A prime sieve is an algorithm that finds the primes up to a bound n. We say that a prime sieve is incremental, if it can quickly determine if n+1 is prime after having found all primes up to n. We say a sieve is compact if it uses roughly √ n space or less. In this paper we present two new results: • We describe the rolling sieve, a practical, incremental prime sieve that takes O(n log logn) ti...

Bringmann and Dousse recently established a conjecture of Dyson dealing with the limiting asymptotics of the Andrews-Garvan crank statistic for integer partitions. A direct “sieving” technique is used to establish this conjecture and establish the range of validity. Unlike the approach of Bringmann and Dousse, the technique readily yields the analogous result for Dyson’s partition rank and all ...

We present the pseudosquares prime sieve, which finds all primes up to n. Define p to be the smallest prime such that the pseudosquare Lp > n/(π(p)(logn)); here π(x) is the prime counting function. Our algorithm requires only O(π(p)n) arithmetic operations and O(π(p) log n) space. It uses the pseudosquares primality test of Lukes, Patterson, and Williams. Under the assumption of the Extended Ri...

Tate pairings has found several new applications in cryptography. However, how to compute Tate pairing is a research focus in all kinds of applications of pairing-based cryptosystems (PBC). In the paper, the structure of Miller’s algorithm is firstly analyzed, which is used to implement Tate pairing. Based on the characteristics that Miller’s algorithm will be improved tremendous if the order o...

In the previous two papers with the same title ([LLY05] by W.C. Li, L. Long, Z. Yang and [ALL05] by A.O.L. Atkin, W.C. Li, L. Long), the authors have studied special families of cuspforms for noncongruence arithmetic subgroups. It was found that the Fourier coefficients of these modular forms at infinity satisfy three-term Atkin and Swinnerton-Dyer congruence relations which are the p-adic anal...