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In this paper, a higher order differential attack on the hash function Luffa v1 is discussed. We confirmed that the algebraic degree of the permutation Qj which is an important non-linear component of Luffa grows slower than an ideal case both by the theoretical and the experimental approaches. According to our estimate, we can construct a distinguisher for step-reduced variants of Luffa v1 up to 7 out of 8 steps by using a block message. The attack for 7 steps requires 2216 messages. As far as we know, this is the first report which investigates the algebraic property of Luffa v1. Besides, this attack does not pose any threat to the security of the full-step of Luffa v1 nor Luffa v2.

Higher Order Differential Attack on Step-Reduced Variants of Luffa v1