Aug 19 – 23
Santa Barbara




Hash Functions Based on Three Permutations: A Generic Security Analysis


Bart Mennink (KU Leuven, Belgium)

Bart Preneel (KU Leuven, Belgium)


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 pi_1, pi_2, pi_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 2^{2n/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 XOR pi_1(x_1) XOR pi_2(x_2) XOR pi_3(x_1 XOR x_2 XOR pi_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 2^{2n/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.




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