Fast and rigorous factorization under the generalized Riemann hypothesis

The quadratic sieve (QS) algorithm is one of the most powerful algorithms to factor large composite primes used to break RSA cryptographic systems. The hardware structure of the QS algorithm seems to be a good fit for FPGA acceleration. Our new e-QS algorithm further simplifies the hardware architecture making it an even better candidate for C2H acceleration. This paper shows our design results in FPGA resource and performance when implementing very long arithmetic on the Nios microprocessor platform with C2H acceleration for different libraries (GMP, LIP, FLINT, NRMP) and QS architecture choices for factoring 32-2048 bit RSA numbers.

The aim of this paper is to justify the common cryptographic practice of selecting elliptic curves using their order as the primary criterion. We can formalize this issue by asking whether the discrete log problem (DLOG) has the same difficulty for all curves over a given finite field with the same order. We prove that this is essentially true by showing polynomial time random reducibility of DLOG among such curves, assuming the Generalized Riemann Hypothesis (GRH). We do so by constructing certain expander graphs, similar to Ramanujan graphs, with elliptic curves as nodes and low degree isogenies as edges. The result is obtained from the rapid mixing of random walks on this graph. Our proof works only for curves with (nearly) the same endomorphism rings. Without this technical restriction such a DLOG equivalence might be false; however, in practice the restriction may be moot, because all known polynomial time techniques for constructing equal order curves produce only curves with nearly equal endomorphism rings.

Elliptic curves cryptography was introduced independently by Victor Miller (Miller, 1986) and Neal Koblitz (Koblitz, 1987(Koblitz, ) in 1985. .At that time elliptic curve cryptography was not actually seen as a promising cryptographic technique.As time progress and further research and intensive development done especially on the implementation side, elliptic curve cryptography is now being implemented widely.Elliptic curves cryptography offers smaller key size, bandwidth savings and faster in implementations when compared to the RSA (Rivest-Shamir-Adleman) cryptography which based its security on the integer factorization problem.The most interesting feature of the elliptic curves is the group structure of the points generated by the curves, where points on the elliptic curves form a group.The security of elliptic curves cryptography relies on the elliptic curves discrete logarithm problem.The elliptic curve discrete logarithm problem is analogous to the ordinary algebraic discrete logarithm problem, l = g x , where given the l and g, it is infeasible to compute the x.Elliptic curve discrete logarithm problem deals with solving for n the relation P = nG.Given the point P and the point G, then it is very hard to find the integer n.To implement the discrete logarithm problem in elliptic curve cryptography, the main task is to compute the order of group of the curves or in other words the number of points on the curve.Computation to find the number of points on a curve, has given rise to several point counting algorithms.The Schoof and the SEA (Schoof-Elkies-Atkin) point counting algorithms will be part of the discussion in this chapter.This chapter is organized as follows: Section 2, gives some preliminaries on elliptic curves, and in section 3, elliptic curve discrete logarithm problem is discussed.Some relevant issues on elliptic curve cryptography is discussed in section 4, in which the Diffie-Hellman key exchange scheme, ElGamal elliptic curve cryptosystem and elliptic curve digital signature scheme are discussed here accompanied with some examples.Section 5 discussed the two point counting algorithms, Schoof algorithm and the SEA (Schoof-Elkies-Atkin) algorithm.Following the discussion in section 5, section 6 summaries some similarities and the differences between these two algorithms.Section 7 gives some brief literature on these two point counting algorithms.Finally, section 8 is the concluding remarks for this chapter. Elliptic curvesElliptic curves obtained their name from their relation to elliptic integrals that arise from the computation of the arc length of ellipses (Lawrence & Wade, 2006).Elliptic curves are www.intechopen.com Cryptography and Security in Computing1 1 2 3 4 2 2 4 Fig. 1.1. 2(1 ) (1 ) yx x x   www.intechopen.com

