Discoveries / algebra

A Conjecture on Triple Counts for the Kasami APN Function

For the Kasami APN function $F(x) = x^{4^k - 2^k + 1}$ on $\mathrm{GF}(2^n)$ with $\gcd(k, n) = 1$, the conjecture asserts that for $\Delta = \{F(b) + F(b+1) + 1\}$ and all distinct nonzero $v_1, v_2$, the number of triples in $\Delta^3$ with $v_1 x + v_2 y + (v_1 + v_2) z = 0$ is exactly $2^{2n-3}$. Proved for $k \bmod n \in \{1, 2, n-2, n-1\}$ and verified exhaustively for $n \le 13$; the general case remains open.