Ken Nakamula

Constructing Pairing-Friendly Elliptic Curves Using Factorization of Cyclotomic Polynomials

The problem of constructing pairing-friendly elliptic curves has received a lot of attention. To find a suitable field for the construction we propose a method to find a polynomial u(x), by the method of indeterminate coefficients, such that i¾? k (u(x)) factors. We also refine the algorithm by Brezing and Weng using a factor of i¾? k (u(x)). As a result, we produce new families of parameters using our algorithm for pairing-friendly elliptic curves with embedding degree 8, and we compute some explicit curves as numerical examples.

Some Properties of Nonstar Steps in Addition Chains and New Cases Where the Scholz Conjecture Is True

Let ?(n) be the smallest possible length of addition chains for a positive integer n. Then Scholz conjectured that ?(2n?1)?n+?(n)?1, which still remains open. It is known that the Scholz conjecture is true when ?(n)?4, where ?(n) is the number of 1s in the binary representation of n. In this paper, we give some properties of nonstar steps in addition chains and prove that the Scholz conjecture is true for infinitely many new integers including the case where ?(n)=5.

Elliptic units and the class numbers of non-galois fields

0.1. In the field @ of complex numbers, let K be a subfield of a finite abelian extension of an imaginary quadratic field F. Our problem is to prove an explicit algebraic formula connecting the class number of K with a group of elliptic units and to give an effective algorithm of computing the class number and fundamental units of K. When K contains F, it has been solved in our previous paper [7]. We now assume that K does not contain F. Then R. Schertz [12] has derived such algebraic formulas from a result of C. Meyer [4] by ingenious matrix calculation. Embedding units by a logarithmic map into a commutative group algebra over the field [w of real numbers, we shall prove a relined formula for our algorithm in a simpler way which also makes the meaning of the matrix calculation clear. A summary of the paper has been published in [lo]. Let L be the composition of K and F, i.e., L = KF, Then the assumption above implies that L is finite abelian over F and quadratic over K. It is easy to show

NZMATH 1.0

This is an announcement of the first official release (version 1.0) of the system NZMATH for number theory by Python [18]. We review all functions in NZMATH 1.0, show its main properties added after the report [11] about NZMATH 0.5.0, and describe new features for stable development. The most important point of the release is that we can now treat number fields. The second major change is that new types of polynomial programs are provided. Elliptic curve primality proving and its related programs are also available, where we partly use a library outside NZMATH as an advantage of writing the system only by Python. A new feature is that NZMATH is registered on SourceForge [19] as an open source project in order to ensure continuous development of the project. This is a unique among existing systems for number theory.

Some Results for Some Conjectures in Addition Chains

An addition chain for a positive integer n is a sequence of positive integers 1 = a 0 < a 1 <… < a r = n, such that for each i ≥ 1, a i = a j + a k for some 0 ≤ j, k < i. The smallest length r for which an addition chain for n exists is denoted by l(n). Scholz conjectured that l(2 n − 1) ≤ n + l(n) − 1. Aiello and Subbarao proposed a stronger conjecture which is “for each integer n ≥ 1, there exists an addition chain for 2 n − 1 with length equals n+l(n) − 1.” This paper improves Brauer’s result for the Scholz conjecture. We propose a special class of addition chain called M B-chain, we conjecture that it is equivalent to l ° -chain and we prove that this conjecture is true for integers n ≤ 8 × 104. Also, we prove that the Scholz and Aiello-Subbarao conjectures are true for integers n ≤ 8 × 104.