Breaking and repairing an approximate message authentication scheme
Traditional hash functions are designed to protect against even the slightest modification of a message. Thus, one bit changed in a message would result in a totally different message digest when a hash function is applied. This feature is not suitable for applications whose message spaces admit a certain fuzziness, such as multimedia communications or biometric authentication applications. In these applications, approximate hash functions must be designed so that the distance between messages are proportionally reflected in the distance between message digests. Most of the previous designs of approximate hash functions employ traditional hash functions. In an ingenious approximate message authentication scheme for an N-ary alphabet recently proposed by Ge, Arce and Crescenzo, the approximate hash functions are based on the majority selection function. This scheme is suitable for N-ary messages with arbitrary alphabet size N. In this paper, we show a hidden property of the majority selection function, which allows us to successfully break this scheme. We show that an adversary, by observing just one message and digest pair, without any knowledge of the secret information, can generate N − 1 new valid message and digest pairs. In order to resist the attack, we propose some modifications to the original design. The corrected scheme is as efficient as the original scheme and it is secure against the attack. By a new combinatorial approach, we calculate explicitly the security parameters of the corrected scheme.