Public-key cryptographic systems often involve raising elements of some group (e.g. GF(2 n), Z/NZ, or elliptic curves) to large powers. An important question is how fast this exponentiation can be done, which often determines whether a given system is practical. The best method for exponentiation depends strongly on the group being used, the hardware the system is implemented on, and whether one element is being raised repeatedly to different powers, different elements are raised to a fixed power, or both powers and group elements vary. This problem has received much attention, but the results are scattered through the literature. In this paper we survey the known methods for fast exponentiation, examining their relative strengths and weaknesses. 1

Abstract For many years mathematicians have searched for a fast and reliable primality test. This is especially relevant nowadays, because the RSA public-key cryptosystem requires very large primes in order to generate secure keys. I will describe some efficient randomised algorithms that are useful in practice, but have the defect of occasionally giving the wrong answer, or taking a very long time to give an answer. Recently Agrawal, Kayal and Saxena found a deterministic polynomial-time primality test. I will describe their algorithm, mention some improvements by Bernstein and Lenstra, and explain why this is not the end of the story.

Computer algebra systems (CASs) have become an essential computational tool in the last decade. General purpose CASs, which are designed to solve a wide variety of problems, have gained special prominence. In this chapter, the capabilities of seven major general purpose CASs (Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD and Reduce) are reviewed on 542 short problems covering a broad range of (primarily) symbolic mathematics.

We consider the classical problem of testing if a given (large) number n is prime or composite. First we outline some of the efficient randomised algorithms for solving this problem. For many years it has been an open question whether a deterministic polynomial time algorithm exists for primality testing, i.e. whether "PRIMES is in P". Recently Agrawal, Kayal and Saxena answered this question in the affirmative. They gave a surprisingly simple deterministic algorithm. We describe their algorithm, mention some improvements by Bernstein and Lenstra, and consider whether the algorithm is useful in practice. Finally, as a topic for future research, we mention a conjecture that, if proved, would give a fast and practical deterministic primality test.