The Error Linear Complexity Spectrum as a Cryptographic Criterion of Boolean Functions
The error linear complexity spectrum constitutes a well-known cryptographic criterion for sequences, indicating how the linear complexity of the sequence decreases as the number of bits allowed to be modified per period increases. In this paper, via defining an association between <inline-formula...
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| Published in: | IEEE transactions on information theory Vol. 65; no. 12; pp. 8345 - 8356 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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IEEE
01.12.2019
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 0018-9448, 1557-9654 |
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| Abstract | The error linear complexity spectrum constitutes a well-known cryptographic criterion for sequences, indicating how the linear complexity of the sequence decreases as the number of bits allowed to be modified per period increases. In this paper, via defining an association between <inline-formula> <tex-math notation="LaTeX">2^{n} </tex-math></inline-formula>-periodic binary sequences and Boolean functions on <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> variables, it is shown that the error linear complexity spectrum also provides useful cryptographic information for the corresponding Boolean function <inline-formula> <tex-math notation="LaTeX">f </tex-math></inline-formula> - namely, it yields an upper bound on the minimum Hamming distance between <inline-formula> <tex-math notation="LaTeX">f </tex-math></inline-formula> and the set of functions depending on fewer number of variables. Therefore, the prominent Lauder-Paterson algorithm for computing the error linear complexity spectrum of a sequence may also be used for efficiently determining approximations of a Boolean function that depend on fewer number of variables. Moreover, it is also shown that, through this approach, low-degree approximations of a Boolean function can be also obtained in an efficient way. |
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| AbstractList | The error linear complexity spectrum constitutes a well-known cryptographic criterion for sequences, indicating how the linear complexity of the sequence decreases as the number of bits allowed to be modified per period increases. In this paper, via defining an association between <inline-formula> <tex-math notation="LaTeX">2^{n} </tex-math></inline-formula>-periodic binary sequences and Boolean functions on <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> variables, it is shown that the error linear complexity spectrum also provides useful cryptographic information for the corresponding Boolean function <inline-formula> <tex-math notation="LaTeX">f </tex-math></inline-formula> - namely, it yields an upper bound on the minimum Hamming distance between <inline-formula> <tex-math notation="LaTeX">f </tex-math></inline-formula> and the set of functions depending on fewer number of variables. Therefore, the prominent Lauder-Paterson algorithm for computing the error linear complexity spectrum of a sequence may also be used for efficiently determining approximations of a Boolean function that depend on fewer number of variables. Moreover, it is also shown that, through this approach, low-degree approximations of a Boolean function can be also obtained in an efficient way. The error linear complexity spectrum constitutes a well-known cryptographic criterion for sequences, indicating how the linear complexity of the sequence decreases as the number of bits allowed to be modified per period increases. In this paper, via defining an association between [Formula Omitted]-periodic binary sequences and Boolean functions on [Formula Omitted] variables, it is shown that the error linear complexity spectrum also provides useful cryptographic information for the corresponding Boolean function [Formula Omitted] - namely, it yields an upper bound on the minimum Hamming distance between [Formula Omitted] and the set of functions depending on fewer number of variables. Therefore, the prominent Lauder-Paterson algorithm for computing the error linear complexity spectrum of a sequence may also be used for efficiently determining approximations of a Boolean function that depend on fewer number of variables. Moreover, it is also shown that, through this approach, low-degree approximations of a Boolean function can be also obtained in an efficient way. |
| Author | Limniotis, Konstantinos Kolokotronis, Nicholas |
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| SubjectTerms | Algorithms Approximation algorithms Approximations Boolean algebra Boolean functions Ciphers Codes Communication channels Complexity Complexity theory Correlation Criteria Cryptography error linear complexity spectrum Input variables Lauder-Paterson algorithm Upper bounds |
| Title | The Error Linear Complexity Spectrum as a Cryptographic Criterion of Boolean Functions |
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