Trapdoor one-way functions on elliptic curves and their application to shorter signatures and asymmetric encryption

摘要

A new trapdoor one-way function is provided. In a general sense, some quadratic algebraic integer z is used. One then finds a curve E and a rational map defining [z] on E. The rational map [z] is the trapdoor one-way function. A judicious selection of z will ensure that [z] can be efficiently computed, that it is difficult to invert, that determination of [z] from the rational functions defined by [z] is difficult, and knowledge of z allows one to invert [z] on a certain set of elliptic curve points.

主权项

1. A method of generating a digital signature performed by one or more processors, the method comprising:
obtaining a plurality of messages; generating a plurality of elliptic curve points by applying a hash function to each of the plurality of messages and converting each hash to a respective one of the plurality of elliptic curve points; generating a summed elliptic curve point by adding together the plurality of elliptic curve points; and generating the digital signature by applying an inverse of an endomorphism to the summed elliptic curve point, the endomorphism corresponding to a quadratic algebraic integer z that satisfies z2+uz+v=0, u and v being secret integers, and v being relatively prime to n.