Perfect nonlinear functions and cryptography

Blondeau, Céline; Nyberg, Kaisa

doi:10.1016/j.ffa.2014.10.007

Cited by 57 publications

(38 citation statements)

References 86 publications

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“…Designing powerful S-boxes is an important critereon for realisation of secure cryptosystems and it is a major component of nonlinear transformations, which are the fulcrum of confusion and diffusion analysis for assessing well-designed ciphers 41 . Therefore, designing S-boxes based on secure mechanisms plays an important part in modern cryptographic tasks 42,43 . Consequently, it is widely investigated.…”

Section: S-box Constructionmentioning

confidence: 99%

Quantum-inspired cascaded discrete-time quantum walks with induced chaotic dynamics and cryptographic applications

El‐Latif

Abd-El-Atty

Amin

et al. 2020

Sci Rep

116

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Designing efficient and secure cryptosystems has been a preoccupation for many scientists and engineers for a long time wherein they use chaotic systems to design new cryptosystems. While one dimensional (1-D) chaotic maps possess powerful properties compared to higher dimension ones, they are vulnerable to various attacks due to their small key space, chaotic discontinuous ranges, and degradation in chaotic dynamical behaviours. Moreover, when simulated on a computer, every such chaotic system produces a periodic cycle. Meanwhile, quantum random walks exhibit the potential for deployment in efficient cryptosystem design, which makes it an excellent solution for this problem. In this context, we present a new method for constructing substitution boxes (S-boxes) based on cascaded quantum-inspired quantum walks and chaos inducement. the performance of the proposed S-box scheme is investigated via established S-box evaluation criterion and outcomes suggest that the constructed S-box has significant qualities for viable applications information security. Further, we present an efficient scheme for pseudo-random numbers generation (PRNG) whose sustainability over long periods remedies the periodicity problem associated with traditional cryptographic applications. Furthermore, by combining the two mechanisms, an atypical image encryption scheme is introduced. Simulation results and analysis validate that the proposed image encryption algorithm will offer gains in many cryptographic applications. Chaotic systems have attracted a great deal of attention across different scientific and engineering disciplines, especially in designing new cryptosystems and cryptanalysis. A chaotic system is an evolution map of a deterministic dynamical system that reconstructs the state of a system S 0 to a new state S 1 depending on the initial state of S 0 , a control parameter C, and time T 1. Chaotic maps exhibit the desired properties of ergodicity, unpredictability, and sensitivity to their control parameter(s) and initial value(s) that satisfy the requirements for cryptosystem confusion-diffusion properties 2-4. In fact, an inappropriate initial control parameter of a chaotic system can lead to non-chaotic behaviours, which implies the reduction in nonlinearity levels as well as circumvention of insecurity pitfalls 5,6. Currently, chaotic dynamical systems play a vital role in designing modern cryptographic applications, such as constructing S-boxes, generating pseudo-random numbers, designing image encryption algorithms and so on 7-16 , which are based on the unproven assumptions pertaining to computational complexity and that their constructions are based on mathematical models. However, with the development of quantum technologies, some of these traditional security mechanisms, and cryptographic applications may be effortlessly violated and abused 17-19. Among the computational models developed in quantum computation, quantum walks (QWs), which is a universal model of quantum computation that has been traditionally employed to de...

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Section: S-box Constructionmentioning

confidence: 99%

Quantum-inspired cascaded discrete-time quantum walks with induced chaotic dynamics and cryptographic applications

El‐Latif

Abd-El-Atty

Amin

et al. 2020

Sci Rep

116

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“…First, we recall the definition of CCZequivalence (where CCZ stands for Carlet-Charpin-Zinoviev [13]) as it is defined e.g. in [14].…”

Section: The Butterfly Structurementioning

confidence: 99%

“…An up to date overview of known APN functions can be found in [14]. As APN functions operating on an even number of bits are still to be found for even block sizes larger than 6, differentially 4-uniform permutations have received a lot of attention from researchers.…”

Section: Differentially 4-uniform Permutations Of Larger Blocksmentioning

confidence: 99%

Cryptanalysis of a Theorem: Decomposing the Only Known Solution to the Big APN Problem

Perrin

Udovenko

Biryukov

2016

Advances in Cryptology – CRYPTO 2016

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Abstract. The existence of Almost Perfect Non-linear (APN) permutations operating on an even number of bits has been a long standing open question until Dillon et al., who work for the NSA, provided an example on 6 bits in 2009. In this paper, we apply methods intended to reverse-engineer S-Boxes with unknown structure to this permutation and find a simple decomposition relying on the cube function over (2 3 ). More precisely, we show that it is a particular case of a permutation structure we introduce, the butterfly. Such butterflies are 2 -bit mappings with two CCZ-equivalent representations: one is a quadratic non-bijective function and one is a degree + 1 permutation. We show that these structures always have differential uniformity at most 4 when is odd. A particular case of this structure is actually a 3-round Feistel Network with similar differential and linear properties. These functions also share an excellent non-linearity for = 3, 5, 7. Furthermore, we deduce a bitsliced implementation and significantly reduce the hardware cost of a 6-bit APN permutation using this decomposition, thus simplifying the use of such a permutation as building block for a cryptographic primitive.

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“…has at most k solutions. Having applications in cryptography, differentially 2-uniform functions F , called almost perfect nonlinear (APN) functions are of particular interest, see [1,15]. In some applications it is required that F is a permutation.…”

Section: Introductionmentioning

confidence: 99%

Determining the Walsh spectra of Taniguchi's and related APN-functions

Anbar

Kalaycı

Meidl

2019

Finite Fields and Their Applications

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We introduce a method based on Bezout's theorem on intersection points of two projective plane curves, for determining the nonlinearity of some classes of quadratic functions on F 2 2m . Among those are the functions of Taniguchi 2019, Carlet 2011, and Zhou and Pott 2013, all of which are APN under certain conditions. This approach helps to understand why the majority of the functions in those classes have solely bent and semibent components, which in the case of APN functions is called the classical spectrum. More precisely, we show that all Taniguchi functions have the classical spectrum independent from being APN. We determine the nonlinearity of all functions belonging to Carlet's class and to the class of Zhou and Pott, which also confirms with comparatively simple proofs earlier results on the Walsh spectrum of APN-functions in these classes. Using the Hasse-Weil bound, we show that some simple sufficient conditions for the APN-ness of the Zhou-Pott functions, which are given in the original paper, are also necessary.

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Perfect nonlinear functions and cryptography

Cited by 57 publications

References 86 publications

Quantum-inspired cascaded discrete-time quantum walks with induced chaotic dynamics and cryptographic applications

Quantum-inspired cascaded discrete-time quantum walks with induced chaotic dynamics and cryptographic applications

Cryptanalysis of a Theorem: Decomposing the Only Known Solution to the Big APN Problem

Determining the Walsh spectra of Taniguchi's and related APN-functions

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