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Four Measures of Nonlinearity

2013; Springer Science+Business Media; Linguagem: Inglês

10.1007/978-3-642-38233-8_6

1611-3349

Joan Boyar, Magnus Gausdal Find, René Peralta,

Cryptographic Implementations and Security

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, μ 1,μ 2, there exist functions f 1,f 2 with μ 1(f 1) > μ 1(f 2) but μ 2(f 1) < μ 2(f 2). We also present new connections between two of these measures. Additionally, we give a lower bound on the multiplicative complexity of collision-free functions.