The Weight and Nonlinearity of 2-rotation Symmetric Cubic Boolean Function

The conceptions of χ-value and K-rotation symmetric Boolean functions are introduced by Cusick. K-rotation symmetric Boolean functions are a special rotation symmetric functions, which are invariant under the k − th power of ρ. In this paper, we discuss cubic 2-value 2-rotation symmetric Boolean function with 2n variables, which denoted by F 2n (x 2n). We give the recursive formula of weight of F 2n (x 2n), and prove that the weight of F 2n (x 2n) is the same as its nonlinearity.


Introduction
Boolean functions have many applications in coding theory and cryptography.Rotation symmetric Boolean functions(RSBF) as invariant Boolean functions under rotation transform have been widely studied.Higher nonlinearity is a very important character of Boolean functions which are widely used in coding theory and S-box design.Rotation symmetric Boolean functions as a subclass of K − rotation symmetric have not higher nonlinearity.So, K − rotation symmetric Boolean functions which are the generalization of notion of rotation symmetric function were proposed by Selc ¸k Kavut.The applications of the k − rotation symmetric(k ≥ 2) to coding theory and S-box design can be found in some papers.Cusick gave the definition of cubic 2 − rotation symmetric Boolean functions and used the notation {2 − (1, r, s) 2n : 2n ≥ s} as the cubic monomial 2-rotation symmetric functions (denoted by 2 − f unctions).Cusick also described the affine equivalence of cubic MRS 2-rotation symmetric, and proved that the sequence of Hamming weights of {2 − (1, r, s) 2n : 2n ≥ s} satisfies a linear recursion with integer coefficients.In this paper, we will give the recursion formula of Hamming weight of {2 − (1, 2, 3) 2n (2n ≥ 10)} and prove that the nonlinearity of {2 − (1, 2, 3) 2n (2n ≥ 10)} is the same as its weight.

Preliminaries
Let F 2 = {0, 1} be the binary field, F n 2 be the n−dimensional vector space of over F 2 .A Boolean function in n variables can be defined as a map from F n 2 into F 2 , denoted by f n (x n ), or f n in brief, where Every Boolean function f n has a unique polynomial representation (usually called the algebraic normal form (ANF)), and the degree of f n is the degree of this polynomial(deg( f n ) in brief).If every term in the algebraic normal form of f n has the same degree, then the function is said to be homogeneous.
is affine and homogeneous(i.e.the constant term is 0), f n is said to be linear.The truth table of f n is defined to be the binary sequence The Hamming weight of a Boolean function f n is defined as the number of nonzero coordinates in its truth table, denoted by wt( f n ).The Hamming distance d( f n , g n ) between two Boolean functions f n and g n is defined as the number of their different coordinates, which equals the Hamming weight of their sum f + g, where + denotes the addition on F 2 .Two Boolean functions f n and g n in n variables are said to be a f f ine equivalent if there exists an invertible matrix A with entries in F 2 and b ∈ F n 2 such that f n (x) = g n (Ax + b).
Definition 1 The nonlinearity NL( f n ) of a Boolean function f n (x n ) is defined as where • is the vector dot product.
It is easy to see that if f n and g n are affine equivalent, then wt( f n ) = wt(g n ) and NL( f n ) = NL(g n ).We say that the weight and nonlinearity are a f f ine invariants.
Definition 2 For a Boolean function f n (x n ).The Fourier transform of f n at c n ∈ F n 2 is defined as If a monomial x 1 x 2 x 3 appears in a rotation symmetric Boolean function as a term then all monomials in the orbit of x 1 x 2 x 3 should appear in the function as terms.A rotation symmetric function is said to be monomial rotation symmetric(MRS ) if it is generated by applying powers of ρ to a single monomial.We use the notation (1, r, s) n for the cubic MRS function in n variables generated by the monomial x 1 x r x s .A Boolean function is said to be k − rotation symmetric if it is invariant under the k − th power of ρ but not under any smaller power.A Boolean function is said to be monomial k − rotation symmetric if it is generated by applying powers of ρ k to a single monomial.For brevity, we refer to these functions as k − f unctions.In this paper, the cubic 2-functions shall be discussed.We use the notation 2 − (1, r, s) 2n for the cubic 2-function in 2n variables generated by the monomial is called a standard f orm of the above 2-function.
A monomial [a, b, c] in a cubic 2−function is said to be pure f orm, if a, b, c are all even or odd.A monomial that is not pure form is said to be mixed f orm.It is obvious that every monomial of 2 − (1, r, s) 2n has the same form.A 2-function is said to be mixed f orm 2 − f unction if it's terms are mixed form.Otherwise, it is said to be pure f orm 2 − f unction.
Assume a is even(odd) and b, c are odd(even).Then the χ − value for 2 − (1, r, s) is defined as χ = c − b .
Theorem 1 Two 2-functions 2 − (1, r, s) 2n and 2 − (1, p, q) 2n are affine equivalent by some permutation for all n if and only if their χ−values are equal.
Theorem 1 tells us that all 2-values functions with 2n variables have the same weights and nonlinearity.So, in the following section, we will discuss the weight and nonlinearity of 2-values function 2 − (1, 2, 3) 2n .
If T is a string , then T denotes the complemented string with 0 and 1 interchanged.If X is a 4−bit block or a string of blocks, then (X) s or X s is the string obtained by concatenation of s copies of X.The concatenation of two strings u, v will be denoted by uv or u||v.Now we define two sets of 4−bit strings and We give the following result about the truth tables of monomials for F 2n (x 2n ).
Lemma 2 The truth table of any monomial for F 2n (x 2n ) is , and i is odd.
From Lemma 2, we give the following algorithm as the output of truth table for F 2n (x 2n ) .Algorithm 1 , where hs i is the string obtained from h s i by complementing its last 2 s−2 bits.Write From the above algorithm, we give the recursive relationship of weight for F 2n (x 2n ).
Proof.Using Algorithm 1, we have (1) and (2) Similarly, we have From ( 1), ( 2) and ( 3), we have 4. The Nonliearity of F 2n (x 2n ) Cusick and Stȃnicȃ conjectured that the nonlinearity of cubic 1-values function F n (x n ) is the same as the weight, and Zhang et al. proved the conjecture.In this section, we shall prove the same result for F 2n (x 2n ), that is, By the definitions of Fourier transform and Hamming weight, we can easily deduce that Therefore, we can restate (4) as (5) On the other hand, the recursion formula of F 2n (0) can be obtained by applying the recursion formula of wt(F 2n (x 2n )).
Before giving the proof of ( 5), we need some notation: Firstly, we give the following recursive relations about f 2n−1 i (c 2n−1 ).
proof We prove the relation for i = 1, since the proof of the others are similar.
From lemma 4, we can easily deduce the following corollary.
Corolary 5 For every c The following lemma give the properties of F 2n (0).
proof We prove it by math induction.From Table 2, we can see the two results are true for 2n = 6, 8, 10, 12, 14, 16, 18. Assume that, for an arbitrary 2n, the result is also true.Let's derive the correctness of conclusion for 2n + 2 from this assumption.
Which exactly means that the result holds for 2n + 2.