Some topics related to resilient functions

A resilient function is a randomness extractor which outputs truly random bits from bit fixing sources. In this paper, we discuss on some topics related to resilient functions: almost resilient functions, covering radius of Reed-Muller codes for resilient functions, and PC(l) of order k Boolean functions. T. Johansson, D. Stinson and the author introduced the notion of almost resilient functions, and showed its construction and a bound. In modern terminology, it is the first construction of a deterministic extractor, where a deterministic extractor outputs almost truly random bits from bit fixing sources. By using recently developed deterministic extractors, we can show a more efficient reduction of m-bit (1, 2)-OT to 1-bit (1, 2)-OT than the previously known ones. We then raise it as an open problem to construct almost zig-zag functions which can improve the OT reduction further more. The covering radius of Reed-Muller codes for resilient functions was introduced by the author as a security criterion of stream ciphers against both correlation attack and low degree approximation attack (for example, the linear attack). Its lower bounds and upper bounds were also derived. We now discuss on its extension to deterministic extractors, i.e., covering radius of Reed-Muller codes for deterministic extractors, A Boolean function satisfies PC(l) of order k if its autocorrelation function is a resilient function, and the first general construction method was given by the author. The author also introduced the notion of almost PC(l) of order k, and presented its construction. Now we finally discuss on its extension to multiple output bit Boolean functions F(X) such that the autocorrelation functions of F(X) are deterministic extractors