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Equivalence Classes of Boolean Functions for

TL;DR: This paper presents a complete characterization of the first order correlation immune Boolean functions that includes the functions that are -resilient, and it is conjectured that the exact complete enumeration for general is intractable.
Abstract: This paper presents a complete characterization of the first order correlation immune Boolean functions that includes the functions that are -resilient. The approach consists in defining an equivalence relation on the full set of Boolean functions with a fixed number of variables. An equivalence class in this relation, called a first-order correlation class, provides a measure of the distance between the Boolean functions it contains and the correlation-immune Boolean functions. The key idea consists on manipulating only the equivalence classes instead of the set of Boolean functions. To achieve this goal, a class operator is introduced to construct a class with variables from two classes of variables. In particular, the class of -resilient functions on variables is considered. An original and efficient method to enumerate all the Boolean functions in this class is proposed by performing a recursive decomposition of classes with less variables. A bottom up algorithm provides the exact number of -resilient Boolean functions with seven variables which is 23478015754788854439497622689296. A tight estimation of the number of -resilient functions with eight variables is obtained by performing a partial enumeration. It is conjectured that the exact complete enumeration for general is intractable.
Citations
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Proceedings ArticleDOI
14 May 2012
TL;DR: It follows, that Fourier-sweet functions are capable of capturing the differences among the classes of bent functions, and at the same time link them to quadratic forms.
Abstract: The paper studies binary and ternary functions that have decision diagrams of identical shape in the original and spectral (Fourier) domain. These functions are called Fourier-sweet functions. This class of functions involves certain classes of bent functions and quadratic forms in both binary and ternary cases. Bent functions and quadratic forms have applications in cryptography and error-correcting codes. Not all bent functions are Fourier-sweet functions. It follows, that Fourier-sweet functions are capable of capturing the differences among the classes of bent functions, and at the same time link them to quadratic forms. Representation by shape invariant decision diagrams in the original and spectral domain might provide some better insight into features of bent functions and quadratic forms. The functions represented by the disjoint quadratic forms in the binary case and diagonal forms in the ternary case are elementary Fourier-sweet functions. In both binary and ternary cases, the application of affine transformations, under certain precisely specified restrictions, to the elementary Fourier-sweet functions produces other Fourier-sweet functions.

4 citations


Cites background or methods from "Equivalence Classes of Boolean Func..."

  • ...This paper is a continuation of the work related to classification of logic functions by decision diagrams [19] and representation and construction of bent functions (binary and multiple-valued) by decision diagrams [20], [21]....

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  • ...These operations partition the set of binary functions of a given number of variables into equivalence classes with identical Walsh spectra under the change of the sign and permutation of precisely defined subsets of Walsh coefficients [14], [19]....

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  • ...For example, BDDs and Walsh decision diagrams (WDDs) are used for the classification of switching functions [19], [24]....

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Proceedings ArticleDOI
22 May 2013
TL;DR: It is pointed out that establishing the links between beads, functions, and their decision diagram representations can be useful in classification of ternary functions, checking the equivalence of functions, as well as their circuit implementations.
Abstract: The paper studies ternary logic functions that have decision diagrams of identical shape. The concept of beads, a special class of binary sequences, is extended to ternary sequences and are used to describe the shape of the decision diagrams representing functions that are mathematical models of such sequences. We point out that establishing the links between beads, functions, and their decision diagram representations can be useful in classification of ternary functions, checking the equivalence of functions, as well as their circuit implementations.

Cites background or methods from "Equivalence Classes of Boolean Func..."

  • ...See discussions about representing representative functions in the NPN - classification by BDDs and Functional decision diagrams (FDDs) in [13] and LP -representatives by Walsh decision diagrams (WDDs) in [10]....

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  • ...See discussions about representing representative functions in the NPN classification by BDDs and Functional decision diagrams (FDDs) in [13] and LP -representatives by Walsh decision diagrams (WDDs) in [10]....

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  • ...This leads to the classification of switching functions in terms of the shape of WDDs [10]....

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  • ...In [10], it is shown that different LP -representative functions can be represented by the Walsh decision diagrams (WDDs) of the same shape....

