CM55: special prime-field elliptic curves almost optimizing den Boer's reduction between Diffie-Hellman and discrete logs.
Daniel R. L. Brown · 2014
Using the Pohlig–Hellman algorithm, den Boer reduced the discrete logarithm prob-lem 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 × 2288, whose special form may provide some efficiency advantages over random fields of similar sizes. The curve has a point of Proth prime order 1 + 55×2286, 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