Luby-Rackoff Variants Over Finite Groups
Feistel permutations appear in a number of well-known block ciphers, including DES. In 1977 the Data Encryption Standard (DES) became the first symmetric key cryptosystem accepted as a standard by NIST. Triple-DES (three consecutive DES encryptions) is commonly used in electronic financial transactions. In 1988, Luby and Rackoff formalized the notion of a provably-secure block cipher and introduced the Luby-Rackoff ciphers. Luby-Rackoff ciphers also use Feistel permutations. They showed that composition of four independently keyed Feistel permutations yields a cipher secure against adaptive chosen plain text and cipher text attacks. The bitwise exclusive-or operation underlies the mentioned ciphers. There is a strong relationship between the algebraic properties of a cryptosystem and its security. Current techniques to discover the algebraic properties of block ciphers based on bitwise exclusive-or are computationally intensive. Our research objective is to investigate well-publicized problems related to the algebraic structure of these ciphers, but based on arbitrary finite groups. Using non-computational methods, we showed that the above mentioned Luby-Rackoff ciphers over any finite group do not form groups. Thus iterated, Luby-Rackoff encryption can increase security. Using a method of Coppersmith we showed for a large number of finite groups that all published simplified versions of DES the set of encryption permutations do not form a group. We also showed that DES ciphers over finite groups can generate, besides the alternating group, a permutation group that contains an odd permutation, solving an open problem from 1980's.