In this paper a probabilistic algorithm is exhibited that factors any positive integer n n into prime factors in expected time at most L n [ 1 2 , 1 + o ( 1 ) ] {L_n}[\tfrac {1}{2},1 + o(1)] for n → ∞ n \to \infty , where L x [ a , b ] = exp ( b ( log ⁡ x ) a ( log log ⁡ x ) 1 − a ) {L_x}[a,b] = {\text {exp}}(b{(\log x)^a}{({\text {log}}\log x)^{1 - a}}) . Many practical factoring algorithms, including the quadratic sieve and the elliptic curve method, are conjectured to have an expected running time that satisfies the same bound, but this is the first algorithm for which the bound can be rigorously proved. Nevertheless, this does not close the gap between rigorously established time bounds and merely conjectural ones for factoring algorithms. This is due to the advent of a new factoring algorithm, the number field sieve, which is conjectured to factor any positive integer n n in time L n [ 1 3 , O ( 1 ) ] {L_n}[\tfrac {1}{3},O(1)] . The algorithm analyzed in this paper is a variant of the class group relations method, which makes use of class groups of binary quadratic forms of negative discriminant. This algorithm was first suggested by Seysen, and later improved by A. K. Lenstra, who showed that the algorithm runs in expected time at most L n [ 1 2 , 1 + o ( 1 ) ] {L_n}[\tfrac {1}{2},1 + o(1)] if one assumes the generalized Riemann hypothesis. The main device for removing the use of the generalized Riemann hypothesis from the proof is the use of multipliers. In addition a character sum estimate for algebraic number fields is used, with an explicit dependence on possible exceptional zeros of the corresponding L L -functions. Another factoring algorithm using class groups that has been proposed is the random class groups method. It is shown that there is a fairly large set of numbers that this algorithm cannot be expected to factor as efficiently as had previously been thought.

We study which groups with pairing can occur as the Jacobian of a finite graph. We provide explicit constructions of graphs whose Jacobian realizes a large fraction of odd groups with a given pairing. Conditional on the generalized Riemann hypothesis, these constructions yield all groups with pairing of odd order, and unconditionally, they yield all groups with pairing whose prime factors are sufficiently large. For groups with pairing of even order, we provide a partial answer to this question, for a certain restricted class of pairings. Finally, we explore which finite abelian groups occur as the Jacobian of a simple graph. There exist infinite families of finite abelian groups that do not occur as the Jacobians of simple graphs.

Let N be a large odd integer. We show how to produce a long sequence {(X,, Y,)}t=12n of integers modulo N which satisfy Xi2 ? Yi modulo N, where Xi, > Nl/2 and | Yi | < cN1/2, Our sequence corresponds to a Hainiltonian path on the n-dimensiotial hypercube Cn, where n is ?(log N/ log log N). One application of these techniques is that, at each vertex of the hypercube, it is possible to search for equations of the form U2 ? V modulo N with V a product of small primes. The search is as in the quadratic sieve algorithm and therefore very fast. This yields a faster way of changing polynomials in the Multiple Polynomial Quadratic Sieve algorithm, since moving along the hypercube turns out to be very cheap.

Elliptic curve cryptosystem is used in the process of embedded systems, the selection and generation algorithm of the elliptic curve will directly affect the efficiency of systems. From Elliptic Curve's selection, Elliptic Curve's structure, Elliptic Curve's generation, this paper discussed the realization of a random elliptic curve method of Embedded Mobile Device, the SEA algorithm and its improved algorithm. The results show that this method can achieve a quick implementation of the elliptic curve method to improve the operating efficiency of embedded systems in the same security guarantees.

The paper proves that an odd composite integer N can be factorized in O(logN) time if N's big divisor is in the form 2α u + 1 or 2α u − 1 and its small divisor does not exceed 2α + 1 Theorems and corollaries around the conclusion are proved with detail mathematical reasoning. The results in the paper again demonstrate the significance of applying the valuated binary tree on analyzing the odd integers.

