59\ 0xFD\ 0x44\ 0xC0\ 0x5A\ 0x3C\ 0x31\ 0x27\ 0xF0\\ &&0x65\ 0xE1\ 0x62\ 0x3F\ 0xD2\ 0x4D\ 0xCF\ 0x78\ 0x7C\ 0x81\ 0x93\ 0xA2\ 0xB0\ 0x0B\ 0xFA\ 0x3B\\ &&0x94\ 0x25\ 0xC7\ 0x09\ 0x5C\ 0x22\ 0xF8\ 0x35\ 0x18\ 0x3E\ 0x97\ 0xEE\ 0x8A\ 0x52\ 0xE0\ 0x1A\\ &&0xB2\ 0x2C\ 0x1D\ 0xCE\ 0x5F\ 0x84\ 0x12\ 0x80\ 0x21\ 0xB9\ 0xD4\ 0xEC\ 0xDF\ 0xA9\ 0x47\ 0xAA\\ &&0xFE\ 0xDB\ 0xBC\ 0x9C\ 0x0E\ 0xB6\ 0xD3\ 0x1F\ 0x56\ 0xA6\ 0xEA\ 0xF4\ 0xAB\ 0x74\ 0x03\ 0xFC\\ &&0x01\ 0xC1\ 0x8C\ 0x0D\ 0x85\ 0x7E\ 0x6E\ 0x23\ 0x67\ 0x8B\ 0xC9\ 0xAD\ 0x42\ 0xA7\ 0xD1\ 0x9E\\ &&0x75\ 0x46\ 0xDE\ 0x77\ 0xAF\ 0xAE\ 0xC8\ 0x51\ 0x11\ 0x92\ 0x2E\ 0xE8\ 0x43\ 0x57\ 0xD6\ 0xE6\\ &&0x33\ 0xB5\ 0x20\ 0x73\ 0x87\ 0x37\ 0x48\ 0x9D\ 0x7A\ 0x2D\ 0x16\ 0x0A\ 0x4F\ 0x0F\ 0x32\ 0xBB\\ &&0x66\ 0x53\ 0x29\ 0xD8\ 0xC2\ 0xE5\ 0x58\ 0x0C\ 0xE3\ 0x39\ 0x7D\ 0x13\ 0x36\ 0xBE\ 0x7F\ 0x5E\\ &&0xDC\ 0xC5\ 0x90\ 0xF2\ 0x55\ 0xD0\ 0x49\ 0xBD\ 0xA3\ 0x10\ 0x98\ 0xF3\ 0xEF\ 0x60\ 0x6A\ 0x24 \end{array} $$
$$\begin{array}{@{}rcl@{}} &&0x40\ 0xF5\ 0xA1\ 0x2F\ 0x61\ 0x96\ 0x71\ 0xD5\ 0x38\ 0x82\ 0x64\ 0x1B\ 0xA5\ 0xD7\ 0xC3\ 0x8F\\ &&0x14\ 0x3A\ 0x6C\ 0x34\ 0x4A\ 0xDA\ 0x45\ 0x02\ 0x99\ 0x91\ 0x05\ 0x26\ 0xCD\ 0xCB\ 0x41\ 0xEB\\ &&0xCC\ 0xF7\ 0xAC\ 0x50\ 0x79\ 0x1C\ 0xCA\ 0xF6\ 0x8E\ 0xE4\ 0x00\ 0x5B\ 0x08\ 0x68\ 0xDD\ 0x9B\\ &&0xFF\ 0x4B\ 0x89\ 0x28\ 0xA8\ 0x76\ 0x9A\ 0xC4\ 0x2A\ 0x4C\ 0x9F\ 0x54\ 0xA4\ 0xE7\ 0x70\ 0xED\} \end{array} $$

