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A computer algebra system risa/asir consists of a command asir for interactive use and several subroutine libraries which can be used as the parts of other programs. The grammar of the user language of asir is a variant of that of C and asir has a built-in dbx-like debugger. Risa's subroutine libraries include basic arithmetic subroutines, parser, evaluator and storage manager, and each of them can be used individually.This paper describes the characteristics and structure of risa system. We also show some sample programs, usage of the debugger and some timing data of fundamental calculations.

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Schoof's algorithm is used to find a secure elliptic curve for cryptosystems, as it can compute the number of rational points on a randomly selected elliptic curve defined over a finite field. By realizing efficient combination of several improvements, such as Atkin-Elkies's method, the isogeny cycles method, and trial search by match-and-sort techniques, we can count the number of rational points on an elliptic curve over GF (p) in a reasonable time, where p is a prime whose size is around 240-bits.