Then the discrete logarithm problem in the groups of rational points of elliptic curves over. Pdf the discrete logarithm problem on elliptic curves. In 2006, cheon proposed a novel algorithm for solving dlpwai cheons algorithm, 8,9, which is the center topic of. In our previous work die11b we have shown that there exist sequences of finite fields over which the elliptic curve discrete logarithm problem. Paper two discrete log algorithms for superanomalous. This problem is the fundamental building block for elliptic curve cryptography and pairingbased cryptography, and has been a major area of research in computational number. The main source for suitable groups are divisor class groups of carefully chosen curves over finite fields. If and, then, so is a solution to the discrete logarithm problem if has order or or is a product of reasonably small primes, then there are some methods for attacking the discrete log problem on, which are beyond the scope of this book. The paper is about the discrete logarithm problem for elliptic curves over characteristic 2. On the discrete logarithm problem for primefield elliptic curves.
We define three hard problems in the theory of elliptic di visibility sequences eds association, eds residue and eds discrete. The smallest integer m satisfying h gm is called the logarithm or index of h with respect to g, and is denoted. Introduction discrete logarithm problem motivations discrete logarithm problem dlp given g group and g. Ecc is called the elliptic curve discrete logarithm problem ecdlp. In publickey cryptography, each participant possesses two keys. The elliptic curve discrete logarithm problem and equivalent. Q2efq to nd an integer a, if it exists, such that q ap. May 09, 2018 14 videos play all elliptic curves in simple weierstrass form trustica blockchain tutorial 9. We study the elliptic curve discrete logarithm problem over finite extension fields. How can there be insecure elliptic curves if the discrete. Let p and q be two points on an elliptic curve such that kp q, where k is a scalar.
Di ehellman key exchange protocol elgamal encryption and signature scheme, dsa. Curves over finite and local fields see also 14h25 keywords elliptic curves discrete logarithm problem. We also relate the problem of eds association to the tate pairing and the mov, freyruc k and shipsey eds attacks on the elliptic curve discrete logarithm problem in the cases where these apply. In this research project the relevant theory of elliptic. Pdf on the discrete logarithm problem for primefield. The elliptic curve discrete logarithm problem ecdlp is the following computational problem. Is the term elliptic curve discrete logarithm problem a misnomer. Cryptography stack exchange is a question and answer site for software developers, mathematicians and others interested in cryptography. Pdf solving elliptic curve discrete logarithm problems. Recent progress on the elliptic curve discrete logarithm. Any answer would be most beneficial for me if it remained light on the math, if thats at all possible. Correspondences on hyperelliptic curves and applications to. Elliptic curves see also 11g05, 11g07, 14kxx 11g20. One way of attacking a discrete log problem is simple brute force.
The discrete logarithm problem is the mathematical trap door function underpinning elliptic curve cryptography. In ecc the group operation is addition and not multiplication. It turns out that for this problem a smaller quantum computer can solve problems further beyond current computing than for integer factorisati. On partial lifting and the elliptic curve discrete logarithm. Elliptic curve cryptosystems ecc have been widely recognized since 19853,6. The dlp and elliptic curves the group g is going to be ef q for some elliptic curve, in which case g and h are points on e and we are trying to nd an integer k with kg h. Q w p the best known algorithms for the ecdlp are algorithms that work on. Shors discrete logarithm quantum algorithm for elliptic curves. The discrete logarithm is an important crypto primitive for public key cryptography. On the discrete logarithm problem in elliptic curves claus diem university of leipzig on the discrete logarithm problem in elliptic curves p. Shors discrete logarithm quantum algorithm for elliptic. Elliptic curve discrete logarithm 1 introduction emis.
At present it appears that given the discrete log problem in is much harder than the discrete log problem in the multiplicative group. We show in some detail how to implement shors efficient quantum algorithm for discrete logarithms for the particular case of elliptic curve groups. For some problems, such as the discrete logarithm problem on general elliptic curves, generic attacks are currently the best known though better algorithms exist for curves of particular forms, e. Recent progress on the elliptic curve discrete logarithm problem. For arbitrary nite groups the problem is dened as fol. In the multiplicative group zp, the discrete logarithm problem is. In 2006, cheon proposed a novel algorithm for solving dlpwai cheons algorithm, 8,9, which is the center topic of this paper. We consider practical issues about index calculus attacks using summation polynomials in this setting. In this short note we describe an elementary technique which leads to a linear algorithm for solving the discrete logarithm problem on elliptic curves of trace one. May 23, 2015 this problem, which is known as the discrete logarithm problem for elliptic curves, is believed to be a hard problem, in that there is no known polynomial time algorithm that can run on a classical computer. Nist recommended elliptic curves, previously specified in fips 1864 appendix d, are now included in draft special publication sp 800186, recommendations for discrete logarithmbased cryptography. We provide explicit formulae for isogenies with kernel isomorphic to z2z3 over an algebraic closure of the base. An oracle is a theoretical constanttime \black box function. If its naturally hard to climb back through the trap door, how can there be insecure elliptic curves.
