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**1 - 1**of**1**### Higher Order Universal One-Way Hash Functions from the Subset Sum Assumption ∗

, 2006

"... Universal One-Way Hash Functions (UOWHFs) may be used in place of collision-resistant functions in many public-key cryptographic applications. At Asiacrypt 2004, Hong, Preneel and Lee introduced the stronger security notion of higher order UOWHFs to allow construction of long-input UOWHFs using the ..."

Abstract
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Universal One-Way Hash Functions (UOWHFs) may be used in place of collision-resistant functions in many public-key cryptographic applications. At Asiacrypt 2004, Hong, Preneel and Lee introduced the stronger security notion of higher order UOWHFs to allow construction of long-input UOWHFs using the Merkle-Damg˚ard domain extender. However, they did not provide any provably secure constructions for higher order UOWHFs. We show that the subset sum hash function is a kth order Universal One-Way Hash Function (hashing n bits to m < n bits) under the Subset Sum assumption for k = O(log m). Therefore we strengthen a previous result of Impagliazzo and Naor, who showed that the subset sum hash function is a UOWHF under the Subset Sum assumption. We believe our result is of theoretical interest; as far as we are aware, it is the first example of a natural and computationally efficient UOWHF which is also a provably secure higher order UOWHF under the same well-known cryptographic assumption, whereas this assumption does not seem sufficient to prove its collisionresistance. A consequence of our result is that one can apply the Merkle-Damg˚ard extender to the subset sum compression function with ‘extension factor ’ k + 1, while losing (at most) about k bits of UOWHF security relative to the UOWHF security of the compression function. The method also leads to a saving of up to m log(k + 1) bits in key length relative to the Shoup XOR-Mask domain extender applied to the subset sum compression function.