Pdf the discrete logarithm problem on elliptic curves. The discrete logarithm is an important crypto primitive for public key cryptography. In this research project the relevant theory of elliptic. Introduction discrete logarithm problem motivations discrete logarithm problem dlp given g group and g. Discrete logarithm problem the security of ecc depends on the difficulty of elliptic curve discrete logarithm problem. We say a call to an oracle is a use of the function on a speci ed input, giving us our desired output. I have just started studying elliptic curve cryptography, and i have this doubt. We often use the idea that we have an oracle to show rough computational.
In ecc the group operation is addition and not multiplication. In 2006, cheon proposed a novel algorithm for solving dlpwai cheons algorithm, 8,9, which is the center topic of this paper. Due to w ork of menezes, ok amoto and v anstone, 2, it. In 2006, cheon proposed a novel algorithm for solving dlpwai cheons algorithm, 8,9, which is the center topic of. Ecc is called the elliptic curve discrete logarithm problem ecdlp. The best known algorithm to solve the ecdlp is exponential, which is why elliptic curve groups are used for cryptography. Algorithms for the elliptic curve discrete logarithm and the. The ecdlp is similar to the oneway function on which dsa and diffiehellman are based, and hence, elliptic curve analogs of each of. Q2efq to nd an integer a, if it exists, such that q ap. On the discrete logarithm problem for primefield elliptic curves. Shors discrete logarithm quantum algorithm for elliptic curves. We show that for any sequences of prime powers q i i and natural numbers n i i with n i.
It turns out that for this problem a smaller quantum computer can solve problems further beyond current computing than for integer factorisation. If the elliptic curve groups is described using multiplicative notation, then the elliptic curve discrete logarithm problem is. On the discrete logarithm problem in elliptic curves p. Is the term elliptic curve discrete logarithm problem a. Any answer would be most beneficial for me if it remained light on.
For most elliptic curves, there is no known analogue of index calculus attacks on the discrete log problem. 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. In the multiplicative group zp, the discrete logarithm problem is. 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. On the discrete logarithm problem in elliptic curves ii. We report on our implementation of indexcalculus methods for hyperelliptic curves over characteristic two finite fields, and discuss the. For an elliptic curve e defined over a finite field k, an instance of the ecdlp is the following. We define three hard problems in the theory of elliptic di visibility sequences eds association, eds residue and eds discrete. Elliptic curves see also 11g05, 11g07, 14kxx 11g20. The paper is about the discrete logarithm problem for elliptic curves over characteristic 2.
On the discrete logarithm problem for primefield elliptic. May 09, 2018 14 videos play all elliptic curves in simple weierstrass form trustica blockchain tutorial 9. The dlp for elliptic curves defined over certain finite fields is believed to be hard. 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. Indeed, the theorem implies that restricted to instances as in the corollary, the elliptic curve discrete logarithm problem can be solved. An introduction to the theory of elliptic curves the discrete logarithm problem fix a group g and an element g 2 g. So, why is ecdlp stated as a variation of the discrete log problem. Correspondences on hyperelliptic curves and applications. An oracle is a theoretical constanttime \black box function. We study the elliptic curve discrete logarithm problem over finite extension fields. The discrete logarithm problem on elliptic curves of trace one.
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. 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 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. Nist requests comments on the set of recommended and allowed elliptic curves included in draft nist sp 800186. There are, however, no mathematical proofs for this belief. Recent progress on the elliptic curve discrete logarithm problem. 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. Pdf solving elliptic curve discrete logarithm problems. Cryptography and elliptic curves this chapter provides an overview of the use of elliptic curves in cryptography. For arbitrary nite groups the problem is dened as fol.
The main reasons of this list are to enhance research on. Recent progress on the elliptic curve discrete logarithm. The elliptic curve discrete logarithm problem ecdlp is the following computational problem. On the discrete logarithm problem in elliptic curves. The discrete logarithm problem on elliptic curves of trace. In our previous work die11b we have shown that there exist sequences of finite fields over which the elliptic curve discrete logarithm problem. There are two kinds of attack on the discrete logarithm problem. Wouldnt discrete multiplier problem or discrete factor problem be more apt.
