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Title:
CM55: special prime-field elliptic curves almost optimizing den Boer’s reduction between Diffie-Hellman and discrete logs
Authors: Daniel R. L. Brown
Abstract:Using the Pohlig–Hellman algorithm, den Boer reduced the discrete logarithm problem to the Diffie–Hellman problem in groups of an order whose prime factors were each one plus a smooth number. This report reviews some related general conjectural lower bounds on the Diffie-Hellman problem in elliptic curve groups that relax the smoothness condition into a more commonly true condition. This report focuses on some elliptic curve parameters defined over a prime field size of size 9+55(2^288), whose special form may provide some efficiency advantages over random fields of similar sizes. The curve has a point of Proth prime order 1+55(2^286), which helps to nearly optimize the den Boer reduction. This curve is constructed using the CM method. It has cofactor 4, trace 6, and fundamental discriminant -55. This report also tries to consolidate the variety of ways of deciding between elliptic curves (or other algorithms) given the efficiency and security of each.
ePrint: https://eprint.iacr.org/2014/877
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