Group law computations on Jacobians of hyperelliptic curves

We derive an explicit method of computing the composition step in Cantor’s algorithm for group operations on Jacobians of hyperelliptic curves. Our technique is inspired by the geometric description of the group law and applies to hyperelliptic curves of arbitrary genus. While Cantor’s general composition involves arithmetic in the polynomial ring F_q[x], the algorithm we propose solves a linear system over the base field which can be written down directly from the Mumford coordinates of the group elements. We apply this method to give more efficient formulas for group operations in both affine and projective coordinates for cryptographic systems based on Jacobians of genus 2 hyperelliptic curves in general form

From Substitution Box To Threshold

With the escalating demand for lightweight ciphers as well as side channel protected implementation of those ciphers in recent times, this work focuses on two aspects. First, we present a tool for automating the task of finding a Threshold Implementation (TI) of a given Substitution Box (SBox). Our tool returns `with decomposition\u27 and `without decomposition\u27 based TI. The `with decomposition\u27 based implementation returns a combinational SBox; whereas we get a sequential SBox from the `without decomposition\u27 based implementation. Despite being high in demand, it appears that this kind of tool has been missing so far. Second, we show an algorithmic approach where a given cipher implementation can be tweaked (without altering the cipher specification) so that its TI cost can be significantly reduced. We take the PRESENT cipher as our case study (our methodology can be applied to other ciphers as well). Indeed, we show over 31 percent reduction in area and over 52 percent reduction in depth compared to the basic threshold implementation

Generating S-Boxes from Semi-fields Pseudo-extensions

Block ciphers, such as the AES, correspond to a very important family of secret-key cryptosystems. The security of such systems partly relies on what is called the S-box. This is a vectorial Boolean function f : F n 2 ֒→ F n 2 , where n is the size of the blocks. It is often the only non linear opera-tion in the algorithm. The most well-known attacks against block ciphers algorithms are the known-plaintext attacks called differential cryptanal-ysis [4, 10] and linear cryptanalysis [11]. To protect such cryptosystems against linear and differential attacks, S-boxes are designed to fulfill some cryptographic criteria (balancedness, high nonlinearity, high algebraic de-gree, avalanche, or transparency [2, 12]) and are usually defined on finite fields, like F2n [7, 3]. Unfortunately, it seems difficult to find good S-Boxes, at least for bijective ones: random generation does not work [8, 9] and the one used in the AES or Camellia are actually variations around a single function, the inverse function in F2n . Would the latter function have an unforeseen weakness (for instance if more practical algebraic attacks are developped), it would be desirable to have some replacement candidates. For that matter, we propose to weaken a little bit the algebraic part of the design of S-Boxes and use finite semi-fields instead of finite fields to build such S-Boxes. Finite semi-fields relax the associativity and com-mutativity of the multiplication law. While semi-fields of a given order are unique up to isomorphism, on the contrary semi-fields of a given order can be numerous: nowadays, on the one hand, it is for instance easy to generate all the 36 semi-fields of order 2 4 , but, on the other hand, it is not even known how many semi-fields are there of order 2 8 . Therefore, we propose to build S-Boxes via semi-fields pseudo extensions of the form S 2 2 4 , where S 2 4 is any semi-field of order 2 4 , and mimic in this structure the use of the inverse function in a finite field. We report here the construction of 10827 S-Boxes, 7052 non CCZ-equivalent, with maximal nonlinearity, differential invariants, degrees and bit interdependency. Among the latter 2963 had fix points, and among the ones without fix points, 3846 had the avalanche level of AES and 243 1 the better avalanche level of Camellia. Among the latter 232 have a better transparency level than the inverse function on a finite field

Reversed Genetic Algorithms for Generation of Bijective S-boxes with Good Cryptographic Properties

Often S-boxes are the only nonlinear component 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 $(8 \times 8)$ to $(16 \times 16)$, 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

An Improved Affine Equivalence Algorithm for Random Permutations

In this paper we study the affine equivalence problem, where given two functions $\vec{F},\vec{G}: \{0,1\}^n \rightarrow \{0,1\}^n$, the goal is to determine whether there exist invertible affine transformations $A_1,A_2$ over $GF(2)^n$ such that $\vec{G} = A_2 \circ \vec{F} \circ A_1$. Algorithms for this problem have several well-known applications in the design and analysis of Sboxes, cryptanalysis of white-box ciphers and breaking a generalized Even-Mansour scheme. We describe a new algorithm for the affine equivalence problem and focus on the variant where $\vec{F},\vec{G}$ are permutations over $n$-bit words, as it has the widest applicability. The complexity of our algorithm is about $n^3 2^n$ bit operations with very high probability whenever $\vec{F}$ (or $\vec{G})$ is a random permutation. This improves upon the best known algorithms for this problem (published by Biryukov et al. at EUROCRYPT 2003), where the first algorithm has time complexity of $n^3 2^{2n}$ and the second has time complexity of about $n^3 2^{3n/2}$ and roughly the same memory complexity. Our algorithm is based on a new structure (called a \emph{rank table}) which is used to analyze particular algebraic properties of a function that remain invariant under invertible affine transformations. Besides its standard application in our new algorithm, the rank table is of independent interest and we discuss several of its additional potential applications

C-DIFFERENTIALS AND GENERALIZED CRYPTOGRAPHIC PROPERTIES OF VECTORIAL BOOLEAN AND P-ARY FUNCTIONS

This dissertation investigates a newly defined cryptographic differential, called a c-differential, and its relevance to the nonlinear substitution boxes of modern symmetric block ciphers. We generalize the notions of perfect nonlinearity, bentness, and avalanche characteristics of vectorial Boolean and p-ary functions using the c-derivative and a new autocorrelation function, while capturing the original definitions as special cases (i.e., when c=1). We investigate the c-differential uniformity property of the inverse function over finite fields under several extended affine transformations. We demonstrate that c-differential properties do not hold in general across equivalence classes typically used in Boolean function analysis, and in some cases change significantly under slight perturbations. Thus, choosing certain affine equivalent functions that are easy to implement in hardware or software without checking their c-differential properties could potentially expose an encryption scheme to risk if a c-differential attack method is ever realized. We also extend the c-derivative and c-differential uniformity into higher order, investigate some of their properties, and analyze the behavior of the inverse function's second order c-differential uniformity. Finally, we analyze the substitution boxes of some recognizable ciphers along with certain extended affine equivalent variations and document their performance under c-differential uniformity.Commander, United States NavyApproved for public release. Distribution is unlimited