Amicable Pairs for Elliptic Curves
Computational issues regarding elliptic curve groups became popular after N. Koblitz and V.Miller independently proposed in 1985 to use elliptic curves to design cryptographic systems. The security of such systems is connected to the order of the elliptic curve group.
We continue work by J. Silverman, K. Stange and others by studying specific properties of orders of elliptic curve groups. In particular, we study elliptic amicable pairs and elliptic curve groups E(F_p) generated by the equation of the form y^2 = x^3 + b or y^2 = x^3 + ax, where F_p is a finite field of prime order p.
An amicable pair (p,q) for an elliptic curve E is a pair of prime numbers for which #E(F_q) = p and #E(F_p) = q. We developed software to collect and analyze data on the frequency of elliptic amicable pairs and properties of the corresponding elliptic curves. We give several new conjectural results about the elliptic curve groups E(F_p) of prime order generated by equations of the form y^2 = x^3+b. Our conjectures include: (1) If each integer b
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