Appendix D: S-boxes generated with GA3

1.1 D.1 S-box No 1: N S =104,d e g(S)=6,A C(S) m a x =56,δ = 6

$$\begin{array}{@{}rcl@{}} &&\{0x90\ 0xA5\ 0x57 \ 0x06 \ 0x2E \ 0x67 \ 0xF5 0x3D \ 0xE1 \ 0xD3 \ 0x36 \ 0xFC \ 0x7F \ 0x8A \ 0x1B \ 0xE5\\ &&0x75\ 0x9B \ 0x6C 0x0D\ 0xED \ 0x97 \ 0x88 \ 0xBE \ 0x3A \ 0x34 \ 0xB3 \ 0x89 \ 0x08 \ 0x31 \ 0x77 \ 0xE3\\ &&0x3F\ 0x99 \ 0x48 \ 0xE7 \ 0xC0 \ 0x39 \ 0x03 \ 0xD7 \ 0x73 \ 0xCA \ 0x4E \ 0xD4 \ 0xB2 \ 0xBF \ 0xA9 \ 0x7E\\ &&0xEB\ 0x6F \ 0xEC \ 0xB1 \ 0x5C \ 0xC3 \ 0x41 \ 0xF6 \ 0xF2 \ 0x0F \ 0x1D \ 0x2C \ 0x3E \ 0x85 \ 0x74 \ 0x5A\\ &&0x1A\ 0xAB \ 0x49 \ 0x87 \ 0xD2 \ 0xAC \ 0x76 \ 0xBB \ 0x96 \ 0xB0 \ 0x19 \ 0x60 \ 0x04 \ 0xDC \ 0x6E \ 0x94\\ &&0x3C\ 0xA2 \ 0x93 \ 0x40 \ 0xD1 \ 0x0A \ 0x9C \ 0x0E \ 0xAF \ 0x37 \ 0x11\ 0x62 \ 0x51 \ 0x27 \ 0xC9 \ 0x24\\ &&0x70\ 0x55 \ 0x32 \ 0x12 \ 0x80 \ 0x38 \ 0x5D \ 0x91 \ 0xD8 \ 0x28 \ 0x64 \ 0x7A \ 0x25 \ 0xFA \ 0x8D \ 0x72\\ &&0x8F\ 0x4F \ 0x02 \ 0x83 \ 0x0B \ 0xF0 \ 0xE0 \ 0xAD \ 0xE9 \ 0x05 \ 0x47 \ 0x23 \ 0xCC \ 0x29 \ 0x5F \ 0x10\\ &&0xFB\ 0xC8 \ 0x50 \ 0xF9 \ 0x21 \ 0x20 \ 0x46 \ 0xDF \ 0x9F \ 0x1C \ 0xA0 \ 0x66 \ 0xCD \ 0xD9 \ 0x58 \ 0x68\\ &&0xBD\ 0x3B \ 0xAE \ 0xFD \ 0x09 \ 0xB9 \ 0xC6 \ 0x13 \ 0xF4 \ 0xA3 \ 0x98 \ 0x84 \ 0xC1 \ 0x81\ 0xBC \ 0x35\\ &&0xE8\ 0xDD \ 0xA7 \ 0x56 \ 0x4C \ 0x6B \ 0xD6 \ 0x82 \ 0x6D \ 0xB7 \ 0xF3 \ 0x9D \ 0xB8 \ 0x30 \ 0xF1 \ 0xD0\\ &&0x52\ 0x4B \ 0x1E \ 0x7C \ 0xDB \ 0x5E \ 0xC7 \ 0x33 \ 0x2D \ 0x9E \ 0x16 \ 0x7D \ 0x61 \ 0xEE \ 0xE4 \ 0xAA\\ &&0xCE\ 0x7B \ 0x2F \ 0xA1 \ 0xEF \ 0x18 \ 0xFF \ 0x5B \ 0xB6 \ 0x0C \ 0xEA \ 0x95 \ 0x2B \ 0x59 \ 0x4D \ 0x01\\ &&0x9A\ 0xB4 \ 0xE2 \ 0xBA \ 0xC4 \ 0x54 \ 0xCB \ 0x8C \ 0x17 \ 0x1F \ 0x8B \ 0xA8 \ 0x43 \ 0x45 \ 0xCF \ 0x65\\ &&0x42\ 0x79 \ 0x22 \ 0xDE \ 0xF7 \ 0x92 \ 0x44 \ 0x78 \ 0x00 \ 0x6A \ 0x8E \ 0xD5 \ 0x86 \ 0xE6 \ 0x53 \ 0x15\\ &&0x71\ 0xC5 \ 0x07 \ 0xA6 \ 0x26 \ 0xF8 \ 0x14 \ 0x4A \ 0xB5 \ 0xA4 \ 0x2A \ 0x69 \ 0xDA \ 0xC2 \ 0x63 \ 0xFE\} \end{array} $$

