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Publication Citation: Four Measures of Nonlinearity

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Author(s): Joan Boyar; Magnus Find; Rene C. Peralta;
Title: Four Measures of Nonlinearity
Published: June 23, 2013
Abstract: Cryptographic applications, such as hashing, block ciphers and stream ciphers, make use of functions which are simple by some criteria (such as circuit implementations), yet hard to invert almost everywhere. A necessary condition for the latter property is to be "sufficiently distant" from linear, and cryptographers have proposed several measures for this distance. In this paper, we show that four common measures, nonlinearity, algebraic degree, annihilator immunity, and multiplicative complexity, are incomparable in the sense that for each pair of measures, υ1, υ2, there exist functions f1, f2 with υ1(f1) > υ1(f2) but υ2(f1) < υ2(f2). We also present new connections between two of these measures. Additionally, we give a lower bound on the multiplicative complexity of collision-free functions.
Conference: 8th International Conference on Algorithms and Complexity (CIAC 2013)
Proceedings: 8th International Conference on Algorithms and Complexity (CIAC 2013)- Springer Lecture Notes in Computer Science
Volume: 7878
Pages: pp. 61 - 72
Location: Barcelona, -1
Dates: May 22-24, 2013
Keywords: nonlinearity measures, annihilator immunity, multiplicative complexity, algebraic degree, collision-free, one-wayness, SHA-3
Research Areas: Math, Computer Security
DOI: http://dx.doi.org/10.1007/978-3-642-38233-8_6  (Note: May link to a non-U.S. Government webpage)
PDF version: PDF Document Click here to retrieve PDF version of paper (691KB)