Use of elliptic curves in cryptography
Lecture notes in computer sciences; 218 on Advances in cryptology---CRYPTO 85
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Designs, Codes and Cryptography
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The previously best attack known on elliptic curve cryptosystems used in practice was the parallel collision search based on Pollard's ρ-method. The complexity of this attack is the square root of the prime order of the generating point used. For arbitrary curves, typically defined over GF(p) or GF(2m), the attack time can be reduced by a factor or √2, a small improvement. For subfield curves, those defined over GF(2ed) with coefficients defining the curve restricted to GF(2e), the attack time can be reduced by a factor of √2d. In particular for curves over GF(2m) with coefficients in GF(2), called anomalous binary curves or Koblitz curves, the attack time can be reduced by a factor of √2m. These curves have structure which allows faster cryptosystem computations. Unfortunately, this structure also helps the attacker. In an example, the time required to compute an elliptic curve logarithm on an anomalous binary curve over GF(2163) is reduced from 281 to 277 elliptic curve operations.