1.2 D.2 S-box No 2: N S =106,d e g(S)=7,A C(S) m a x =48,δ = 4

$$\begin{array}{@{}rcl@{}} &&\{0xC1 \ 0xC0 \ 0xAA \ 0x63 \ 0x4B \ 0x02 \ 0x90 \ 0x58 \ 0x84 \ 0xB6 \ 0x53 \ 0x99 \ 0x1A\ 0xEF \ 0x7E \ 0x80\\ &&0x10 \ 0xFE \ 0x09 \ 0x68 \ 0x88 \ 0xF2 \ 0xED \ 0xDB \ 0x5F \ 0x51 \ 0xD6\ 0xEC \ 0x6D \ 0x54 \ 0x12 \ 0x86\\ &&0x2D \ 0xFC \ 0x5A \ 0x82 \ 0xA5 \ 0x5C \ 0x66 \ 0xB2 \ 0x16 \ 0xAF \ 0x2B \ 0xB1 \ 0xD7 \ 0xDA \ 0xCC \ 0x1B\\ &&0x8E \ 0x0A \ 0x89 \ 0xD4 \ 0x39 \ 0xA6 \ 0x24 \ 0x93 \ 0x97 \ 0x6A \ 0x78 \ 0x49 \ 0x5B \ 0xE0 \ 0x11 \ 0xD0\\ &&0x7F \ 0xCE \ 0x2C \ 0xE2 \ 0xB7 \ 0xC9 \ 0x13 \ 0xDE \ 0xF3 \ 0xD5 \ 0x7C \ 0x05 \ 0x61 \ 0xB9 \ 0x0B \ 0xF1\\ &&0x59 \ 0xC7 \ 0xF6 \ 0x25 \ 0xF9 \ 0x6F \ 0xB4 \ 0x6B \ 0xCA \ 0x52 \ 0x3F \ 0x07 \ 0x34 \ 0x42 \ 0xAC \ 0x41\\ &&0x15 \ 0x30 \ 0x57 \ 0x77 \ 0xE5 \ 0x5D \ 0x38 \ 0xF4 \ 0xBD \ 0x4D \ 0x01 \ 0x1F \ 0x40 \ 0x9F \ 0xE8 \ 0x17\\ &&0xEA \ 0x2A \ 0x67 \ 0xE6 \ 0x6E \ 0x95 \ 0x85 \ 0xC8 \ 0x8C \ 0x60 \ 0x22 \ 0x46 \ 0xA9 \ 0x4C \ 0x3A \ 0x75\\ &&0x9E \ 0xAD \ 0x35 \ 0x9C \ 0x44 \ 0x45 \ 0x23 \ 0xBA \ 0xFA \ 0x79 \ 0xC5 \ 0x03 \ 0xA8 \ 0xBC \ 0x3D \ 0x0D\\ &&0xD8 \ 0x5E \ 0xCB \ 0x98 \ 0x6C \ 0xDC \ 0xA3 \ 0x76 \ 0x91 \ 0xC6 \ 0xFD \ 0xE1 \ 0xA4 \ 0xE4 \ 0xD9 \ 0x50\\ &&0x8D \ 0xB8 \ 0xC2 \ 0x33 \ 0x29 \ 0x0E \ 0xB3 \ 0xE7 \ 0x08 \ 0xD2 \ 0x96 \ 0xF8 \ 0xDD \ 0x55 \ 0x94 \ 0xB5\\ &&0x37 \ 0x2E \ 0x7B \ 0x19 \ 0xBE \ 0x3B \ 0xA2 \ 0x56 \ 0x48 \ 0xFB \ 0x73 \ 0x18 \ 0x04 \ 0x8B \ 0x81 \ 0xCF \end{array} $$
$$\begin{array}{@{}rcl@{}} &&0xAB \ 0x1E \ 0x4A \ 0xC4 \ 0x8A \ 0x7D \ 0x9A \ 0x3E \ 0xD3 \ 0x69 \ 0x8F \ 0xF0 \ 0x4E \ 0x3C \ 0x28 \ 0x64\\ &&0xFF \ 0xD1 \ 0x87 \ 0xDF \ 0xA1 \ 0x31 \ 0xAE \ 0xE9 \ 0x72 \ 0x7A \ 0xEE \ 0xCD \ 0x26 \ 0x20 \ 0x32 \ 0x00\\ &&0x27 \ 0x1C \ 0x47 \ 0xBB \ 0x92 \ 0xF7 \ 0x21 \ 0x1D \ 0x65 \ 0x0F \ 0xEB \ 0xB0 \ 0xE3 \ 0x83 \ 0x36 \ 0x70\\ &&0x14 \ 0xA0 \ 0x62 \ 0xC3 \ 0x43 \ 0x9D \ 0x71 \ 0x2F \ 0xF5 \ 0xA7 \ 0x74 \ 0xBF \ 0x4F \ 0x0C \ 0x9B \ 0x06\} \end{array} $$

