## View abstract

### Session S26 - Finite fields and applications

Thursday, July 15, 13:00 ~ 13:20 UTC-3

## Constructions of APN permutations

### Petr Lisonek

#### Simon Fraser University, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakc043ef4141822051bb22d32e1fccf66d').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyc043ef4141822051bb22d32e1fccf66d = 'pl&#105;s&#111;n&#101;k' + '&#64;'; addyc043ef4141822051bb22d32e1fccf66d = addyc043ef4141822051bb22d32e1fccf66d + 'sf&#117;' + '&#46;' + 'c&#97;'; var addy_textc043ef4141822051bb22d32e1fccf66d = 'pl&#105;s&#111;n&#101;k' + '&#64;' + 'sf&#117;' + '&#46;' + 'c&#97;';document.getElementById('cloakc043ef4141822051bb22d32e1fccf66d').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyc043ef4141822051bb22d32e1fccf66d + '\'>'+addy_textc043ef4141822051bb22d32e1fccf66d+'<\/a>';

Let ${\mathbb F_q}$ denote the finite field with $q$ elements. Almost perfect nonlinear (APN) function is a mapping $f:{\mathbb F_{2^n}}\rightarrow{\mathbb F_{2^n}}$ such that the equation $f(x+a)+f(x)=b$ has at most two solutions for any fixed $a\in{\mathbb F_{2^n}^*}$ and $b\in{\mathbb F_{2^n}}$.

APN functions are of great interest in the design of cryptographic block ciphers as they provide the best resistance against differential cryptanalysis. Some block cipher designs require that the APN function is a permutation of ${\mathbb F_{2^n}}$. Several infinite families of APN permutations are known, but not much is known in general. The Big APN Problem asks whether the exist any APN permutations of ${\mathbb F_{2^n}}$ for even $n>6$.

In this talk we survey several recent results (our as well as due to other authors) on constructing APN permutations using vector spaces of Walsh zeros, as well as on the non-existence of APN permutations of certain specific forms (known as Kim functions).

Joint work with Benjamin Chase (Simon Fraser University, Canada).