On the Covering Radii of Binary Reed–Muller Codes in the Set of Resilient Boolean Functions

Abstract

Let Rt,n denote the set of t-resilient Boolean functions of n variables, and letρ(t, r, n) be the maximum distance between t-resilient functions and r-th order Reed-Muller code RM (r, n). We prove (analytically) thatρ(t, 2, 6) = 16 for t = 0, 1, 2,ρ(3, 2, 7) = 32. Using a result from coding theory on the covering radius of (n−3)-rd order Reed-Muller codes, we establish exact values of the covering radius of RM (n − 3, n) in the set of 1-resilient Boolean functions of n variables, when n/2 = 1 mod 2. We also present new lower bounds forρ(t, 2, 7) with t = 0, 1, 2, and generalize our methods to obtain lower bounds forρ(t, 2, n) with t ≤ n − 4,ρ(t, 3, n) with t ≤ n − 5, andρ(t, 4, n) with t ≤ n − 6.

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“…For i = 1 the result is true by expression (2). Assume that the result is true for i − 1 < k, we will prove that it is also true for i.…”

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On the construction of new bent functions from the max-weight and min-weight functions of old bent functions

“…Boolean functions are components of S-boxes used in different types of cryptographic applications such as block ciphers, stream ciphers and hash functions [3,5,22] as well as in coding theory [1,17], among others.…”

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“…Boolean functions are widely used in different types of cryptographic applications, such as block ciphers, stream ciphers, and hash functions [1][2][3], and in coding theory [4,5], among others. For example, the implementation of an S-box needs nonlinear Boolean functions to resist attacks such as the linear and differential cryptanalysis [6][7][8][9].…”