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References
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Journal ArticleDOI
TL;DR: A theory of secrecy systems is developed on a theoretical level and is intended to complement the treatment found in standard works on cryptography.
Abstract: THE problems of cryptography and secrecy systems furnish an interesting application of communication theory.1 In this paper a theory of secrecy systems is developed. The approach is on a theoretical level and is intended to complement the treatment found in standard works on cryptography.2 There, a detailed study is made of the many standard types of codes and ciphers, and of the ways of breaking them. We will be more concerned with the general mathematical structure and properties of secrecy systems.

8,777 citations

Book
01 Jan 2009
TL;DR: This text can be used as the basis for an advanced undergraduate or a graduate course on the subject, or for self-study, and is certain to become the definitive reference on the topic.
Abstract: Analytic Combinatorics is a self-contained treatment of the mathematics underlying the analysis of discrete structures, which has emerged over the past several decades as an essential tool in the understanding of properties of computer programs and scientific models with applications in physics, biology and chemistry. Thorough treatment of a large number of classical applications is an essential aspect of the presentation. Written by the leaders in the field of analytic combinatorics, this text is certain to become the definitive reference on the topic. The text is complemented with exercises, examples, appendices and notes to aid understanding therefore, it can be used as the basis for an advanced undergraduate or a graduate course on the subject, or for self-study.

3,616 citations

Journal ArticleDOI
TL;DR: It is shown in this paper that the iterative algorithm introduced by Berlekamp for decoding BCH codes actually provides a general solution to the problem of synthesizing the shortest linear feedback shift register capable of generating a prescribed finite sequence of digits.
Abstract: It is shown in this paper that the iterative algorithm introduced by Berlekamp for decoding BCH codes actually provides a general solution to the problem of synthesizing the shortest linear feedback shift register capable of generating a prescribed finite sequence of digits. The shift-register approach leads to a simple proof of the validity of the algorithm as well as providing additional insight into its properties. The equivalence of the decoding problem for BCH codes to a shift-register synthesis problem is demonstrated, and other applications for the algorithm are suggested.

2,269 citations

Book ChapterDOI
01 Jun 2010
TL;DR: Encryption-decryption is the most ancient cryptographic activity, but its nature has deeply changed with the invention of computers, because the cryptanalysis (the activity of the third person, the eavesdropper, who aims at recovering the message) can use their power.
Abstract: Introduction A fundamental objective of cryptography is to enable two persons to communicate over an insecure channel (a public channel such as the internet) in such a way that any other person is unable to recover their message (called the plaintext ) from what is sent in its place over the channel (the ciphertext ). The transformation of the plaintext into the ciphertext is called encryption , or enciphering. Encryption-decryption is the most ancient cryptographic activity (ciphers already existed four centuries b.c.), but its nature has deeply changed with the invention of computers, because the cryptanalysis (the activity of the third person, the eavesdropper, who aims at recovering the message) can use their power. The encryption algorithm takes as input the plaintext and an encryption key K E , and it outputs the ciphertext. If the encryption key is secret, then we speak of conventional cryptography , of private key cryptography , or of symmetric cryptography . In practice, the principle of conventional cryptography relies on the sharing of a private key between the sender of a message (often called Alice in cryptography) and its receiver (often called Bob). If, on the contrary, the encryption key is public, then we speak of public key cryptography . Public key cryptography appeared in the literature in the late 1970s.

943 citations

Journal ArticleDOI
TL;DR: A new class of combining functions is presented, which provides better security against correlation attacks, and the security is quantified by the smallest number m + 1 of subsequences that must be simultaneously considered in a correlation attack.
Abstract: Pseudonoise generators for cryptographic applications consisting of several linear feedback shift registers with a nonlinear combining function have been proposed as running key generators in stream ciphers. These running key generators eau sometimes be broken by (ciphertext-only) correlation attacks on individual subsequences. A new class of combining functions is presented, which provides better security against such attacks. The security is quantified by the smallest number m + 1 of subsequences that must be simultaneously considered in a correlation attack. A necessary condition for such m th-order correlation-immunity is proved. A recursive construction is given that permits the construction of an m th-order immune combining function for n subsequences for any m and n with 1 \leq m . Finally, the trade-off between the length of the linear equivalent of the nonlinear generator and the order m of its immunity against correlation attacks is considered.

764 citations