Using an RSA Accelerator for Modular Inversion

We present a very simple new algorithm for modular inversion. Modular inversion can be done by the extended Euclidean algorithm. We substitute the extended Euclidean algorithm by a standard (non-extended) Euclidean algorithm that works on integers of approximately double the length of the modulus. This substitution can be very useful on smart card coprocessors, since in some cases computations with longer numbers than necessary can be done at no extra cost. Many smart card coprocessors have been designed for the RSA algorithm of, say, 1024 bits length. On the other hand, elliptic curve algorithms work with much smaller numbers, and modular inversion is a much more important primitive in elliptic curve cryptography than in RSA cryptography. On one smart card coprocessor the new algorithm is more than twice as fast as the classical algorithm.