A characterisation of S-box fitness landscapes in cryptography

Substitution Boxes (S-boxes) are nonlinear objects often used in the design of cryptographic algorithms. The design of high quality S-boxes is an interesting problem that attracts a lot of attention. Many attempts have been made in recent years to use heuristics to design S-boxes, but the results were often far from the previously known best obtained ones. Unfortunately, most of the effort went into exploring different algorithms and fitness functions while little attention has been given to the understanding why this problem is so difficult for heuristics. In this paper, we conduct a fitness landscape analysis to better understand why this problem can be difficult. Among other, we find that almost each initial starting point has its own local optimum, even though the networks are highly interconnected

Current implementation of advance encryption standard (AES) S-Box

Although the attack on cryptosystem is still not severe, the development of the scheme is stillongoing especially for the design of S-Box. Two main approach has beenused, which areheuristic method and algebraic method. Algebraic method as in current AES implementationhas been proven to be the most secure S-Box design to date. This review paper willconcentrate on two kinds of method of constructing AES S-Box, which are algebraic approachand heuristic approach. The objective is to review a method of constructing S-Box, which arecomparable or close to the original construction of AES S-Box especially for the heuristicapproach. Finally, all the listed S-Boxes from these two methods will be compared in terms oftheir security performance which is nonlinearity and differential uniformity of the S-Box. Thefinding may offer the potential approach to develop a new S-Box that is better than theoriginal one.Keywords: block cipher; AES; S-Bo

On the Evolution of Boomerang Uniformity in Cryptographic S-boxes

S-boxes are an important primitive that help cryptographic algorithms to be resilient against various attacks. The resilience against specific attacks can be connected with a certain property of an S-box, and the better the property value, the more secure the algorithm. One example of such a property is called boomerang uniformity, which helps to be resilient against boomerang attacks. How to construct S-boxes with good boomerang uniformity is not always clear. There are algebraic techniques that can result in good boomerang uniformity, but the results are still rare. In this work, we explore the evolution of S-boxes with good values of boomerang uniformity. We consider three different encodings and five S-box sizes. For sizes $4\times 4$ and $5\times 5$, we manage to obtain optimal solutions. For $6\times 6$, we obtain optimal boomerang uniformity for the non-APN function. For larger sizes, the results indicate the problem to be very difficult (even more difficult than evolving differential uniformity, which can be considered a well-researched problem).Comment: 15 pages, 3 figures, 4 table

Bounds on the nonlinearity of differentially uniform functions by means of their image set size, and on their distance to affine functions

We revisit and take a closer look at a (not so well known) result of a 2017 paper, showing that the differential uniformity of any vectorial function is bounded from below by an expression depending on the size of its image set. We make explicit the resulting tight lower bound on the image set size of differentially δ -uniform functions (which is the only currently known non-trivial lower bound on the image set size of such functions). We also significantly improve an upper bound on the nonlinearity of vectorial functions obtained in the same reference and involving their image set size. We study when the resulting bound is sharper than the covering radius bound. We obtain as a by-product a lower bound on the Hamming distance between differentially δ -uniform functions and affine functions, which we improve significantly with a second bound. This leads us to study what can be the maximum Hamming distance between vectorial functions and affine functions. We provide an upper bound which is slightly sharper than a bound by Liu, Mesnager and Chen when m<n , and a second upper bound, which is much stronger in the case (happening in practice) where m is near n ; we study the tightness of this latter bound; this leads to an interesting question on APN functions, which we address (negatively). We finally derive an upper bound on the nonlinearity of vectorial functions by means of their Hamming distance to affine functions and make more precise the bound on the differential uniformity which was the starting point of the paper.acceptedVersio