Hi, the sage reference on http://www.sagemath.org/doc/reference/sage/schemes/elliptic_curves/constructor.html reads that the EllipticCurve constructor can either be used with N prime (>> EllipticCurve(GF(N), [a, b])) or with N composite (>> EllipticCurve(Zmod(N),[a,b])), where [a,b] are the curve parameters in simplified Weierstrass form. However, computing the cardinality only works when N is prime (in both cases: GF and Zmod). Hence Zmod/GF/EllipticCurve must be doing an is_prime or at least is_pseudoprime check. For ECM (Lenstra's Elliptic Curve Method) and particularly Goldwasser-Kilian/ECPP (elliptic curve primality proving), it would be useful to do computations on an elliptic curve and just pretend that N was prime. As the whole purpose of ECPP is to prove primality, an is_prime() call within ECPP would be useless.
Is there a way to use EllipticCurve() for composite N and to call cardinality() without calling is_prime() ? Alternatively, is there a way to call EllipticCurve() with a pseudoprimality test only ? (calling is_pseudoprime() would be acceptable as pseudoprimality tests only require polynomial time). For N composite, the cardinality/Schoof-Elkies-Atkin algorithm should crash or return a curve order which doesn't lie within the Hasse interval (and thus proves compositeness). Thanks for your help! Georg -- To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
