Hyper-bent Boolean functions were introduced in 2001 by Youssef and Gong (and initially proposed by Golomb and Gong in 1999 as a component of S-boxes) to ensure the security of symmetric cryptosystems but no cryptographic attack has been identified till 2016.

Hyper-bent functions have properties still stronger than the well-known bent functions which were already studied by Dillon and

Rothaus more than four decades ago. Hyper-bent functions are very rare and whose classification is still elusive. Therefore, not only their characterization, but also their generation are challenging problems.

In the context of filtered LFSRs, Canteaut and Rotella showed at the 2016 FSE conference that when considering fast correlation attacks, the relevant criterion should no longer be nonlinearity, but rather generalized nonlinearity. Indeed, they showed that if $f+ Tr(\lambda x^k)$ (where ``$Tr$'' stands for the absolute trace function over $F_{2^n}$) is biased, then we can apply a fast correlation attack to recover $x_0^k$ where $x_0$ denotes the initial state. If $k$ is coprime to $2^n-1$, then the attack recovers the initial state. Moreover, the case when $k$ is not coprime to $2^n-1$ also leads to another attack and a new criterion to evaluate the security of filtered LFSR. The new criterion given on filtered LFSRs has thus revived interest in the topic of hyperbent functions.

In this talk, we shall give a complete survey on all what is known on hyper-bent Boolean functions. We will also present very recent results (2018) on hyper-bent functions in arbitrary characteristic as well as generalized hyper-bent functions.

# Hyper-bent and generalized hyper-bent functions

Friday, 27 July, 2018 - 14:30