1.3 D.3 S-box No 3: N S =108,d e g(S)=7,A C(S) m a x =48,δ = 4

$$\begin{array}{@{}rcl@{}} &&\{0x3B \ 0x2B \ 0x90 \ 0xC1 \ 0xE9 \ 0xA0 \ 0x32 \ 0xFA \ 0x26 \ 0x14 \ 0xF1 \ 0x57 \ 0xB8 \ 0x4D \ 0xDC \ 0x22\\ &&0xB2 \ 0x5C \ 0xAB \ 0xCA \ 0x2A \ 0x50 \ 0x4F \ 0x79 \ 0xFD \ 0xF3 \ 0x74 \ 0x4E \ 0xCF \ 0xF6 \ 0xB0 \ 0x24\\ &&0xF8 \ 0x5E \ 0x8F \ 0x20 \ 0x07 \ 0xFE \ 0xC4 \ 0x10 \ 0xB4 \ 0x0D \ 0x89 \ 0x13 \ 0x75 \ 0x78 \ 0x6E \ 0xB9\\ &&0x2C \ 0xA8 \ 0x62 \ 0x76 \ 0x9B \ 0x04 \ 0x86 \ 0x31 \ 0x35 \ 0xC8 \ 0xDA \ 0xEB \ 0xF9 \ 0x42 \ 0xB3 \ 0x72\\ &&0xDD \ 0x6C \ 0x8E \ 0x40 \ 0x15 \ 0x6B \ 0xB1 \ 0x7C \ 0x51 \ 0x77 \ 0xDE \ 0xA7 \ 0xC3 \ 0x1B \ 0xA9 \ 0x53\\ &&0xFB \ 0x65 \ 0x54 \ 0x87 \ 0x16 \ 0xCD \ 0x5B \ 0xC9 \ 0x68 \ 0xF0 \ 0x9D \ 0xA5 \ 0x96 \ 0xE0 \ 0x0E \ 0xE3\\ &&0xB7 \ 0x92 \ 0xF5 \ 0xD5 \ 0x47 \ 0xFF \ 0x9A \ 0x56 \ 0x1F \ 0xEF \ 0xA3 \ 0xBD \ 0xE2 \ 0x3D \ 0x4A \ 0xB5\\ &&0x48 \ 0x88 \ 0xC5 \ 0x44 \ 0xCC \ 0x37 \ 0x27 \ 0x6A \ 0x2E \ 0xC2 \ 0x80 \ 0xE4 \ 0x0B \ 0xEE \ 0x98 \ 0xD7\\ &&0x3C \ 0x0F \ 0x97 \ 0x3E \ 0xE6 \ 0xE7 \ 0x81 \ 0x18 \ 0x58 \ 0xDB \ 0x67 \ 0xA1 \ 0x0A \ 0x1E \ 0x9F \ 0xAF\\ &&0x7A \ 0xFC \ 0x69 \ 0x3A \ 0xCE \ 0x7E \ 0x01 \ 0xD4 \ 0x33 \ 0x64 \ 0x5F \ 0x43 \ 0x06 \ 0x46 \ 0x7B \ 0xF2\\ &&0x2F \ 0x1A \ 0x60 \ 0x91 \ 0x8B \ 0xAC \ 0x11 \ 0x45 \ 0xAA \ 0x70 \ 0x34 \ 0x5A \ 0x7F \ 0xF7 \ 0x36 \ 0x17\\ &&0x95 \ 0x8C \ 0xD9 \ 0xBB \ 0x1C \ 0x99 \ 0x00 \ 0xF4 \ 0xEA \ 0x59 \ 0xD1 \ 0xBA \ 0xA6 \ 0x29 \ 0x23 \ 0x6D\\ &&0x09 \ 0xBC \ 0xE8 \ 0x66 \ 0x28 \ 0xDF \ 0x38 \ 0x9C \ 0x71 \ 0xCB \ 0x2D \ 0x52 \ 0xEC \ 0x9E \ 0x8A \ 0xC6\\ &&0x5D \ 0x73 \ 0x25 \ 0x7D \ 0x03 \ 0x93 \ 0x0C \ 0x4B \ 0xD0 \ 0xD8 \ 0x4C \ 0x6F \ 0x84 \ 0x82 \ 0x08 \ 0xA2\\ &&0x85 \ 0xBE \ 0xE5 \ 0x19 \ 0x30 \ 0x55 \ 0x83 \ 0xBF \ 0xC7 \ 0xAD \ 0x49 \ 0x12 \ 0x41 \ 0x21 \ 0x94 \ 0xD2\\ &&0xB6 \ 0x02 \ 0xC0 \ 0x61 \ 0xE1 \ 0x3F \ 0xD3 \ 0x8D \ 0x63 \ 0x05 \ 0xD6 \ 0x1D \ 0xED \ 0xAE \ 0x39 \ 0xA4\} \end{array} $$

