Montgomery Curves Research Articles

Supersingularity of elliptic and Edwards curves over $F_{p^n}$

We consider the algebraic curves which have form of Edwards and elliptic curves with coordinates in $F_ p^n $. These curves are most effective support for a cyclic group of points which have many applications now. In this paper it was found conditions of supersingularity for Edwards curve and same case of twisted Edwards and Montgomery curves over $F_ p^n $ where $p\equiv 3 mod 4$. There was proved that projective twisted Edwards curve is not elliptic. Some properties of this curve were investigated. Conditions of minimal cofactor of twisted Edwards curve were founded. Construction of a curve with given order and minimal cofactor was made.

Montgomery curves and their arithmetic

Three decades ago, Montgomery introduced a new elliptic curve model for use in Lenstra's ECM factorization algorithm. Since then, his curves and the algorithms associated with them have become foundational in the implementation of elliptic curve cryptosystems. This article surveys the theory and cryptographic applications of Montgomery curves over non-binary finite fields, including Montgomery's x-only arithmetic and Ladder algorithm, x-only Diffie–Hellman, y-coordinate recovery, and 2-dimensional and Euclidean differential addition chains such as Montgomery's PRAC algorithm.

Using Montgomery curve arithmetic over F2p for point scalar multiplication on short Weierstrass curve over Fp with exactly one 2-torsion point and order not divisible by 4

Montgomery curves are well known because of their efficiency and side channel attacks vulnerability. In this article it is showed how Montgomery curve arithmetic may be used for point scalar multiplication on short Weierstrass curve ESW over Fp with exactly one 2-torsion point and #ESW (Fp) not divisible by 4. If P ∈ ESW (Fp) then also P ∈ ESW (Fp2). Because ESW (Fp2) has three 2-torsion points (because ESW (Fp) has one 2-torsion point) it is possible to use 2-isogenous Montgomery curve EM (Fp2) to the curve ESW (Fp2) for counting point scalar multiplication on ESW (Fp). However arithmetic in (Fp2) is much more complicated than arithmetic in Fp, in hardware implementations this method may be much more useful than standard methods, because it may be nearly 45% faster.

ECM using Edwards curves

This paper introduces EECM-MPFQ, a fast implementation of the elliptic-curve method of factoring integers. EECM-MPFQ uses fewer modular multiplications than the well-known GMP-ECM software, takes less time than GMP-ECM, and finds more primes than GMP-ECM. The main improvements above the modular-arithmetic level are as follows: (1) use Edwards curves instead of Montgomery curves; (2) use extended Edwards coordinates; (3) use signed-sliding-window addition-subtraction chains; (4) batch primes to increase the window size; (5) choose curves with small parameters and base points; (6) choose curves with large torsion.