For every pair of positive integers n,k with gcd(k,n)=1, let F:F_2^n→F_2^n be defined by F(x)=x^4^k-2^k+1, and let Delta=F(b)+F(b+1)+1:b∈F_2^n. Then, for all distinct nonzero v_1,v_2∈F_2^n, |(x,y,z)∈Delta^3:v_1x+v_2y+(v_1+v_2)z=0|=2^2n-3.

Kasami APN-function conjecture

The Kasami APN-function conjecture was posed at NSUCRYPTO in 2019. Recent work proves several boundary parameter cases and checks all admissible small dimensions, but does not settle the full conjecture.

Known results

Hernando and McGuire settled the monomial Kasami-Welch case within the broader exceptional-APN conjecture.
Aubry, McGuire, and Rodier settled the odd-degree polynomial case outside the Gold and Kasami-Welch degrees.
Férard, Oyono, and Rodier proved further non-exceptionality results for perturbations of Kasami-Welch functions.
Later work established partial absolute-irreducibility results while explicitly leaving pending cases.

August 2026 partial proof and verification

A new arXiv preprint proves the cases $k \bmod n\in\{1,2,n-2,n-1\}$ , including a complete proof for $k=2$ , and exhaustively verifies all admissible dimensions $n\le 13$ . These results substantially narrow the remaining range, but the paper states that the general conjecture remains open elsewhere.

Current status (as of August 2026): The cases $k \bmod n\in\{1,2,n-2,n-1\}$ and all admissible $n\le 13$ are settled, while the remaining general parameter range is open.

Kasami APN-function conjecture

Primary source

Additional references

Progress summary

Solutions 0