| Abstract: |
| An almost perfect nonlinear (APN) function $f(x)$ is a mapping from the finite field $GF(2^n)$ with $2^n$ elements to itself that has the property that $f(x+a) + f(a) = b$ has at most 2 solutions $x$ for any nonzero $a$ and any $b$ in the finite field. APN functions have many applications in cryptography to construct optimal S-boxes, in coding theory to construct optimal error-correcting codes, and in discrete mathematics. Many basic constructions of APN functions are known but there are still many open and challenging problems.
This talk will provide an introduction and an overview over known results on APN functions as well as some recent new constructions of infinite classes of APN functions. These are generalizations of sporadic binomial APN functions constructed by Edel and Pott in 2006. An open question has been to generalize these to an infinite family. We present a generalization to quadrinomial APN functions. Some of the remaining open problems in this area will also be discussed. |
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