The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm.

In this paper we describe our distributed implementation of two factoring algorithms. the elliptic curve method (ecm) and the multiple polynomial quadratic sieve algorithm (mpqs). Since the summer of 1987. our erm-implementation on a network of MicroVAX processors at DEC’s Systems Research Center has factored several most and more wanted numbers from the Cun-ningham project. In the summer of 1988. we implemented the multiple polynomial quadratic sieve algorithm on rhe same network On this network alone. we are now able to factor any!@I digit integer, or to find 35 digit factors of numbers up to 150 digits long within one month. To allow an even wider distribution of our programs we made use of electronic mail networks For the distribution of the programs and for inter-processor communicatton. Even during the mitial stage of this experiment machines all over the United States and at various places in Europe and Ausnalia conhibuted 15 percent of the total factorization effort. At all the sites where our program is running we only use cycles that would otherwise have been idle. This shows that the enormous computational task of factoring 100 digit integers with the current algoritluns can be completed almost for free. Since we use a negligible fraction of the idle cycles of alI the machines on the worldwide elecnonic mail networks. we could factor 100 digit integers within a few days with a little more help.

The problem of finding the prime factors of large composite numbers has always been of mathematical interest. With the advent of public key cryptosystems it is also of practical importance, because the security of some of these cryptosystems, such as the Rivest-Shamir-Adelman (RSA) system, depends on the difficulty of factoring the public keys. In recent years the best known integer factorisation algorithms have improved greatly, to the point where it is now easy to factor a 60-decimal digit number, and possible to factor numbers larger than 120 decimal digits, given the availability of enough computing power. We describe several algorithms, including the elliptic curve method (ECM), and the multiple-polynomial quadratic sieve (MPQS) algorithm, and discuss their parallel implementation. It turns out that some of the algorithms are very well suited to parallel implementation. Doubling the degree of parallelism (i.e. the amount of hardware devoted to the problem) roughly increases the size of a number which can be factored in a fixed time by 3 decimal digits. Some recent computational results are mentioned – for example, the complete factorisation of the 617-decimal digit Fermat number F11 = 2211 + 1 which was accomplished using ECM.

Abstract. This paper explains the design and implementation of a highsecurity elliptic-curve-Diffie-Hellman function achieving record-setting speeds: e.g., 832457 Pentium III cycles (with several side benefits: free key compression, free key validation, and state-of-the-art timing-attack protection), more than twice as fast as other authors ’ results at the same conjectured security level (with or without the side benefits). 1

Using the parametrizations of Kubert, we show how to produce infinite families of elliptic curves which have prescribed nontrivial torsion over Q and rank at least one. These curves can be used to speed up the ECM factorization algorithm of Lenstra. We also briefly discuss curves with complex multiplication in this context. 1 Introduction 1.1 The ECM method of Lenstra [5] for finding a prime factor p of a number N uses a "random" elliptic curve E : y 2 = f(x) = x 3 + ax + b: If the number k of points on E modulo p is smooth, the method succeeds. Suyama [9] and Montgomery [7] developed infinite classes of curves E for which k has some prescribed small factors; on reasonable probabilistic assumptions (borne out in practice) this should lead to a slight improvement in the method. Specifically, Montgomery and Suyama each force a factor of 12 in k, and Montgomery forces a factor of 16 but only on the assumption that p is congruent to 1 modulo 4. In this paper, we show how to force a...

Abstract. This paper proposes a fast elliptic curve multiplication algorithm applicable for any types of curves over finite fields Fp (p a prime), based on [Mon87], together with criteria which make our algorithm resistant against the side channel attacks (SCA). The algorithm improves both on an addition chain and an addition formula in the scalar multiplication. Our addition chain requires no table look-up (or a very small number of pre-computed points) and a prominent property is that it can be implemented in parallel. The computing time for n-bit scalar multiplication is one ECDBL + (n − 1) ECADDs in the parallel case and (n − 1) ECDBLs + (n − 1) ECADDs in the single case. We also propose faster addition formulas which only use the x-coordinates of the points. By combination of our addition chain and addition formulas, we establish a faster scalar multiplication resistant against the SCA in both single and parallel computation. The improvement of our scalar multiplications over the previous method is about 37 % for two processors and 5.7 % for a single processor. Our scalar multiplication is suitable for the implementation on smart cards. 1

Abstract. Edwards recently introduced a new normal form for elliptic curves. Every elliptic curve over a non-binary field is birationally equivalent to a curve in Edwards form over an extension of the field, and in many cases over the original field. This paper presents fast explicit formulas (and register allocations) for group operations on an Edwards curve. The algorithm for doubling uses only 3M + 4S, i.e., 3 field multiplications and 4 field squarings. If curve parameters are chosen to be small then the algorithm for mixed addition uses only 9M + 1S and the algorithm for non-mixed addition uses only 10M + 1S. Arbitrary Edwards curves can be handled at the cost of just one extra multiplication by a curve parameter. For comparison, the fastest algorithms known for the popular “a4 = −3 Jacobian ” form use 3M + 5S for doubling; use 7M + 4S for mixed addition; use 11M + 5S for non-mixed addition; and use 10M + 4S for non-mixed addition when one input has been added before. The explicit formulas for non-mixed addition on an Edwards curve can be used for doublings at no extra cost, simplifying protection against side-channel attacks. Even better, many elliptic curves (approximately 1/4 of all isomorphism classes of elliptic curves over a non-binary finite field) are birationally equivalent — over the original field — to Edwards curves where this addition algorithm works for all pairs of curve points, including inverses, the neutral element, etc. This paper contains an extensive comparison of different forms of elliptic curves and different coordinate systems for the basic group operations (doubling, mixed addition, non-mixed addition, and unified addition) as well as higher-level operations such as multi-scalar multiplication.

Abstract. The heart of the improvements by Elkies to Schoof’s algorithm for computing the cardinality of elliptic curves over a finite field is the ability to compute isogenies between curves. Elkies ’ approach is well suited for the case where the characteristic of the field is large. Couveignes showed how to compute isogenies in small characteristic. The aim of this paper is to describe the first successful implementation of Couveignes’s algorithm. In particular, we describe the use of fast algorithms for performing incremental operations on series. We also insist on the particular case of the characteristic 2. 1.

Elliptic curve cryptography (ECC) was introduced by Victor Miller and Neal Koblitz in 1985. ECC proposed as an alternative to established public-key systems such as DSA and RSA, have recently gained a lot attention in industry and academia. The main reason for the attractiveness of ECC is the fact that there is no sub-exponential algorithm known to solve the discrete logarithm problem on a properly chosen elliptic curve. This means that significantly smaller parameters can be used in ECC than in other competitive systems such RSA and DSA, but with equivalent levels of security. Some benefits of having smaller key sizes include faster computations, and reductions in processing power, storage space and bandwidth. This makes ECC ideal for constrained environments such as pagers, PDAs, cellular phones and smart cards. The implementation of ECC, on the other hand, requires several choices such as the type of the underlying finite field, algorithms for implementing the finite field arithmetic and so on. In this paper we give we presen an selective overview of the main methods.

Abstract. We study the complexity of computing one or several terms (not necessarily consecutive) in a recurrence with polynomial coefficients. As applications, we improve the best currently known upper bounds for factoring integers deterministically and for computing the Cartier–Manin operator of hyperelliptic curves.