Almost security of cryptographic Boolean functions

The propagation criterion, PC(ℓ) of order k, is one of the most general cryptographic criteria of secure Boolean functions f. In this paper, we formalize its ε-almost version. The new definition requires that f(X)+f(X+Δ) is almost uniformly distributed while in the original definition, it must be strictly uniformly distributed. Better parameters are then obtained than the strict PC(ℓ) of order k functions. To construct ε-almost PC(ℓ) of order k functions, we introduce a notion of domain distance.