In this paper, we focus on the relation collection step of the Function Field Sieve (FFS), which is to date the best algorithm known for computing discrete logarithms in small-characteristic finite fields of cryptographic sizes. Denoting such a finite field by Fpn, where p is much smaller than n, the main idea behind this step is to find polynomials of the form a(t)-b(t)x in Fp[t][x] which, when considered as principal ideals in carefully selected function fields, can be factored into products of low-degree prime ideals. Such polynomials are called "relations", and current record-sized discrete-logarithm computations need billions of those. Collecting relations is therefore a crucial and extremely expensive step in FFS, and a practical implementation thereof requires heavy use of cache-aware sieving algorithms, along with efficient polynomial arithmetic over Fp[t]. This paper presents the algorithmic and arithmetic techniques which were put together as part of a new public implementation of FFS, aimed at medium-to record-sized computations.

We present the properties of a new class of Boolean functions defined as the sum of m symmetric functions with decreasing number of variables and degrees. The choice of this construction is justified by the possibility to study these functions by using tools existing for symmetric functions. On the one hand we show that the synthesis is well understood and give an upper bound on the gate complexity. On the other hand, we investigate the Walsh spectrum of the sum of two functions and get explicit formulae for the case of degree at most three.

We present an extensive study of symmetric Boolean functions, especially of their cryptographic properties. Our main result establishes the link between the periodicity of the simplified value vector of a symmetric Boolean function and its degree. Besides the reduction of the amount of memory required for representing a symmetric function, this property has some consequences from a cryptographic point of view. For instance, it leads to a new general bound on the order of resiliency of symmetric functions, which improves Siegenthaler's bound. The propagation characteristics of these functions are also addressed and the algebraic normal forms of all their derivatives are given. We finally detail the characteristics of the symmetric functions of degree at most 7, for any number of variables. Most notably, we determine all balanced symmetric functions of degree less than or equal to 7.

This paper the link between the periodicity of the value vectors symmetric Boolean functions and their degrees. It also deduce new results concerning balancedness, resiliency and propagation criteria of symmetric Boolean functions.