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N − 2i)! i! 2n/2 3n/8≤i ≤n/2 and B(3n/8, n/2) ≤ n! 3n/8≤i ≤n/2 (n − 2i )! i! 18) = 0. A simple calculation shows that for 0 ≤ i ≤ 3n/8, the integer n! 19) (n − 2i)! i! 19) in the range 0 ≤ i ≤ 3n/8 is asymptotic to n! (n/4)! (3n/8)! < 6n 3n/8 . e Hence we get the following estimate for the sum of the low order terms in B(0, n/2): B(0, 3n/8) < 3n 6n 8 e 3n/8 . 20) The last term in A(0, n/2), which is not even the largest one, by Stirling’s formula is asymptotic to n! (n/2)! 20). 23. 5 PROPAGATION CRITERIA A Boolean function f (x) in n variables is said to satisfy the propagation criterion of degree k (P C (k) for short) if changing any i (1 ≤ i ≤ k) of the n bits in the input x results in the output of the function being changed for exactly half of the 2n vectors x.

See [39]. The proof involves detailed geometric analysis inside cubes of high dimension, among other things. 3 COUNTING BALANCED SAC FUNCTIONS Boolean functions in cryptographic applications almost always need to be balanced, or nearly so. Therefore it is of interest to see if results like those above can be proved for the number of balanced SAC Boolean functions in n variables. We let Un denote this number. We are also interested in the size of the ratio Bn defined by Bn = 2−n log2 Un . As above, it is clear that Bn ≤ 1 and it is natural to conjecture that lim Bn exists.

The Berlekamp–Massey algorithm runs according to the following pseudocode: Input: binary sequence s = s0 , s1 , . . , sn−1 of length n Output: linear complexity 0 ≤ L(sn ) ≤ n begin C (x) = 1 L=0 m = −1 B(x) = 1 N =0 while (N < n) m−1 d = sN ⊕ i=0 ci sN −1−i (computes the next discrepancy) if (d = 1) T (x) = C (x) C (x) = C (x) + B(x) · x N −m if L ≤ N /2 L=N +1−L m=N B(x) = T (x) end if else (N = N + 1) end if end while end The Berlekamp–Massey algorithm is based on the next result [300,322].