1.4 D.4 S-box No 4: N S =110,d e g(S)=7,A C(S) m a x =40,δ = 4

$$\begin{array}{@{}rcl@{}} &&\{0x5C \ 0xC0 \ 0x32 \ 0x63 \ 0x4B \ 0x02 \ 0x90 \ 0x58 \ 0x84 \ 0xB6 \ 0x53 \ 0x99 \ 0x1A\ 0xEF \ 0x7E \ 0x80\\ &&0x10 \ 0xFE \ 0x09 \ 0x68 \ 0x88 \ 0xF2 \ 0xED \ 0xDB \ 0x5F \ 0x51 \ 0xD6\ 0xEC \ 0x6D \ 0x54 \ 0x12 \ 0x86\\ &&0x5A \ 0xFC \ 0x2D \ 0x82 \ 0xA5 \ 0xBD \ 0x66 \ 0xB2 \ 0x16\ 0xAF \ 0x2B \ 0xB1 \ 0xD7 \ 0xDA \ 0xCC \ 0x1B\\ &&0x8E \ 0x0A \ 0x89 \ 0xD4 \ 0x39 \ 0xA6 \ 0x24\ 0x93 \ 0x97 \ 0x6A \ 0x78 \ 0x49 \ 0x5B \ 0xE0 \ 0x11 \ 0xD0\\ &&0x7F \ 0xCE \ 0x2C \ 0xE2 \ 0xB7\ 0xC9 \ 0x13 \ 0xDE \ 0xF3 \ 0xD5 \ 0x7C \ 0x05 \ 0x61 \ 0xB9 \ 0x0B \ 0xF1\\ &&0x59 \ 0xC7 \ 0xC1\ 0x25 \ 0xB4 \ 0x6F \ 0xF9 \ 0x6B \ 0xCA \ 0x52 \ 0x3F \ 0x07 \ 0x34 \ 0x42 \ 0xAC \ 0x41\\ &&0x15\ 0x30 \ 0x57 \ 0x77 \ 0xE5 \ 0x5D \ 0x38 \ 0xF4 \ 0xF6 \ 0x4D \ 0x01 \ 0x1F \ 0x40 \ 0x9F \ 0xE8 \ 0x17\\ &&0xEA \ 0x2A \ 0x67 \ 0xE6 \ 0x6E \ 0x95 \ 0x85 \ 0xC8 \ 0x8C \ 0x60 \ 0x22 \ 0x46 \ 0xA9 \ 0x4C \ 0x3A \ 0x75\\ &&0x9E \ 0xAD \ 0x35 \ 0x9C \ 0x44 \ 0x45 \ 0x23 \ 0xBA \ 0xFA \ 0x79 \ 0xC5 \ 0x03 \ 0xA8 \ 0xBC \ 0x3D \ 0x0D\\ &&0xD8 \ 0x5E \ 0xCB \ 0x98 \ 0x6C \ 0xDC \ 0xA3 \ 0x76 \ 0x91 \ 0xC6 \ 0xFD \ 0xE1 \ 0xA4 \ 0xE4 \ 0xD9 \ 0x50\\ &&0x8D \ 0xB8 \ 0xC2 \ 0x33 \ 0x29 \ 0x0E \ 0xB3 \ 0xE7 \ 0x08 \ 0xD2 \ 0x96 \ 0xF8 \ 0xDD \ 0x55 \ 0x94 \ 0xB5 \end{array} $$
$$\begin{array}{@{}rcl@{}} &&0x37 \ 0x2E \ 0x7B \ 0x19 \ 0xBE \ 0x3B \ 0xA2 \ 0x56 \ 0x48 \ 0xFB \ 0x73 \ 0x18 \ 0x04 \ 0x8B \ 0x81 \ 0xCF\\ &&0xAB \ 0x1E \ 0x4A \ 0xC4 \ 0x8A \ 0x7D \ 0x9A \ 0x3E \ 0xD3 \ 0x69 \ 0x8F \ 0xF0 \ 0x4E \ 0x3C \ 0x28 \ 0x64\\ &&0xFF \ 0xD1 \ 0x87 \ 0xDF \ 0xA1 \ 0x31 \ 0xAE \ 0xE9 \ 0x72 \ 0x7A \ 0xEE \ 0xCD \ 0x26 \ 0x20 \ 0xAA \ 0x00\\ &&0x27 \ 0x1C \ 0x47 \ 0xBB \ 0x92 \ 0xF7 \ 0x21 \ 0x1D \ 0x65 \ 0x0F \ 0xEB \ 0xB0 \ 0xE3 \ 0x83 \ 0x36 \ 0x70\\ &&0x14 \ 0xA0 \ 0x62 \ 0xC3 \ 0x43 \ 0x9D \ 0x71 \ 0x2F \ 0xF5 \ 0xA7 \ 0x74 \ 0xBF \ 0x4F \ 0x0C \ 0x9B \ 0x06\} \end{array} $$

