Four Measures of Nonlinearity

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, μ1, μ2, there exist functions f1, f2 with μ1(f1) > μ1(f2) but μ2(f1) < μ2(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.