Sensitivities and block sensitivities of elementary symmetric Boolean functions

: Boolean functions have important applications in molecular regulatory networks, engineering, cryptography, information technology, and computer science. Symmetric Boolean functions have received a lot of attention in several decades. Sensitivity and block sensitivity are important complexity measures of Boolean functions. In this paper, we study the sensitivity of elementary symmetric Boolean functions and obtain many explicit formulas. We also obtain a formula for the block sensitivity of symmetric Boolean functions and discuss its applications in elementary symmetric Boolean functions.


Introduction
In 1938, Shannon [28] recognized that symmetric functions had efficient switch network implementation. Since then, a lot of research has been carried out on symmetric or partially symmetric Boolean functions, and detection of symmetry has become important in logic synthesis, technology mapping, binary decision diagram minimization, and testing [1,10,22].
For the applications of symmetric Boolean functions in cryptography, Canteaut and Videau [3] presented an extensive study in 2005, and more results on (totally) symmetric Boolean functions can be found in other papers [2,5,8,17,21,23,27].
It is clear that any symmetric Boolean function can be written as a sum of some elementary symmetric Boolean functions. Hence, it is a fundamental question to have a comprehensive understanding about elementary symmetric Boolean functions. In ref. [8], the authors studied the balancedness of elementary symmetric Boolean functions and they proposed a conjecture which has received a lot of attention [4][5][6]8,9,13,14,31].
In ref. [7], Cook et al. introduced the definition of sensitivity as a combinatorial measure for Boolean functions by providing lower bounds on the time needed by CREW PRAM (Concurrent Read Exclusive Write, Parallel Random Access Machine). The concept was extended by Nisan [24] to block sensitivity. The study of sensitivity and block sensitivity of Boolean functions has been an active research topic for many years [11,12,16,18,19,25,26,29,30,32].
Recently, Huang proved the long standing Sensitivity Conjecture [15]: for any Boolean function f , ( ) ≤ ( ) bs f s f 2 4 , where ( ) bs f is the block sensitivity of f and ( ) s f is the sensitivity of f .
In Section 2 of this paper, we introduce the algebraic normal form (ANF) of Boolean functions and the definition of symmetric Boolean functions. In Section 3, we first recall definitions used in the paper, then obtain many explicit formulas of the sensitivities of elementary symmetric Boolean functions by using some elementary combinatorial results. The main idea of Section 3 is motivated by ref. [8,33]. In Section 4, we prove a formula for the block sensitivity of symmetric Boolean functions. Based on our knowledge, this is the first study about the block sensitivity of symmetric Boolean functions. We apply this formula to elementary symmetric Boolean functions and show that the block sensitivity can be strictly greater than the sensitivity for some elementary symmetric Boolean functions. The conclusion is included in Section 5.

Preliminaries
In this section, we introduce the definitions and notations. Let = = { } 0, 1 2 . If f : ⟶ n is a function with n variables and values in , it is well known [20] that f can be expressed as a polynomial, called the ANF: We use x i to denote the word obtained by flipping the i-th bit of x.
Definition 3.2. [16,26] The sensitivity ( ) The average sensitivity of f, denoted by ( ) s f , is In the above definition, .
First, we will calculate the sensitivities and average sensitivities of ( ) X n 1, and ( ) X n n , .
is equivalent to Proof. The sufficiency is obvious. We show the necessity in the following.
Since for every word x in n , ( ) = s f n x ; , we know that every variable x i is essential in f . If ( ) > f deg 1, without loss of generality, we may assume that the term with degree ( ) f deg contains variable x 1 . Hence, we can write f as , , , , , are both n. In other words, ( ( )) = ( ( )) = s X n s X n n 1,¯1, .
We need some lemmas to calculate the sensitivities of ( ) X d n , for ≤ ≤ − d n 2 1.
, then the sensitivity of f is n. Otherwise, and 0 otherwise, then we have , be the value of ( ) X d n , when x has Hamming weight j, then Using Lemma 3.12 for = p 2 and Lemma 3.14, a straightforward calculation shows We first consider the sensitivities of ( ) X d n , for odd d.     , the sensitivity of ( ) X d n , is n: Proof. If n is odd, we already know ( ( )) = s X n n n , . For < < d n 0 and d odd, we have and by Lemma 3.9, we have . □ In order to compute the sensitivities of ( ) X d n , for even d, we need to improve Lemma 3.9. Proof. We first show that there is no i such that ( 1 , which is a contradiction.
We show that there is no i such that On the other hand, let To simplify the notation, sometimes, we write the value vector⟨ Proof. By Lemma 3.16, the value vector is the first + n 1 numbers of the sequence . The calculation is identical to the case when q is even. We are done. □ A bound for the sensitivities of elementary symmetric Boolean functions can be received by Lemma 3.18.
by definition. From Lemma 3.18, for even d, we have for odd d, we are done. □ , then It shows that the lower bound can also be reached sometimes.
To understand and prove more general formulas, we will first calculate the sensitivities of ( ) X d n , for some small even d. The same techniques will be used to obtain more general formulas later. .   It is clear that one can continue to compute the explicit formulas of ( ) X d n , for fixed d. In the following, we will consider the situation that d is very close to n. We already know ( ( )) = s X n n n , .

  
Proof. By Lemma 3.11, the value vector is  is m 2 . Hence, we may assume We only need to show we have | ( − ) C j d 2 2 , 2 1 . Therefore, 1, 2 2 , 2 2 , 2 1 2 , 2 mod 2 , , and ≥ t 1, we have and ≥ n d, the sensitivity of ( ) X d n , is , the value vector is , is even for even R and odd S). We only need to show that ( + + + ) . By the Lucas theorem, We are done since the least period of   , , j is even, the value vector is , j is odd, the value vector is The calculation is identical to the case of even j. The theorem is proved. □ Example 3.34. In Theorem 3.30, if = k 2, = t 1, then = d 6. If = k 2, = t 2, then = d 12.
One can check that these results are consistent with the previous lemmas.
We have be the value vector of symmetric Boolean Proof. This follows from Lemmas 3.9 and 3.10. □ Generally, we have be the value vector of symmetric Boolean  In the following, we assume ≥ k 2 i for = … i t 1, , . Since f is symmetric, we only need to calculate the sensitivities of f over the + n 1 words n 0, 1, , and the greatest sensitivity will be ( ) bs f . We divide this + n 1 words into t groups. For each group, we do straightforward calculation as we did in Example 4.2. We list the results below. For easy notation, we write word ( … ) Group 1: Group 2: We will first find the maximal sensitivity number in each group.
In Group 1, it is clear that In Group j, ≤ ≤ − j t 2 1, we will show the maximal number will be the first or the last one. Namely, Take the maximal value of this set, we prove the formula of ( ) bs f . □ Since ( ) ≤ ( ) ≤ s f bs f nfor any Boolean function f with n variables, by definition, we have the following.
Theorem 4.5. For odd d and ≥ n d, the block sensitivity of ( ) X d n , is n.
Proof. This follows from Theorem 3.17. □ Theorem 4.6. If = d 2 k , ∈ k and ≥ n d, then ( ( )) = ( ( )) = bs X d n bs X n , 2, 2 be the value vector of ( ) X n 2 , . From the proof of Theorem 3.19, we know when q is even and In the following, we see the sensitivity is strictly less than the block sensitivity for some elementary symmetric Boolean functions.