1.5 D.5 S-box No 5: N S =112,d e g(S)=7,A C(S) m a x =32,δ = 6

$$\begin{array}{@{}rcl@{}} &&\{0x2F \ 0xC0 \ 0x32 \ 0x63 \ 0x4B \ 0x02 \ 0x90 \ 0x58 \ 0x84 \ 0xB6 \ 0x53 \ 0x99 \ 0x1A\ 0xEF \ 0x7E \ 0x80\\ &&0x10 \ 0xFE \ 0x09 \ 0x68 \ 0x88 \ 0xF2 \ 0xED \ 0xDB \ 0x5F \ 0x51 \ 0xD6\ 0xEC \ 0x6D \ 0x54 \ 0x12 \ 0x86\\ &&0x5A \ 0xFC \ 0x2D \ 0x82 \ 0xA5 \ 0x5C \ 0x66 \ 0xB2 \ 0x16\ 0xAF \ 0x2B \ 0xB1 \ 0xD7 \ 0xDA \ 0xCC \ 0x1B\\ &&0x8E \ 0x0A \ 0x89 \ 0xD4 \ 0x39 \ 0xA6 \ 0x24\ 0x93 \ 0x97 \ 0x6A \ 0x78 \ 0x49 \ 0x5B \ 0xE0 \ 0x11 \ 0xD0\\ &&0x7F \ 0xCE \ 0x2C \ 0xE2 \ 0xB7\ 0xC9 \ 0x13 \ 0xDE \ 0xF3 \ 0xD5 \ 0x7C \ 0x05 \ 0x61 \ 0xB9 \ 0x0B \ 0xF1\\ &&0x59 \ 0xC7 \ 0xF6\ 0x25 \ 0xB4 \ 0x6F \ 0xF9 \ 0x6B \ 0xCA \ 0x52 \ 0x3F \ 0x07 \ 0x34 \ 0x42 \ 0xAC \ 0x41\\ &&0x15\ 0x30 \ 0x57 \ 0x77 \ 0xE5 \ 0x5D \ 0x38 \ 0xF4 \ 0xBD \ 0x4D \ 0x01 \ 0x1F \ 0x40 \ 0x9F \ 0xE8 \ 0x17\\ &&0xEA \ 0x2A \ 0x67 \ 0xE6 \ 0x6E \ 0x95 \ 0x85 \ 0xC8 \ 0x8C \ 0x60 \ 0x22 \ 0x46 \ 0xA9 \ 0x4C \ 0x3A \ 0x75\\ &&0x9E \ 0xAD \ 0x35 \ 0x9C \ 0x44 \ 0x45 \ 0x23 \ 0xBA \ 0xFA \ 0x79 \ 0xC5 \ 0x03 \ 0xA8 \ 0xBC \ 0x3D \ 0x0D\\ &&0xD8 \ 0x5E \ 0xCB \ 0x98 \ 0x6C \ 0xDC \ 0xA3 \ 0x76 \ 0x91 \ 0xC6 \ 0xFD \ 0xE1 \ 0xA4 \ 0xE4 \ 0xD9 \ 0x50\\ &&0x8D \ 0xB8 \ 0xC2 \ 0x33 \ 0x29 \ 0x0E \ 0xB3 \ 0xE7 \ 0x08 \ 0xD2 \ 0x96 \ 0xF8 \ 0xDD \ 0x55 \ 0x94 \ 0xB5\\ &&0x37 \ 0x2E \ 0x7B \ 0x19 \ 0xBE \ 0x3B \ 0xA2 \ 0x56 \ 0x48 \ 0xFB \ 0x73 \ 0x18 \ 0x04 \ 0x8B \ 0x81 \ 0xCF\\ &&0xAB \ 0x1E \ 0x4A \ 0xC4 \ 0x8A \ 0x7D \ 0x9A \ 0x3E \ 0xD3 \ 0x69 \ 0x8F \ 0xF0 \ 0x4E \ 0x3C \ 0x28 \ 0x64\\ &&0xFF \ 0xD1 \ 0x87 \ 0xDF \ 0xA1 \ 0x31 \ 0xAE \ 0xE9 \ 0x72 \ 0x7A \ 0xEE \ 0xCD \ 0x26 \ 0x20 \ 0xAA \ 0x00\\ &&0x27 \ 0x1C \ 0x47 \ 0xBB \ 0x92 \ 0xF7 \ 0x21 \ 0x1D \ 0x65 \ 0x0F \ 0xEB \ 0xB0 \ 0xE3 \ 0x83 \ 0x36 \ 0x70\\ &&0x14 \ 0xA0 \ 0x62 \ 0xC3 \ 0x43 \ 0x9D \ 0x71 \ 0xC1 \ 0xF5 \ 0xA7 \ 0x74 \ 0xBF \ 0x4F \ 0x0C \ 0x9B \ 0x06\} \end{array} $$

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Ivanov, G., Nikolov, N. & Nikova, S. Reversed genetic algorithms for generation of bijective s-boxes with good cryptographic properties. Cryptogr. Commun. 8, 247–276 (2016). https://doi.org/10.1007/s12095-015-0170-5

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