We provide a simple and exact formula for the minimum Miller loop length in Ate _i pairing based on Brezing–Weng curves, in terms of the involved parameters, under a mild condition on the parameters. It will also be shown that almost all cryptographically useful/meaningful parameters satisfy the mild condition. Hence the simple and exact formula is valid for them. It will also turn out that the formula depends only on essentially two parameters, providing freedom to choose the other parameters to address the design issues other than minimizing the loop length.

We are motivated by Ramanujan’s recursion formula for sums of the product of two Eisenstein series (Berndt in Ramanujan’s Notebook, Part II, 1989, Entry 14, p.332) and its proof, and also by Besge-Liouville’s convolution identity for the ordinary divisor function σ k−1 (n)  (Williams in Number Theory in the Spirit of Liouville, vol. 76, 2011, Theorem 12.3). The objective of this paper is to introduce and prove convolution identities for the twisted divisor functions σ ∗ k−1 (n)  as well as for the twisted Eisenstein series S 2k+2,χ 0    and S 2k+2,χ 1    , S ∗ 2k+2   , S ∗ 2k+2,χ 0    , and S ∗ 2k+2,χ 1    . As applications based on our main results, we establish many interesting identities for pyramidal, triangular, Mersenne, and perfect numbers. Moreover, we show how our main results can be used to obtain arithmetical formulas for the number of representations of an integer n as the sums of s squares.

In this paper, we study the convolution sums involving restricted divisor functions, their generalizations, their relations to Bernoulli numbers, and some interesting applications.

This paper presents results of an all-sky searches for periodic gravitational waves in the frequency range $[50, 1190] \text{ Hz}$ and with frequency derivative ranges of $\sim[-2 \times 10^{-9}, 1.1 \times 10^{-10}] \text{ Hz}/s$ for the fifth LIGO science run (S5). The novelty of the search lies in the use of a non-coherent technique based on the Hough-transform to combine the information from coherent searches on timescales of about one day. Because these searches are very computationally intensive, they have been deployed on the Einstein@Home distributed computing project infrastructure. The search presented here is about a factor 3 more sensitive than the previous Einstein@Home search in early S5 LIGO data. The post-processing has left us with eight surviving candidates. We show that deeper follow-up studies rule each of them out. Hence, since no statistically significant gravitational wave signals have been detected, we report upper limits on the intrinsic gravitational wave amplitude $h_0$. For example, in the 0.5 Hz-wide band at 152.5 Hz, we can exclude the presence of signals with $h_0$ greater than $7.6 \times 10^{-25}$ with a 90% confidence level.