A lower bound for the affinity level for almost all Boolean functions

2008 ◽  
Vol 18 (5) ◽  
Author(s):  
O. A. Logachev
Keyword(s):  
Lower Bound ◽  
Boolean Functions ◽  
Almost All

scholarly journals Improving bounds on probabilistic affine tests to estimate the nonlinearity of Boolean functions

2021 ◽  
Author(s):  
Ana Sălăgean ◽  
Pantelimon Stănică
Keyword(s):  
Lower Bound ◽  
Boolean Functions ◽  
Good Accuracy ◽  
Arbitrary Order ◽  
Probability Of Failure ◽  
Probabilistic Methods ◽  
Walsh Transform ◽  
The One ◽  
Almost All ◽  
Linearity Test

AbstractIn this paper we want to estimate the nonlinearity of Boolean functions, by probabilistic methods, when it is computationally very expensive, or perhaps not feasible to compute the full Walsh transform (which is the case for almost all functions in a larger number of variables, say more than 30). Firstly, we significantly improve upon the bounds of Zhang and Zheng (1999) on the probabilities of failure of affinity tests based on nonhomomorphicity, in particular, we prove a new lower bound that we have previously conjectured. This new lower bound generalizes the one of Bellare et al. (IEEE Trans. Inf. Theory 42(6), 1781–1795 1996) to nonhomomorphicity tests of arbitrary order. Secondly, we prove bounds on the probability of failure of a proposed affinity test that uses the BLR linearity test. All these bounds are expressed in terms of the function’s nonlinearity, and we exploit that to provide probabilistic methods for estimating the nonlinearity based upon these affinity tests. We analyze our estimates and conclude that they have reasonably good accuracy, particularly so when the nonlinearity is low.

scholarly journals On diagnostic tests of contact break for contact circuits

2020 ◽  
Vol 30 (2) ◽  
pp. 103-116 ◽  
Author(s):  
Kirill A. Popkov
Keyword(s):  
Boolean Function ◽  
Diagnostic Test ◽  
Boolean Functions ◽  
Diagnostic Tests ◽  
Almost All

AbstractWe prove that, for n ⩾ 2, any n-place Boolean function may be implemented by a two-pole contact circuit which is irredundant and allows a diagnostic test with length not exceeding n + k(n − 2) under at most k contact breaks. It is shown that with k = k(n) ⩽ 2n−4, for almost all n-place Boolean functions, the least possible length of such a test is at most 2k + 2.

scholarly journals Exact quantum algorithms have advantage for almost all Boolean functions

2015 ◽  
pp. 435-452
Author(s):  
Andris Ambainis ◽  
Jozef Gruska ◽  
Shenggen Zheng
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Query Complexity ◽  
Exact Quantum ◽  
Quantum Query Complexity ◽  
Almost All ◽  
Quantum Query

It has been proved that almost all n-bit Boolean functions have exact classical query complexity n. However, the situation seemed to be very different when we deal with exact quantum query complexity. In this paper, we prove that almost all n-bit Boolean functions can be computed by an exact quantum algorithm with less than n queries. More exactly, we prove that ANDn is the only n-bit Boolean function, up to isomorphism, that requires n queries.

scholarly journals On the Lyapunov Exponent of Monotone Boolean Networks †

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1035
Author(s):  
Ilya Shmulevich
Keyword(s):  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Boolean Functions ◽  
Boolean Network ◽  
Discrete Dynamical Systems ◽  
Boolean Networks ◽  
Asymptotic Formulas ◽  
Small Perturbations ◽  
Monotone Boolean Functions ◽  
Almost All

Boolean networks are discrete dynamical systems comprised of coupled Boolean functions. An important parameter that characterizes such systems is the Lyapunov exponent, which measures the state stability of the system to small perturbations. We consider networks comprised of monotone Boolean functions and derive asymptotic formulas for the Lyapunov exponent of almost all monotone Boolean networks. The formulas are different depending on whether the number of variables of the constituent Boolean functions, or equivalently, the connectivity of the Boolean network, is even or odd.

scholarly journals New Constructions of Balanced Boolean Functions with Maximum Algebraic Immunity, High Nonlinearity and Optimal Algebraic Degree

2020 ◽  
Vol 17 (7) ◽  
pp. 639-654
Author(s):  
Dheeraj Kumar SHARMA ◽  
Rajoo PANDEY
Keyword(s):  
Lower Bound ◽  
Boolean Function ◽  
Boolean Functions ◽  
Primitive Element ◽  
Galois Field ◽  
Algebraic Degree ◽  
Algebraic Immunity ◽  
Greatest Common Divisor ◽  
Primitive Elements ◽  
High Nonlinearity

This paper consists of proposal of two new constructions of balanced Boolean function achieving a new lower bound of nonlinearity along with high algebraic degree and optimal or highest algebraic immunity. This construction has been made by using representation of Boolean function with primitive elements. Galois Field,  used in this representation has been constructed by using powers of primitive element such that greatest common divisor of power and  is 1. The constructed balanced  variable Boolean functions achieve higher nonlinearity, algebraic degree of , and algebraic immunity of   for odd ,  for even . The nonlinearity of Boolean function obtained in the proposed constructions is better as compared to existing Boolean functions available in the literature without adversely affecting other properties such as balancedness, algebraic degree and algebraic immunity.

scholarly journals A Lower Bound of Fast Algebraic Immunity of a Class of 1-Resilient Boolean Functions

IEEE Access ◽  
2019 ◽  
Vol 7 ◽  
pp. 90145-90151 ◽  
Author(s):  
Yindong Chen ◽  
Liu Zhang ◽  
Jianlong Xu ◽  
Weihong Cai
Keyword(s):  
Lower Bound ◽  
Boolean Functions ◽  
Algebraic Immunity
Sign in / Sign up

Export Citation Format

Share Document