Luby-Rackoff ciphers: Why XOR is not so exclusive

TLDR: A four round Luby-Rackoff cipher is constructed, operating over finite groups of characteristic greater than 2, that is not only completely secure against adaptive chosen plaintext and ciphertext attacks, but has better time / space complexity and uses fewer random bits than all previously considered Luby, Rackoff ciphers of equivalent security in the literature.

Abstract: This work initiates a study of Luby-Rackoff ciphers when the bitwise exclusive-or (XOR) operation in the underlying Feistel network is replaced by a binary operation in an arbitrary finite group. We obtain various interesting results in this context: - First, we analyze the security of three-round Feistel ladders over arbitrary groups. We examine various Luby-Rackoff ciphers known to be insecure when XOR is used. In some cases, we can break these ciphers over arbitrary Abelian groups and in other cases, however, the security remains an open problem. - Next, we construct a four round Luby-Rackoff cipher, operating over finite groups of characteristic greater than 2, that is not only completely secure against adaptive chosen plaintext and ciphertext attacks, but has better time / space complexity and uses fewer random bits than all previously considered Luby-Rackoff ciphers of equivalent security in the literature. Surprisingly, when the group is of characteristic 2 (i.e., the underlying operation on strings is bitwise exclusive-or), the cipher can be completely broken in a constant number of queries. Notably, for the former set of results dealing with three rounds (where we report no difference) we need new techniques. However for the latter set of results dealing with four rounds (where we prove a new theorem) we rely on a generalization of known techniques albeit requires a new type of hash function family, called a monosymmetric hash function family, which we introduce in this work. We also discuss the existence (and construction) of this function family over various groups, and argue the necessity of this family in our construction. Moreover, these functions can be very easily and efficiently implemented on most current microprocessors thereby rendering the four round construction very practical.