Towards a better understanding of one-wayness: Facing linear permutations Conference paper First Online: 25 May 2006 DOI :
10.1007/BFb0054136

Volume
1403 of the book series
Lecture Notes in Computer Science (LNCS) Cite this paper as: Hiltgen A.P. (1998) Towards a better understanding of one-wayness: Facing linear permutations. In: Nyberg K. (eds) Advances in Cryptology — EUROCRYPT'98. EUROCRYPT 1998. Lecture Notes in Computer Science, vol 1403. Springer, Berlin, Heidelberg Abstract The one-wayness of linear permutations, i.e., invertible linear Boolean functions F: {0,1}n → {0, 1}n, is investigated. For linear permutations with a triangular matrix description (tlinear permutations), we prove that one-wayness, C(F−1)/C(F) , is non-trivially upperbounded by 16√n , where C(.) denotes unrestricted circuit complexity. We also prove that this upper bound strengthens as the complexity of the inverse function increases, limiting the one-wayness of t-linear permutations with C(F−1) = n2/(c log2(n )) to a constant, i.e., a value that is independent of n. Direct implications for linear and also non-linear permutations are discussed. Moreover, and for the first time ever, a description is given about where, in the case of linear permutations, practical one-wayness would have to come from, if it exists.

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