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17 USC 105 interim-entered record; under temporary embargo. ; In 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. ; U.S. Government affiliation is unstated in article text.

The article of record as published may be found at http://dx.doi.org/10.1109/TIT.2021.3123104 ; The Difference Distribution Table (DDT) and the differential uniformity play a major role for the design of substitution boxes in block ciphers, since they indicate the func- tion's resistance against differential cryptanalysis. This concept was extended recently to c-DDT and c-differential uniformity, which have the potential of extending differential cryptanalysis. Recently, a new theoretical tool, the Boomerang Connectivity Table (BCT) and the corresponding boomerang uniformity were introduced to quantify the resistance of a block cipher against boomerang-style attacks. Here we concentrate on two classes (introduced recently) of permutation polynomials over finite fields of even characteristic. For one of these, which is an involution used to construct a 4-uniform permutation, we explicitly determine the c-DDT entries and BCT entries. For the second type of function, which is a differentially 4-uniform function, we give bounds for its c-differential and boomerang uniformities. ; The research of Sartaj Ul Hasan is partially supported by MATRICS grant MTR/2019/000744 from the Science and Engineering Research Board, Government of India. Pantelimon Stănică acknowledges the sabbatical support from Naval Postgraduate School from September 2020 to July 2021.

This paper explores the distribution of algebraic thickness of Boolean functions (that is, the minimum number of terms in the ANF of the functions in the orbit of a Boolean function, through all affine transformations), in four and five variables, and the complete distribution is presented. Additionally, a complete analysis of some complexity properties (e.g., nonlinearity, balancedness, etc.) of all relevant orbits of Boolean functions is presented. Some properties of our notion of rigid function (which enabled us to reduce significantly the computation) are shown and some open questions are proposed, providing some further explanation of one of these questions.

This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. As such, it is in the public domain, and under the provisions of Title 17, United States Code, Section 105, may not be copyrighted. ; Proceedings of the 10th International Workshop on Boolean Problems, Freiberg, Germany, Sept. 19-21, 2012, pp. 225-232, 2011 ; Algebraic immunity (Al) is a property of a Boolean function f that measures its susceptibility to an alebraic attack. If f has a low algebraic immunity and f is used in an encryption protocol, then there are ways to successfully cyrptanalyze the system. As a result, it is important to have an efficient means to compute the alebraic immunity of Boolean functions. Unfortunately, algebraic immunity is one of the most complex cryptographic properties to compute. For example, it is significantly more difficult to compute than nonlinearity. Here, we show the advantage of a reconfigurable computer in computating a function's algebraic immunity. For example, we show that a reconfigurable computer is 4.9 times faster than a conventional computer in this computation for 5-variable computer is 4.9 times fater than a conventional computer in this computation for 5-variable functions. Indeed, we compute the distribution of functions to algebraic immunity for all 5-variable functions, a computation that has not been previously accomplished. Interestingly, the problem we address is to design a logic circuit that computes a characteristic of a logic function.

In this paper we define a notion of partial APNness and find various characterizations and constructions of classes of functions satisfying this condition. We connect this notion to the known conjecture that APN functions modified at a point cannot remain APN. In the second part of the paper, we find conditions for some transformations not to be partially APN, and in the process, we find classes of functions that are never APN for infinitely many extensions of the prime field F2, extending some earlier results of Leander and Rodier. ; S.K. was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2016R1D1A1B03931912 and No. 2016R1A5A1008055). The research of the first two authors was supported by Trond Mohn foundation. ; S.K. was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2016R1D1A1B03931912 and No. 2016R1A5A1008055). The research of the first two authors was supported by Trond Mohn foundation.