A Note on 5-bit Quadratic Permutations’ Classification

Božilov, Dušan; Bilgin, Begül; Sahin, Hacı Ali

doi:10.13154/tosc.v2017.i1.398-404

Cited by 3 publications

(3 citation statements)

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“…Then, using the classification of the affine equivalent classes for 3 and 4 bit permutations, Bilgin et al provided threshold implementations for all of these classes. Later, Bozilov et al [14] and De Meyer and Bilgin [30] provided threshold implementations of 5 and 6 bit quadratic permutations. In Table 5.1 we summarise the known results on threshold implementations for 3 and 4 bit permutations and 5 bit quadratic permutations (excluding linear permutations which are known to admit a TI with t+1 = 2 shares) [12,14].…”

Section: Discussing Known Casesmentioning

confidence: 99%

“…Analysing all the data available in [12,14,30], we observed that all permutations admitting t + 1 shares, except one, have 0 nonlinearity (have linear component functions making them the worst with respect to linear cryptanalysis). The only exception C 4 301 (notation 16…”

Section: Two Conjectures On Nonexistence Of Threshold Implementations...mentioning

confidence: 99%

“…There is currently no known universal construction of a threshold implementation for an arbitrary bijective S-box of any size. Instead, research on threshold implementations focuses on finding solutions for small sizes [12,14]. In particular, there are several known TI sharings of the AES S-box, but the most relevant for our considerations is the one started by Wegener and Moradi [15] who gave a decomposition of the AES S-box into two cubic power functions, namely x → x 26 and x → x 49 .…”

Section: Introductionmentioning

confidence: 99%

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An Optimal Universal Construction for the Threshold Implementation of Bijective S-Boxes

Piccione,

Andreoli,

Budaghyan

et al. 2023

IEEE Trans. Inform. Theory

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Threshold implementation is a method based on secret sharing to secure cryptographic ciphers (and in particular S-boxes) against differential power analysis side-channel attacks which was proposed by Nikova, Rechberger, and Rijmen in 2006. Until now, threshold implementations were only constructed for specific types of functions and some small S-boxes, but no generic construction was ever presented. In this paper, we present the first universal threshold implementation with t + 2 shares that is applicable to any bijective S-box, where t is its algebraic degree (or is larger than the algebraic degree). While being universal, our construction is also optimal with respect to the number of shares, since the theoretically smallest possible number, t+1, is not attainable for some bijective S-boxes. Our results enable low latency secure hardware implementations without the need for additional randomness. In particular, we apply this result to find two uniform sharings of the AES S-box. The first sharing is obtained by using the threshold implementation of the inversion in F 2 8 and the second by using two threshold implementations of two cubic power permutations that decompose the inversion. Area and performance figures for hardware implementations are provided.

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Section: Discussing Known Casesmentioning

confidence: 99%

Section: Two Conjectures On Nonexistence Of Threshold Implementations...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An Optimal Universal Construction for the Threshold Implementation of Bijective S-Boxes

Piccione,

Andreoli,

Budaghyan

et al. 2023

IEEE Trans. Inform. Theory

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Generating Cryptographic S-Boxes Using the Reinforcement Learning

Kim

Heo

et al. 2021

IEEE Access

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Substitution boxes (S-boxes) are essential components of many cryptographic primitives. The Dijkstra algorithm, SAT solvers, and heuristic methods have been used to find bitsliced implementations of S-boxes. However, it is difficult to apply these methods for 8-bit S-boxes because of their size. Therefore, in order to implement these S-boxes so that the countermeasure of side-channel attack can be applied efficiently, using structures such as Feistel, Lai-Massey, and MISTY that can be bitsliced implemented with a small number of nonlinear operations has been widely used. Since S-boxes constructed with structures consist of small S-boxes and have specific designs, there are limitations to their cryptographic security and efficiency. In this paper, we propose a new method for generating S-boxes by stacking bitwise operations from the identity function, an approach which is different from existing methods. This method can be expressed in Markov decision process, and reinforcement learning is a suitable solver for Markov decision process. Our goal is to train this method to an agent through reinforcement learning to generate S-boxes to which the masking scheme, which is a countermeasure of side-channel attack, can be efficiently applied. In particular, our method provided various S-boxes superior or comparable to existing S-boxes. We produced 8-bit S-boxes with differential uniformity 16 (resp. 32) and linearity 128 (resp. 128), generated with nine (resp. eight) nonlinear operations, for the first time. To our best knowledge, this is the first study to construct cryptographic S-Box by incorporating reinforcement learning. INDEX TERMSS-box, Masking efficiency, Reinforcement learning, Bitsliced implementation, Linearity, Differential uniformity I. INTRODUCTION To apply security applications to mobile and embedded platforms, lightweight and efficient cryptographic primitives are required. Substitution boxes (S-boxes) are representative nonlinear functions, giving cryptographic primitives Shannon's confusion property [1]. Therefore, finding S-boxes with sufficient security and efficiency is an important issue for the designers of cryptographic algorithms. Side-channel attacks, first published by Kocher in 1996 [2], can draw secret information of cryptographic algorithms from side-channel leakages such as electromagnetic emissions and power consumption. Since mathematical cryptanalysis alone cannot guarantee the security of cryptographic primitives against side-channel attacks, various counter measures have been proposed. Techniques to randomize the intermediate values of ciphers are widely used, and among them, a higher-order Boolean masking techniqueis the most popular approach. 18The method of implementing a S-box as Table Look Up 19 (TLU) requires table storage space, and especially when ap-20 plying the masking scheme, even more flip-flops are required.

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