Affine equivalence for cubic rotation symmetric Boolean functions withn=pqvariables

Abstract

a b s t r a c tRotation symmetric Boolean functions have been extensively studied in the last fifteen years or so because of their importance in cryptography and coding theory. Until recently, very little was known about the basic question of when two such functions are affine equivalent. The simplest case of quadratic rotation symmetric functions which are generated by cyclic permutations of the variables in a single monomial was only settled in a 2009 paper of Kim, Park and Hahn. The much more complicated analogous problem for cubic functions was solved for permutations using a new concept of patterns in a 2011 paper of Cusick, and it is conjectured that, as in the quadratic case, this solution actually applies for all affine transformations. The patterns method enables a detailed analysis of the affine equivalence classes for various special classes of cubic rotation symmetric functions in n variables. Here the case of functions generated by a single monomial and having pq variables, where p and q are primes, is examined in detail, and in particular, a formula for the number of classes is proved. This is significant because it is the first time that a complete enumeration of the number of classes has been found when the number of variables is divisible by two distinct primes.

Circulant matrices and affine equivalence of monomial rotation symmetric Boolean functions

“…We also show a result on dimension which is not a prime, nor a power of a prime (we learned meanwhile that this result is the subject of the new paper [14]). …”

“…To show that the number of cubic MRS in 2 We independently derived the next result (we found out after submitting this work that the recent paper [14] gives this result with no restriction on p, q) that seemed complicated to obtain via the previously published methods, that is, we find the number of equivalence classes for cubic MRS in n = pq (for primes 3 ≤ p < q) variables. if p ≡ 5 (mod 6), and q ≡ 5 (mod 6).…”

“…[7,10,11,14,12,13], but no major advances have been made for general high degree Boolean functions. Berlekamp and Welch [2] in 1972 found explicitly all equivalence classes for functions on 5 variables, and in 1991, Maiorana [29] looked at 6 variables and found 150, 357 such equivalence classes (both of these results also allowed transformations of the output).…”