Weight Recursions for Any Rotation Symmetric Boolean Functions

Let f
_n
(x
₁
, x
₂
,⋯, x
_n
) denote the algebraic normal form (polynomial form) of a rotation symmetric Boolean function of degree d in n ≥ d variables and let wt(f
_n
) denote the Hamming weight of this function. Let (1, α
₂
,
_⋯
, α
_d
)
_n
denote the function f
_n
of degree d in n variables generated by the monomial x
₁
x
_α2
⋯x
_α
d
. Such a function f
_n
is called monomial rotation symmetric (MRS). It was proved in a 2012 paper that for any MRS f
_n
with d = 3, the sequence of weights {w
_k
= w
_t
(f
_k
) : k = 3, 4,
_⋯
} satisfies a homogeneous linear recursion with integer coefficients. In this paper, it is proved that such recursions exist for any rotation symmetric function f
_n
; such a function is generated by some sum of t monomials of various degrees. A Mathematica program is available on arxiv.org which explicitly computes the homogeneous linear recursion for the weights, given any rotation symmetric f
_n
. The reader who is only interested in finding some recursions can use the program and not be concerned with the details of the rather complicated proofs in this paper.