k-th order symmetric SAC boolean functions and bisecting binomial coefficients

The Strict Avalanche Criterion (SAC) and symmetry for Boolean functions are important properties in cryptographic applications. High order SAC was first studied by Forré. Based on bisecting binomial coefficients and S. Lloyd's work, we describe a method to find kth order symmetric SAC functions (SSAC(k)). In this paper, we determine all the SSAC(k)n-variable functions for n⩽30, k=1,2,…,n-2. Also, for infinitely many n, we give some nontrivial binomial coefficient bisections. The existence of nontrivial bisections makes the problem to find all SSAC(k) functions very difficult.