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Unformatted text preview: les  show how difficult the problem really is. Computing discrete logarithms is closely related to factoring. If you can solve the discrete logarithm problem, then you can factor. (The converse has never been proven to be true.) Currently, there are three methods for calculating discrete logarithms in a prime field [370, 934, 648]: the linear sieve, the Gaussian integer scheme, and the number field sieve. The preliminary, extensive computing has to be done only once per field. Afterward, individual logarithms can be quickly calculated. This can be a security disadvantage for systems based on these fields. It is important that different applications use different prime fields. Multiple users in the same application can use a common field, though. In the world of extension fields, GF(2n) hasn’t been ignored by researchers. An algorithm was proposed in . Coppersmith’s algorithm makes finding discrete logarithms in fields such as GF(2127) reasonable and finding them in fields around GF(2400) possible . This was based on work in . The precomputation stage of this algorithm is enormous, but otherwise it is nice and efficient. A practical implementation of a less efficient version of the same algorithm, after a seven-hour precomputation period, found discrete logs in GF(2127) in several seconds each [1130, 180]. (This particular field, once used in some cryptosystems [142, 1631, 1632], is insecure.) For surveys of some of these results, consult [1189, 1039]. More recently, the precomputations for GF(2227), GF(2313), and GF(2401) are done, and significant progress has been made towards GF(2503). These calculations are being executed on an nCube-2 massively parallel computer with 1024 processors [649, 650]. Computing discrete logarithms in GF(2593) is still barely out of reach. Like discrete logarithms in a prime field, the precomputation required to calculate discrete logarithms in a polynomial field has to be done only once. Taher ElGamal  gives an algorithm for calc...
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This note was uploaded on 10/18/2010 for the course MATH CS 301 taught by Professor Aliulger during the Fall '10 term at Koç University.
- Fall '10