Research On The Construction Of Bulgarian Boolean Function And Its Related Properties

Boolean functions play an important role in the cryptography, their prop-erties directly effect the safety of the cipher system. In this paper, from the perspective of the number theory we construct a large family of Boolean func-tions by using the polynomials over finite fields, and study the cryptography properties. The following are the main results.First, a large family of Boolean functions is constructed by using the poly-nomials over Fq. Let Fq be the finite field of order q= pr with odd prime and integer r≥ 1, and letβ0,…,βr-1 be linearly independent elements of Fq over Fp. Define s=「log2p」, and write ki-1=ui1+ui2·2+…+uis· 2s-1 with Uij ∈{0,1} for 1≤ j≤ s and 1≤ i≤ r. Assume that f(x) ∈ Fq[x] has no multiple zero in Fq and 0<deg(/)< p. Define B(u11,…,u1s,…,ur1,…,urs) then we discuss the cryptography properties including the maximum Fourier co-efficient, nonlinearity, average sensitivity, sparsity, collision and avalanche effect.Second, we prove that a large family of pseudorandom binary sequence is collision free and possesses the avalanche property, extend the concepts of collision and avalanche effect to the above Boolean functions, and study their collision and avalanche effect.