For most elliptic curves, there is no known analogue of index calculus attacks on the discrete log problem. Some history at ecc 2004 in bochum, pierrick gaudry presented an. In this short note we describe an elementary technique which leads to a linear algorithm for solving the discrete logarithm problem. Due to w ork of menezes, ok amoto and v anstone, 2, it. Correspondences on hyperelliptic curves and applications. On the discrete logarithm problem in elliptic curves p. Wouldnt discrete multiplier problem or discrete factor problem be more apt. I have just started studying elliptic curve cryptography, and i have this doubt. For an elliptic curve e defined over a finite field k, an instance of the ecdlp is the following. We show that for any sequences of prime powers q i i and natural numbers n i i with n i. Its improvement is evident for the primefield case. We rst provide a brief background to public key cryptography and the discrete logarithm problem, before introducing elliptic curves and the elliptic. The ecc is believed to have higher security than the rsa scheme8, except for special elliptic curves. In practice the method described means that when choosing elliptic curves to use in cryptography one has to eliminate all curves whose group orders are equal to the order of the.
On the discrete logarithm problem for primefield elliptic. The algorithm works for any finite field and it exploits summation polynomials. We provide the first cryptographically interesting instance of the elliptic curve discrete logarithm problem which resists all previously known attacks, but which can be solved with modest computer resources using the weil descent attack methodology of frey. An introduction to the theory of elliptic curves the discrete logarithm problem fix a group g and an element g 2 g. Solving a discrete logarithm problem with auxiliary input. It is thus important to be able to compute efficiently, in order to verify that the elliptic curve one wishes to use for a cryptosystem doesnt have any. Solving a discrete logarithm problem with auxiliary input on. Algorithms for the elliptic curve discrete logarithm and the. Any answer would be most beneficial for me if it remained light on.
We rst provide a brief background to public key cryptography and the discrete logarithm problem, before introducing elliptic curves and the elliptic curve analogue of the discrete logarithm problem. Given p and q, it is computationally infeasible to obtain k, if k is sufficiently large. Summation polynomial algorithms for elliptic curves in. Discrete logarithm problem the security of ecc depends on the difficulty of elliptic curve discrete logarithm problem. The dlp for elliptic curves defined over certain finite fields is believed to be hard. The shanks method and the kangaroo method of pollard can also be used to compute the discrete logarithm of in about j ehg6i steps when this discrete log is known to lie in an interval of. It turns out that for this problem a smaller quantum computer can solve problems further beyond current computing than for integer factorisation. On the discrete logarithm problem in elliptic curves claus diem august 9, 2010 dedicated to gerhard frey abstract we study the elliptic curve discrete logarithm problem over. On the discrete logarithm problem in elliptic curves ii.
Indeed, the theorem implies that restricted to instances as in the corollary, the elliptic curve discrete logarithm problem can be solved. We show that for any sequences of prime powers q i i. The first five or six chapters are not unique but otherwise necessary to establish the groundwork for this text i. A 160 bit elliptic curve cryptographic key could be broken on a quantum computer using around. Nist requests comments on the set of recommended and allowed elliptic curves included in draft nist sp 800186. On the discrete logarithm problem in elliptic curves. So, why is ecdlp stated as a variation of the discrete log problem. The ecdlp is similar to the oneway function on which dsa and diffiehellman are based, and hence, elliptic curve analogs of each of.
There are, however, no mathematical proofs for this belief. We say a call to an oracle is a use of the function on a speci ed input, giving us our desired output. Because of indexcalculus algorithms one has to avoid curves of genus. The elliptic curve discrete logarithm problem and equivalent hard. Cryptography and elliptic curves this chapter provides an overview of the use of elliptic curves in cryptography. Chapter 6 does provide a brief foray into elliptic curve cryptography with sections on the diffiehellman key exchange, the elgamal public. With the basics of public key cryptography in hand, we are now in a position to apply elliptic curves to public key cryptography in order to generate public and private keys. We often use the idea that we have an oracle to show rough computational. There are two kinds of attack on the discrete logarithm problem. For elliptic curvebased protocols, it is assumed that finding the discrete logarithm of a random elliptic curve element with respect to a publicly known base point is infeasible. Is the term elliptic curve discrete logarithm problem a. Just one relation among points of the factor base needs to be found.
We report on our implementation of indexcalculus methods for hyperelliptic curves over characteristic two finite fields, and discuss the. One way to tackle this problem is to try to compute a from xa. Suppose we want to use elliptic curves over fq with q. Elliptic curve discrete logarithm problem ecdlp is the discrete logarithm problem for the group of points on an elliptic curve over a. In practice the method described means that when choosing elliptic curves to use in cryptography one has to eliminate all curves whose group orders are equal to the order of the finite. The main reasons of this list are to enhance research on. The best known algorithm to solve the ecdlp is exponential, which is why elliptic curve groups are used for cryptography. The elliptic curves discrete logarithm problem and an. The discrete logarithm problem on elliptic curves of trace. Several cryptographic schemes base their security upon the hardness of the discrete logarithm problem for elliptic curves ecdlp. Isogenies and the discrete logarithm problem in jacobians. The generalized weil pairing and the discrete logarithm. The discrete logarithm problem with auxiliary input dlpwai is a problem to. Pdf the application of elliptic curves in public key cryptography is relatively recent.
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