Asymptotic enumeration of correlation-immune boolean functions

A boolean function of $n$ boolean variables is {correlation-immune} of order $k$ if the function value is uncorrelated with the values of any $k$ of the arguments. Such functions are of considerable interest due to their cryptographic properties, and are also related to the orthogonal arrays of statistics and the balanced hypercube colourings of combinatorics. The {weight} of a boolean function is the number of argument values that produce a function value of 1. If this is exactly half the argument values, that is, $2^{n-1}$ values, a correlation-immune function is called {resilient}. An asymptotic estimate of the number $N(n,k)$ of $n$-variable correlation-immune boolean functions of order $k$ was obtained in 1992 by Denisov for constant $k$. Denisov repudiated that estimate in 2000, but we will show that the repudiation was a mistake. The main contribution of this paper is an asymptotic estimate of $N(n,k)$ which holds if $k$ increases with $n$ within generous limits and specialises to functions with a given weight, including the resilient functions. In the case of $k=1$, our estimates are valid for all weights.Comment: 18 page

A Lower Bound on the Number of Boolean Functions with Median Correlation Immunity

The number of $n$-ary balanced correlation immune (resilient) Boolean functions of order $\frac{n}{2}$ is not less than $n^{2^{(n/2)-2}(1+o(1))}$ as $n\rightarrow\infty$. Keywords: resilient function, correlation immune function, orthogonal arrayComment: 3 page

Linear recurrences and asymptotic behavior of exponential sums of symmetric boolean functions

In this paper we give an improvement of the degree of the homogeneous linear recurrence with integer coefficients that exponential sums of symmetric Boolean functions satisfy. This improvement is tight. We also compute the asymptotic behavior of symmetric Boolean functions and provide a formula that allows us to determine if a symmetric boolean function is asymptotically not balanced. In particular, when the degree of the symmetric function is a power of two, then the exponential sum is much smaller than $2^n$.Comment: 18 pages, 3 figure

Улучшенные оценки для числа k-эластичных и корреляционно-иммунных двоичных отображений

Получены улучшенные нижние и верхние оценки для числа корреляционно-иммунных порядка к и k-эластичных ((n, m, к)-устойчивых) двоичных отображений. Improved lower and upper bounds for |K (n,m, k)| (the number of correlation-immune of order k binary mappings) and |R (m,n, k)| (the number of (n, m, k)-resilient binary mappings) are obtained. By M (n, k) we denote 8=0 s n — k fn^ ^ ^ ,,, /П n and by T (n, m, k) — the expression (2m - 1) + M (n, k) log^ . If m ^ 5 and k (5 + 2log2n) + 6m ^ ^ n (1/3 — y) for fixed 0 n0, 'm2 — m — 12 2 + 1M M (n, k) — £i ^ log2 |R (n, m, k)| — m2n + T (n, m, k) ^ ^ ((16m — 47) 2m 4 — m + ^ M (n, k) + e2. If m ^ 5 and k (5 + 2log2n) + 6m ^ n (5/18 — 7) for fixed 0 n0, m2 m 12 2 + 1M M (n, k) — e1 ^ log2 |K (n, m, k) | — m2n + m2m 1 + T (n, m, k) — n + 1 + log2 П 2 — ^ (2m — 1) ^ ((16m — 47) 2m 4 — m + 3) M (n, k) + e2

Asymptotic enumeration of correlation-immune boolean functions

A boolean function of n boolean variables is correlation-immune of order k if the function value is uncorrelated with the values of any k of the arguments. Such functions are of considerable interest due to their cryptographic properties, and are also related to the orthogonal arrays of statistics and the balanced hypercube colourings of combinatorics. The weight of a boolean function is the number of argument values that produce a function value of 1. If this is exactly half the argument values, that is, 2n - 1 values, a correlation-immune function is called resilient. An asymptotic estimate of the number N(n,k) of n-variable correlation-immune boolean functions of order k was obtained in 1992 by Denisov for constant k. Denisov repudiated that estimate in 2000, but we show that the repudiation was a mistake. The main contribution of this paper is an asymptotic estimate of N(n,k) which holds if k increases with n within generous limits and specialises to functions with a given weight, including the resilient functions. In the case of k = 1, our estimates are valid for all weights