For every positive integer n, every positive integer k with gcd(k,n)=1, and every finite field K of characteristic 2 with |K|=2^n, let d_k=4^k-2^k+1 and Delta_k=(b+1)^d_k+b^d_k+1:b∈ K. Then, for all distinct nonzero v_1,v_2∈ K, one has |(x,y,z)∈Delta_k^3:v_1x+v_2y+(v_1+v_2)z=0|=2^2n-3.

Carlet’s cyclic-additive conjecture for Kasami monomials

Carlet formulated the cyclic-additive difference-set condition in 2018; the Kasami-family case was posed as an open problem at the NSUCRYPTO 2019 olympiad. The conjecture asks for a specific incidence count for every admissible Kasami parameter.

Known results

Carlet, 2018: formulated the cyclic-additive condition.
Nagy, McNeil, and Vajda, August 19, 2026: proved the conjecture for $k\bmod n\in\{1,2,n-2,n-1\}$ , including the Kasami exponent $d=13$ .
The same preprint exhaustively verified all admissible $(n,k)$ with $n\le 13$ .

October 2026 claimed proof

Nagy, McNeil, and Vajda now claim the required incidence count for all admissible parameters, using twisted root counts and an incidence argument on a Fermat cubic, with Lean formalization. The claim is presented in an unrefereed preprint.

Current status (as of October 2026): The conjecture is claimed solved for all admissible parameters, but the new proof and its formalization have not undergone independent mathematical verification.

Carlet’s cyclic-additive conjecture for Kasami monomials

Primary source

Additional references

Progress summary

Solutions 0