In elliptic curve cryptosystems, scalar multiplications performed on the curves have much effect on the efficiency of the schemes, and many efficient methods have been proposed. In particular, recoding methods of the scalars play an important role in the performance of the algorithm used. For integer radices, non-adjacent form (NAF) and its generalizations (e.g., generalized non-adjacent form (...

This paper introduces fast algorithms for performing group operations on twisted Edwards curves, pushing the recent speed limits of Elliptic Curve Cryptography (ECC) forward in a wide range of applications. Notably, the new addition algorithm uses 8M for suitably selected curve constants. In comparison, the fastest point addition algorithms for (twisted) Edwards curves stated in the literature ...

In the underlying finite field arithmetic of an elliptic curve cryptosystem, field multiplication is the next computational costly operation other than field inversion. We present two novel algorithms for efficient implementation of field multiplication and modular reduction used frequently in an elliptic curve cryptosystem defined over GF (2). We provide a complexity study of the two algorithm...

It is clear that the negation map can be used to speed up the computation of elliptic curve discrete logarithms with the Pollard rho method. However, the random walks defined on elliptic curve points equivalence class {±P} used by Pollard rho will always get trapped in fruitless cycles. We propose an efficient alternative approach to resolve fruitless cycles. Besides the theoretical analysis, w...

Let C be a Q-curve with no complex multiplication. In this note we characterize the number fields K such that there is a curve C′ isogenous to C having all the isogenies between its Galois conjugates defined over K, and also the curves C′ isogenous to C defined over a number field K such that the abelian variety ResK/Q(C/K) obtained by restriction of scalars is a product of abelian varieties of...

Elliptic curve cryptosystems, proposed by Koblitz ((11]) and Miller ((15]), can be constructed over a smaller eld of deenition than the ElGamal cryptosystems ((5]) or the RSA cryptosystems ((19]). This is why elliptic curve cryptosystems have begun to attract notice. In this paper, we investigate eecient elliptic curve exponentiation. We propose a new coordinate system and a new mixed coordinat...

Elliptic curve cryptography is known for its complexity due to its discrete logarithm problem, and this gives advantage to the system used since the formula developed using this concept is hard to break, therefore this criteria has given mathematicians a courage to explore this area of research. Earlier in [10], we have showed a new method for imbedding plaintexts using complementary elliptic c...

We use properties of the division polynomials of an elliptic curve E over a finite field Fq together with a pure result about elliptic divisibility sequences from the 1940s to construct a very simple alternative to the Menezes-Okamoto-Vanstone algorithm for solving the elliptic curve discrete logarithm problem in the case where #E(Fq) = q − 1.

Let E ! P 1 be an elliptic surface deened over a number eld K, or equivalently an elliptic curve deened over K(T). In this note we prove, assuming Tate's conjecture, that the rank of E(K(T 1=n)) is bounded by F (E)d K (n), where F (E) is an explicit constant independent of n and d K (n) is an explicit elementary function. In particular, if K \ Q(d) = Q for all djn, then d K (n) = d(n) is just t...

Elliptic curve cryptosystems, proposed by Koblitz ((12]) and Miller ((16]), can be constructed over a smaller eld of deenition than the ElGamal cryptosystems ((6]) or the RSA cryptosystems ((20]). This is why elliptic curve cryptosystems have begun to attract notice. In this paper, we investigate eecient elliptic curve exponentiation. We propose a new coordinate system and a new mixed coordinat...