Unformatted text preview: o successful cryptanalytic results against the algorithm, it has never gained wide acceptance in the cryptographic community. The scheme is two to three orders of magnitude faster than RSA, but has some problems. The public key is enormous: 219 bits long. The data expansion is large: The ciphertext is twice as long as the plaintext. Some attempts at cryptanalysis of this system can be found in [8, 943, 1559, 306]. None of these were successful in the general case, although the similarity between the McEliece algorithm and knapsacks worried some. In 1991, two Russian cryptographers claimed to have broken the McEliece system with some parameters . Their paper contained no evidence to substantiate their claim, and most cryptographers discount the result. Another Russian attack, one that cannot be used directly against the McEliece system, is in [1447, 1448]. Extensions to McEliece can be found in [424, 1227, 976]. Other Algorithms Based on Linear Error-Correcting Codes
The Niederreiter algorithm  is closely related to the McEliece algorithm, and assumes that the public key is a random parity-check matrix of an error-correcting code. The private key is an efficient decoding algorithm for this matrix. Another algorithm, used for identification and digital signatures, is based on syndrome decoding ; see  for comments. An algorithm based on error-correcting codes  is insecure [698, 33, 31, 1560, 32]. 19.8 Elliptic Curve Cryptosystems
Elliptic curves have been studied for many years and there is an enormous amount of literature on the subject. In 1985, Neal Koblitz and V. S. Miller independently proposed using them for public-key cryptosystems [867, 1095]. They did not invent a new cryptographic algorithm with elliptic curves over finite fields, but they implemented existing public-key algorithms, like Diffie-Hellman, using elliptic curves. Elliptic curves are interesting because they provide a way of constructing “elements” and “rules of combining” that produce groups. These groups have enough familiar properties to build cryptographic algorithms, but they don’t have certain prop...
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