All my scientific publications are listed on this page. You
can click on **[more info]** to see the abstract of
the paper and a list of keywords, on **[bibtex]**
to have the corresponding citation in bibtex format and
on **[pdf]** to download the article (if
available).

## Articles

2013
More Differentially 6-uniform Power Functions,

Céline Blondeau, Léo Perrin
[more info]
[pdf]
[bibtex]
*Pre-proceedings of WCC 2013, p223-233 [link]*

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

Abstract:
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.

2013
On the properties of S-boxes,

Léo Perrin
[more info]
[pdf]
[bibtex]
Master's Thesis from the KTH math department, done in the crypto group at Aalto university.

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

Abstract:
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.

## Slides

2013
On the Properties of S-boxes
[pdf]