9,578 research outputs found
A primality proving using elliptic curves with complex multiplication by imaginary quadratic fields of class number three
In 2015, Abatzoglou, Silverberg, Sutherland, and Wong presented a framework
for primality proving algorithms for special sequences of integers using an
elliptic curve with complex multiplication. For some technical reason, their
framework can not be applied to an elliptic curve with complex multiplication
by an imaginary quadratic field of class number greater than two. In this
paper, we present a method to apply their framework to imaginary quadratic
fields of class number three. As an application, we give two special sequences
of integers to which our method can be applied, and a computational result for
the primality of these sequences
Aliquot Cycles for Elliptic Curves with Complex Multiplication
We review the history of elliptic curves and show that it is possible to form a group law using the points on an elliptic curve over some field L. We review various methods for computing the order of this group when L is finite, including the complex multiplication method. We then define and examine the properties of elliptic pairs, lists, and cycles, which are related to the notions of amicable pairs and aliquot cycles for elliptic curves, defined by Silverman and Stange. We then use the properties of elliptic pairs to prove that aliquot cycles of length greater than two exist for elliptic curves with complex multiplication, contrary to an assertion of Silverman and Stange, proving that such cycles only occur for elliptic curves of j-invariant equal to zero, and they always have length six. We explore the connection between elliptic pairs and several other conjectures, and propose limitations on the lengths of elliptic lists
Barreto-Naehrig Curve With Fixed Coefficient - Efficiently Constructing Pairing-Friendly Curves -
This paper describes a method for constructing Barreto-Naehrig (BN) curves and twists of BN curves that are pairing-friendly and have the embedding degree by using just primality tests without a complex multiplication (CM) method.
Specifically, this paper explains that the number of points of elliptic curves and over \Fp is given by 6 polynomials in , , two of which are irreducible, classified by the value of for a prime with an integer.
For example, elliptic curve over \Fp always becomes a BN curve for any with .
Let be irreducible.
Then, to construct a pairing-friendly elliptic curve, it is enough to find an integer of appropriate size such that and are primes
Analysis of Parallel Montgomery Multiplication in CUDA
For a given level of security, elliptic curve cryptography (ECC) offers improved efficiency over classic public key implementations. Point multiplication is the most common operation in ECC and, consequently, any significant improvement in perfor- mance will likely require accelerating point multiplication. In ECC, the Montgomery algorithm is widely used for point multiplication. The primary purpose of this project is to implement and analyze a parallel implementation of the Montgomery algorithm as it is used in ECC. Specifically, the performance of CPU-based Montgomery multiplication and a GPU-based implementation in CUDA are compared
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