Differential uniformity, Differential spectrum, Kloosterman sum, Power function, Roots of trinomial, $x\to x^{2^t-1}$, Dickson polynomial.

We provide the differential spectra of differentially 6-uniform functions among the family of power functions $x\to x^{2^t-1}$ defined in $F_{2^n}$. We show that the functions $x\to x^{2^t-1}$ when $t=\frac{n-1}{2}, \frac{n+3}{2}$ with odd $n$ and when $t=\frac{\mathrm{kn}+1}{3},\frac{(3-k)n+2}{3}$ with $\mathrm{kn}\equiv 2 \text{mod} 3$ have differential spectra similar to the one of the function $x\to x^7$ which belongs to the same family. To provide the differential spectra for these functions, a recent result of Helleseth and Kholosha regarding the number of roots of polynomials of the form $x^{2^t+1}+x+a$ is used. A discussion regarding the non-linearity and the algebraic degree of this family of exponents is provided.

Differential cryptanalysis, S-box, Differential uniformity, Differential spectrum, Kloosterman sum, Power function, Roots of trinomial, $x\to x^{2^t-1}$, Dickson polynomial.

S-boxes are key components of many symmetric cryptographic primitives. Among them, some block ciphers and hash functions are vulnerable to attacks based on differential cryptanalysis, a technique introduced by Biham and Shamir in the early 90's. Resistance against attacks from this family depends on the so-called differential properties of the S-boxes used.

When we consider S-boxes as functions over finite fields of characteristic 2, monomials turn out to be good candidates. In this Master's Thesis, we study the differential properties of a particular family of monomials, namely those with exponent $2^t-1$. In particular, conjectures from Blondeau's PhD Thesis are proved.

More specifically, we derive the differential spectrum of monomials with exponent $2^t-1$ for several values of $t$ using a method similar to the proof Blondeau et al. made of the spectrum of $x\to x^7$. The first two chapters of this Thesis provide the mathematical and cryptographic background necessary while the third and fourth chapters contain the proofs of the spectra we extracted and some observations which, among other things, connect this problem with the study of particular Dickson polynomials.