Supersingular isogeny elliptic curve cryptography before we start, lets be clear. This chapter shows that ordinary elliptic curves, though widely used in traditional elliptic curve cryptography, do not provide a good foundation for postquantum cryptography. In this paper we give a new algorithm for computing the endomorphism ring of a supersingular elliptic curve e that runs, under certain heuristics, in time olog. The input to the hash function is a binary number of kdigits, and from this one computes a sequence of k2isogenies, starting at e.
For cryptographic purposes one needs non supersingular curves, whose group orders are divisible by a large prime factor. Subsequently, we show that isogenybased public key cryptography can exploit the fast kummer surface arithmetic that arises from the theory of theta functions. Practical postquantum key exchange from the learning with. If grh holds true, the expected run time of our algorithm is oelogq3.
I we found a quantum subexponential attack for ordinary i. Computing isogenies between supersingular elliptic curves. I largest embedding degree for supersingular elliptic curves ef 2n is k 4, and for ef 3n is k 6. Galbraith, \ supersingular curves in cryptography, asiacrypt 2001. Mathematicsdepartment, royalhollowayuniversityoflondon, egham,surreytw200ex,uk. Traditionally, most elliptic curve cryptography uses ordinary curves. I will survey the checkered history of supersingular elliptic curves in cryptography, from their first consideration in the seminal papers of koblitz and miller, to their rejection after the discovery of the weil and tate pairing attacks on the discrete logarithm problem for these curves, and concluding with their resurrection alongside the discovery of pairingbased cryptography. On the security of supersingular isogeny cryptosystems. It has its roots in elliptic curve cryptography ecc, a somewhat older branch of. Curves in the same isogeny class are either all supersingular or all ordinary. As a result supersingular elliptic curves are in general never used in cryptography. An introduction to supersingular elliptic curves and supersingular primes anh huynh abstract in this article, we introduce supersingular elliptic curves over a. In the elliptic curve case it is known that for supersingular curves one.
Unless otherwise stated, all rights belong to the author. Nov 20, 2001 in this paper curves of higher genus are studied. Supersingular isogeny elliptic curve cryptography sage. A constructive application of supersingular curves to cryptography is. Publickey cryptography from supersingular elliptic curve isogenies. The state of elliptic curve cryptography 175 it is well known that e is an additively written abelian group with the point 1serving as its identity element. Supersingular elliptic curves in cryptography springerlink.
Ecc requires smaller keys compared to nonec cryptography based on plain galois fields to provide equivalent security. Chapter 3 builds upon the materials in chapter 2 and elaborates more on the nature of endomorphism rings of supersingular constructing the deuring correspondence with applications to supersingular isogenybased cryptography 1. In this paper, we study a di erent primitive that does not fall into any of the above classes, but is currently believed to o er postquantum resistance. Hardware components for postquantum elliptic curves cryptography. Constructing the deuring correspondence with applications to. Now we hope to show that the endomorphism ring of a supersingular elliptic curve over a finite field is actually a maximal order. However, for some recent interesting cryptographic applications 18,15, 2,3,22,9, supersingular elliptic curves turn out to be very good. Craig costello summer school on realworld crypto and privacy. The jinvariant of eis the output of the hash function. The underlying hard problem for isogenybased cryptography is. Sike is a public key encryption pke, and a key encapsulation mechanism kem.
Elliptic curves and their applications to cryptography. Weil and tate pairings exist and have similar properties for abelian varieties that they have for elliptic curves. In lecture 7 we proved that for any nonzero integer n, the multiplicationbynmap n is separable if and only if n is not divisible by p. Silverberg, \ supersingular abelian varieties in cryptology, crypto 2002. We discuss both the advantages and drawbacks of our constructions, we study their security and we demonstrate their practicality with a proofofconcept implementation. Index terms elliptic curve isogenies, postquantum cryptography. Isogenies of hyperelliptic jacobians of dimension 2 or 3 have also. Jul, 2018 supersingular elliptic curves and is one of the promising schemes for postqua ntum cryptography. Discrete logarithm attacks menezes, okamoto, vanstone pairingbased cryptography joux hash functions from expander graphs charles, goren, lauter. Washington introduction the basic theory weierstrass equations the group law projective space and the point at.
Computational problems in supersingular elliptic curve. Veri able delay functions from supersingular isogenies and. Nist launches the postquantum cryptography standardization project. This chapter discusses some general methods to nd group orders of nite groups. Koblitz, \an elliptic curve implementation of the finite field digital signature algorithm, crypto 1998. Find materials for this course in the pages linked along the left. This gives rise to new possibilities for e cient supersingular isogenybased cryptography. You may download, display and print this publication for your own personal use. Ellipticcurve cryptography ecc is an approach to publickey cryptography based on the algebraic structure of elliptic curves over finite fields.
We assume for the remainder of this paper that we are in the supersingular case. It has its roots in elliptic curve cryptography ecc, a somewhat older branch of publickey cryptographythatwasstartedinthe1980s,whenmillerandkoblitz. Ways to ensure that a curve is not supersingular are also given. I supersingular curves in characteristic 2 or 3 good for pairings. In the elliptic curve case it was shown by menezes, okamoto and vanstone that for supersingular curves one has k. Supersingular isogenies, sidh, kummer surface, richelot isogeny, scholtens. Postquantum cryptography on fpga based on isogenies on. It is important in public key cryptography to find encryption and decryption functions and corre sponding key pairs such that for any key pair ui,ri.
