The National Institute of Standards and Technology (NIST) launched in 2008 a public competition, called the SHA-3 competition, aiming at definining a new standard for hash functions. We study here the algebraic properties of some of the candidates to this contest. Among the functions analyzed, is the Keccak algorithm, which was recently announced to be the winning proposal and thus the new standard SHA-3. We study, in a first time, a new class of distinguishers introduced in 2009 and called the zero-sum distinguishers. We apply these distinguishers to some of the candidates of the SHA-3 competition. Next, we investigate the algebraic degree of iterated permutations. We establish, at a first step, a new bound exploiting the usual structure of the nonlinear layer in SPN constructions. After this, we study the role of the inverse permutation in an iterated construction and we prove a second and more general bound on the degree. Equally, we present a study on a new notion concerning a class of Sboxes, that expresses the fact that some components of an Sbox can be written as an affine function of some variables on a well-chosen subspace and on all its cosets. The analysis of such a property leads to the improvement of an existing attack on the hash function Hamsi. Finally, in a second part, we investigate the security of two SHA-3 candidates against side-channel attacks and we propose some countermeasures in software level.
