Continued fractions of Laurent series with partial quotients from a given set
1. Introduction. Van der Poorten and Shallit's paper [10] begins: " It is notorious that it is damnably difficult to explicitly compute the continued fraction of a quantity presented in some other form ". The quantity is usually presented either as a power series or as the root of a specific equation. There has been some success in the former case for continued fractions of real numbers, such as Euler's famous continued fraction for e [11] and more recent work [10] on " folded " continued fractions; however, other than the well-known results for quadratic real numbers, the only success with the latter has been for continued fractions of Laurent series rather than real numbers. In this paper we continue this line of investigation. We consider families of continued fractions of Laurent series whose partial quotients all lie in a given set. Following ideas of Baum and Sweet [2], we show that one may describe the zeros of certain collections of equations in terms of such families. The paragraphs that follow introduce the notation and definitions necessary to give a fuller description of our results. Let F q be the finite field with q elements and L q denote the field of formal Laurent series in x
Towards factoring in $${SL(2,\,\mathbb{F}_{2^n})}$$
The security of many cryptographic protocols relies on the hardness of some computational problems. Besides discrete logarithm or integer factorization, other problems are regularly proposed as potential hard problems. The factorization problem in finite groups is one of them. Given a finite group G, a set of generators generators for this group and an element $${g\in G}$$, the factorization problem asks for a “short” representation of g as a product of the generators. The problem is related to a famous conjecture of Babai on the diameter of Cayley graphs. It is also motivated by the preimage security of Cayley hash functions, a particular kind of cryptographic hash functions. The problem has been solved for a few particular generator sets, but essentially nothing is known for generic generator sets. In this paper, we make significant steps towards a solution of the factorization problem in the group $${G:=\,SL(2,\,\mathbb{F}_{2^n})}$$, a particularly interesting group for cryptographic applications. To avoid considering all generator sets separately, we first give a new reduction tool that allows focusing on some generator sets with a “nice” special structure. We then identify classes of trapdoor matrices for these special generator sets, such that the factorization of a single one of these matrices would allow efficiently factoring any element in the group. Finally, we provide a heuristic subexponential time algorithm that can compute subexponential length factorizations of any element for any pair of generators of $${SL(2,\,\mathbb{F}_{2^n})}$$. Our results do not yet completely remove the factorization problem in $${SL(2,\,\mathbb{F}_{2^n})}$$ from the list of potential hard problems useful for cryptography. However, we believe that each one of our individual results is a significant step towards a polynomial time algorithm for factoring in $${SL(2,\,\mathbb{F}_{2^n})}$$.
FURUTA INEQUALITY AND ITS RELATED TOPICS
This article is devoted to a brief survey of Furuta inequality and its related topics. It consists of 4 sections: 1. From Lowner-Heinz inequality to Furuta inequality, 2. Ando-Hiai inequality, 3. Grand Furuta inequality, and 4. Chaotic order.