Deterministic polynomial factoring over finite fields: A uniform approach via [formula omitted]-schemes

Mollified moments of cubic Dirichlet L-functions over the Eisenstein field

To prove: that the generalized Riemann hypothesis is true, namely, that all the zeta zeros of L -functions are balanced on the one-half number line, using the scale of the harmonic zeta function. Now that the Riemann hypothesis has been proven (in “Summa Characteristica and the Riemann Hypothesis: Scaling Riemann’s Mountain”, Journal of Interdisciplinary Mathematics, Vol. 11 (6) (December 2008)), it can be extended by analytical continuation to prove the generalized hypothesis concerning L-functions. The initial “gene” or generating (one half) principle that proves the Riemann hypothesis also proves that no L-series has a zero with real part on the complex plane other than one half. As a result of this proof, the Birch/Swinnerton-Dyer conjecture is also resolved. Finally, what impact, if any, do these proofs have on security codes based upon prime numbers?

It has long been known that, under the assumption of the Riemann Hypothesis, one can give upper and lower bounds for the error ∑ p ≤ x log ⁡ p − x \sum _{p \le x} \log p - x in the Prime Number Theorem, such bounds being within a factor of ( log ⁡ x ) 2 (\log x)^{2} of each other and this fact being equivalent to the Riemann Hypothesis. In this paper we show that, provided “Riemann Hypothesis” is replaced by “Generalized Riemann Hypothesis”, results of similar (often greater) precision hold in the case of the corresponding formula for the representation of an integer as the sum of k k primes for k ≥ 4 k \ge 4 , and, in a mean square sense, for k ≥ 3 k \ge 3 . We also sharpen, in most cases to best possible form, the original estimates of Hardy and Littlewood which were based on the assumption of a “Quasi-Riemann Hypothesis”. We incidentally give a slight sharpening to a well-known exponential sum estimate of Vinogradov-Vaughan.

In this paper, we introduce a new method to solve the elliptic curve discrete logarithm probelm (ECDLP) over a finite field by using the elliptic curve lifting problem. Moreover, we propose to find a non-trivial point in E1(Q) in order to get a lifted elliptic curve with rank smaller than the number of lifted points. By this method, we conclude that finding a non-trivial point in E1(Q) implies solving the ECDLP, the discrete logarithm problem (DLP) and the integer factorization problem (IFP). Finally, we find that the minimum of canonical height of a point in E1(Q) is almost O(|Ẽ(Fp)|), which means that it is too large to be found by the brute force search.

Elliptic curves are some specific type of curves known as hyper elliptic curves.Compared to the integer factorization problem(IFP) based systems, using elliptic curve based cryptography will significantly decrease key size of the encryption.Therefore, application of this type of cryptography in systems that need high security and smaller key size has found great attention.Hyperelliptic curves help to make key length shorter.Many investigations are done with regard to improving computations, hardware and software implementation of these curves, their security and resistance against attacks.This paper studies and analyzes researches done about security and efficiency of hyperelliptic curves.

Several mathematical concepts from analytics number theory, especially the counting functions (CFs) are employed to solve some modern problems in mathematics, computer sciences, engineering and others. One of them is the factorization problem for integers into two primes. This work proposes an extended factorization problem using the CFs, as main part from analytics number theory, to factor a real number r into two primes p and q. The computations of the points doubling and addition on the elliptic curves E modulo a composite number n are also used in this extension to give new two versions of the factorization over real field R. On the first version, the primes are obtained from doubling and addition on elliptic curve points that are computed modulo composite number which is resulting as output of the CFs algorithm with the input r. Whereas, the primes, on the second version, are got directly from applying the CFs algorithm with r as input. The successful computation of the scalar multiplication kP on the elliptic curve E modulo primes are determined. The computational complexities of extended elliptic curve factorization versions are determined using the counting operations. These operations are elliptic curve and finite field operations. New numerical results on two versions are computed as study cases. The elliptic curve factorization algorithm over R (ECF-R) with two versions, is more efficient for factoring-based cryptography in comparing to the original ECM.