Related papers: Efficient ECM factorization in parallel with the L…
Quispel-Roberts-Thompson (QRT) maps are a family of birational maps of the plane which provide the simplest discrete analogue of an integrable Hamiltonian system, and are associated with elliptic fibrations in terms of biquadratic curves.…
The elliptic curve method (ECM) is one of the best factorization methods available. It is possible to use hyperelliptic curves instead of elliptic curves but it is in theory slower. We use special hyperelliptic curves and Kummer surfaces to…
We present normal forms for elliptic curves over a field of characteristic $2$ analogous to Edwards normal form, and determine bases of addition laws, which provide strikingly simple expressions for the group law. We deduce efficient…
High-performance techniques to simulate quantum programs on classical hardware rely on exponentially large vectors to represent quantum states. When simulating quantum algorithms, the quantum states that occur are often sparse due to…
The arithmetic of elliptic curves, namely polynomial addition and scalar multiplication, can be described in terms of global sections of line bundles on $E\times E$ and $E$, respectively, with respect to a given projective embedding of $E$…
The complexity of the elliptic curve method of factorization (ECM) is proven under the celebrated conjecture of existence of smooth numbers in short intervals. In this work we tackle a different version of ECM which is actually much more…
Two prominent methods for integer factorization are those based on general integer sieve and elliptic curve. The general integer sieve method can be specialized to quadratic integer sieve method. In this paper, a probability analysis for…
Elliptic curve multiplications can be improved by replacing the standard ladder algorithm's base 2 representation of the scalar multiplicand, with mixed-base representations with power-of-2 bases, processing the n bits of the current digit…
For smooth finite fields $F_q$ (i.e., when $q-1$ factors into small primes) the Fast Fourier Transform (FFT) leads to the fastest known algebraic algorithms for many basic polynomial operations, such as multiplication, division,…
Lenstra's integer factorization algorithm is asymptotically one of the fastest known algorithms, and is ideally suited for parallel computation. We suggest a way in which the algorithm can be speeded up by the addition of a second phase.…
Multiplication is one of the most important operation in Elliptic Curve Cryptography (ECC) arithmetic. For point addition and point doubling in ECC scalar (integer) multiplication is required. In higher order classical (standard)…
The first step in elliptic curve scalar multiplication algorithms based on scalar decompositions using efficient endomorphisms-including Gallant-Lambert-Vanstone (GLV) and Galbraith-Lin-Scott (GLS) multiplication, as well as…
An efficient integer factorization algorithm would reduce the security of all variants of the RSA cryptographic scheme to zero. Despite the passage of years, no method for efficiently factoring large semiprime numbers in a classical…
We introduce the twisted $\boldsymbol{\mu}_4$-normal form for elliptic curves, deriving in particular addition algorithms with complexity $9\mathbf{M} + 2\mathbf{S}$ and doubling algorithms with complexity $2\mathbf{M} + 5\mathbf{S} +…
Pairings have been widely used since their introduction to cryptography. They can be applied to identity-based encryption, tripartite Diffie-Hellman key agreement, blockchain and other cryptographic schemes. The Acceleration of pairing…
We present an algorithm which speeds scalar multiplication on a general elliptic curve by an estimated 3.8 % to 8.5 % over the best known general methods when using affine coordinates. This is achieved by eliminating a field multiplication…
High-order methods gain more and more attention in computational fluid dynamics. However, the potential advantage of these methods depends critically on the availability of efficient elliptic solvers. With spectral-element methods, static…
Elliptic curve cryptography (ECC) has emerged as the dominant public-key protocol, with NIST standardizing parameters for binary field GF(2^m) ECC systems. This work presents a hardware implementation of a Hybrid Multiplication technique…
In this paper, we present several methods for construction of elliptic curves with large torsion group and positive rank over number fields of small degree. We also discuss potential applications of such curves in the elliptic curve…
Elliptic curves have a well-known and explicit theory for the construction and application of endomorphisms, which can be applied to improve performance in scalar multiplication. Recent work has extended these techniques to hyperelliptic…