A Public-Key Cryptosystem Utilizing Cyclotomic Fields

While it is well-known that the RSA public-key cryptosystem can be broken if its modulus N can be factored, it is not known whether there are other ways of breaking RSA. This paper presents a public-key scheme which necessarily requires knowledge of the factorization of its modulus in order to be broken. Rabin introduced the first system whose security is equivalent to the difficulty of factoring the modulus. His scheme is based on squaring (cubing) for encryption and extracting square (cube) roots for decryption. This introduces a 1:4 (1:9) ambiguity in the decryption. Various schemes which overcome this problem have been introduced for both the quadratic and cubic case. We generalize the ideas of Williams' cubic system to larger prime exponents. The cases of higher prime order introduce a number of problems not encountered in the quadratic and cubic cases, namely the existence of fundamental units in the underlying cyclotomic field, the evaluation of higher power residue symbols, and the increased difficulty of Euclidean division in the field. Kcywords: Public-key cryptosystem, cyclotomic field, residue symbol, Euclidean division