One way to tackle this problem is to try to compute a from xa. On the discrete logarithm problem in elliptic curves claus diem university of leipzig on the discrete logarithm problem in elliptic curves p. 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. 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 finite. The elliptic curve discrete logarithm problem and equivalent hard. Solving a discrete logarithm problem with auxiliary input on. Then the discrete logarithm problem in the groups of rational points of elliptic curves over. Elliptic curve discrete logarithm problem ecdlp is the discrete logarithm problem for the group of points on an elliptic curve over a. Elliptic curve cryptosystems ecc have been widely recognized since 19853,6. 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.
Pdf the application of elliptic curves in public key cryptography is relatively recent. 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. The elliptic curve discrete logarithm problem and equivalent. Chapter 6 does provide a brief foray into elliptic curve cryptography with sections on the diffiehellman key exchange, the elgamal public. A las vegas algorithm to solve the elliptic curve discrete logarithm. Cryptography stack exchange is a question and answer site for software developers, mathematicians and others interested in cryptography. Understanding the qsdh problem on elliptic curves 0. Di ehellman key exchange protocol elgamal encryption and signature scheme, dsa. Suppose we want to use elliptic curves over fq with q.
We show that for any sequences of prime powers q i i. The dlp over elliptic curves is called ecdlp elliptic curve discrete logarithm problem. Shors discrete logarithm quantum algorithm for elliptic. 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 algorithm works for any finite field and it exploits summation polynomials. We consider practical issues about index calculus attacks using summation polynomials in this setting.
Isogenies and the discrete logarithm problem in jacobians of. Because of indexcalculus algorithms one has to avoid curves of genus. The discrete logarithm problem with auxiliary input dlpwai is a problem to. Any answer would be most beneficial for me if it remained light on the math, if thats at all possible. Some history at ecc 2004 in bochum, pierrick gaudry presented an. How can there be insecure elliptic curves if the discrete. At present it appears that given the discrete log problem in is much harder than the discrete log problem in the multiplicative group.
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. 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. Correspondences on hyperelliptic curves and applications to. Elliptic curve discrete logarithm 1 introduction emis. Curves over finite and local fields see also 14h25 keywords elliptic curves discrete logarithm problem. 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. Given p and q, it is computationally infeasible to obtain k, if k is sufficiently large.
Solving a discrete logarithm problem with auxiliary input. 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. Pdf on the discrete logarithm problem for primefield. We provide explicit formulae for isogenies with kernel isomorphic to z2z3 over an algebraic closure of the base.
The smallest integer m satisfying h gm is called the logarithm or index of h with respect to g, and is denoted. It turns out that for this problem a smaller quantum computer can solve problems further beyond current computing than for integer factorisati. Let p and q be two points on an elliptic curve such that kp q, where k is a scalar. In publickey cryptography, each participant possesses two keys.
We show in some detail how to implement shors efficient quantum algorithm for discrete logarithms for the particular case of elliptic curve groups. In this short note we describe an elementary technique which leads to a linear algorithm for solving the discrete logarithm problem. We rst provide a brief background to public key cryptography and the discrete logarithm problem, before introducing elliptic curves and the elliptic. Paper two discrete log algorithms for superanomalous. A 160 bit elliptic curve cryptographic key could be broken on a quantum computer using around. The first five or six chapters are not unique but otherwise necessary to establish the groundwork for this text i. The generalized weil pairing and the discrete logarithm. Q w p the best known algorithms for the ecdlp are algorithms that work on. On partial lifting and the elliptic curve discrete. Its improvement is evident for the primefield case. If its naturally hard to climb back through the trap door, how can there be insecure elliptic curves.
1277 931 1259 536 225 687 635 1412 296 1455 1446 1090 407 287 227 1403 1425 1490 340 335 959 474 1282 390 865 1067 768 1300 1448 1171 191 1337 450 757 132 1158 362 274 632 1073 488 1279 1086 363