Hash Functions Based on Three Permutations: A Generic Security Analysis

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We consider the family of 2n-to-n-bit compression functions that are solely based on at most three permutation executions and on XOR-operators, and analyze its collision and preimage security. Despite their elegance and simplicity, these designs are not covered by the results of Rogaway and Steinberger (CRYPTO 2008). By defining a carefully chosen equivalence relation on this family of compression functions, we obtain the following results. In the setting where the three permutations π 1, π 2, π 3 are selected independently and uniformly at random, there exist at most four equivalence classes that achieve optimal 2 n/2 collision resistance. Under a certain extremal graph theory based conjecture, these classes are then proven optimally collision secure. Three of these classes allow for finding preimages in 2 n/2 queries, and only one achieves optimal 22n/3 preimage resistance (with respect to the bounds of Rogaway and Steinberger, EUROCRYPT 2008). Consequently, a compression function is optimally collision and preimage secure if and only if it is equivalent to F(x 1,x 2) = x 1 ⊕ π 1(x 1) ⊕ π 2(x 2) ⊕ π 3(x 1 ⊕ x 2 ⊕ π 1(x 1)). For compression functions that make three calls to the same permutation we obtain a surprising negative result, namely the impossibility of optimal 2 n/2 collision security: for any scheme, collisions can be found with 22n/5 queries. This result casts some doubt over the existence of any (larger) secure permutation-based compression function built only on XOR-operators and (multiple invocations of) a single permutation.