Unfortunately, these fancy terms supersingular, elliptic curve, isogeny are bound to sound magical to the untrained ear. Elliptic curve cryptography was generalised to higher genus curves by. Towards quantumresistant cryptosystems from supersingular. Constructing the deuring correspondence with applications. We are interested in the set of supersingular curves up to isomorphism over a specific field thm mestre. In mathematics, the supersingular isogeny graphs are a class of expander graphs that arise in computational number theory and have been applied in ellipticcurve cryptography. It is analogous to the diffiehellman key exchange, but is based on walks in a supersingular isogeny graph and is designed to resist. Publickey cryptography from supersingular elliptic curve. Supersingular isogeny diffiehellman on edwards curves. Elliptic curves and postquantum cryptography computing. Frey and ruck gave a method to transform the discrete logarithm problem in the divisor class group of a curve over equation into a discrete. We investigate the postquantum security of supersingular cryptography, by considering a more general isogeny problem for supersingular curves. The purpose of this publication is to investigate how they can be used to process points of supersingular elliptic curves.
Hardware components for postquantum elliptic curves. For this reason, in the rest of the thesis we consider only the case of nonordinary, i. Sutherland 14 ordinary and supersingular elliptic curves let ekbe an elliptic curve over a eld of positive characteristic p. Supersingular curves are weak for crypto i when i started working on ecc in 1997 the mantra was. I a keyexchange protocol, similar to di ehellman, using isogenies between supersingular elliptic curves why isogenies. There is a problem with the chapter 2 pdf in the online edition of washington. Pdf an efficient signature scheme from supersingular elliptic. Computing supersingular isogenies on kummer surfaces. It seems like this would be a great drop in replacement for diffiehellman in both openssl and gpg. Block cipher is a concept from symmetric cryptography. Our main result is theorem 3 which states that for supersingular curves there is an upper bound, which depends only on the genus, on the values of the extension degree k. Annals of mathematics, mathematical sciences research institute 126 1986.
It is known that computing endomorphism rings of supersingular curves is equivalent to computing isogenies between supersingular elliptic curves, and it is believed that both these problems are hard 17,6. In this paper we give a new algorithm for computing the endomorphism ring of a supersingular elliptic curve e that runs, under certain heuristics, in time olog p2p1. Such curves can readily be used for pairing based cryptography. The fact that supersingular curves allow for fast group operations, suggests that they might be useful in cryptography. Our goal is to shed some light on this proposed type of postquantum cryptography and bring basic understanding of these mythical isogenies to the masses. Parti elliptic curves and cryptography throughout this part we let kbe a. Supersingular abelian varieties are a special class of abelian varieties. E cient algorithms for supersingular isogeny di ehellman. Let be any subset of the vertices of the graph, and be any vertex in.
Elliptic curves in cryptography factoring ecm, primality proving ecpp simple and fast key exchange digital signatures. Ways to ensure that a curve is not supersingular are also discussed. The prospect of a large scale quantum computer that is capable of implementing shors algorithm 48 has given rise to the eld of postquantum cryptography pqc. Elliptic curve cryptography, quantum safe cryptography, isogenies, supersingular curves 1 introduction the computation of an isogeny between two elliptic curves in an important problem in public key cryptography. A quantum algorithm for computing isogenies between supersingular elliptic curves jeanfran. First use of supersingular isogenies in cryptography. However, for some recent interesting cryptographic applications 18. Two elliptic curves e 1 and e 2 are isogenous if there exists an isogeny from e 1 to e 2. Elliptic curves and postquantum cryptography a quantum computer could e. Supersingular isogeny diffiehellman michael naehrig microsoft research real world cryptography conference new york, 4 january 2017. In particular, we show that chains of 2isogenies between elliptic curves can instead be computed as chains of richelot 2.
Bounds on the possible values for k in the case of supersingular curves are given which imply that supersingular curves are weaker than the general case for cryptography. Computing endomorphism rings of supersingular elliptic curves is an important problem in computational number theory, and it is also closely connected to the security of some of the recently proposed isogenybased cryptosystems. Supersingular isogeny diffiehellman key exchange sidh is a postquantum cryptographic algorithm used to establish a secret key between two parties over an otherwise insecure communications channel. Then we discuss supersingular curves and the weil pairing and see how the pairing can be used. However in chapter 7 a subexponential attack on the dlp for supersingular elliptic curves will be given. Im wondering if that code is available or if someone is working on it. Readings elliptic curves mathematics mit opencourseware. Postquantum cryptography, di ehellman key exchange, supersingular elliptic curves, isogenies, sidh. Resolving this problem acquaints us with a major algorithmic paradigm for computing isogenies, which is to nd a path in an isogeny graph. For standard elliptic curve cryptography, supersingular elliptic curves are known to be weak.
Hardness of supersingularisogeny graphbased cryptography. I because they seem to be quantumresistant why supersingular elliptic curves. Their vertices represent supersingular elliptic curves over finite fields and their edges represent isogenies between curves. Publickey encryption requires a trapdoor oneway function. To the best of our knowledge, we present the first hardware implementation of isogenybased cryptography available in the literature. Avoid supersingular curves, they are weak for crypto. Oct 31, 2016 postquantum cryptography on fpga based on isogenies on elliptic curves abstract. Supersingular curves di er from ordinary curves in many ways, and this has practical implications for algorithms that work with elliptic curves over nite elds, such as algorithms for counting points 16, generating codes 17, computing endomorphism rings 8, and calculating discrete logarithms.
An introduction to sidh sidh supersingular elliptic curves in. Online edition of washington available from oncampus computers. E 2 with a xed, smooth degree that is public which maps e 1 to e 2 supersingular isogeny problem given p. A quantum algorithm for computing isogenies between. Supersingular elliptic curves have many endomorphisms over the algebraic closure. It suffices to prove the maximality for all primes, that is, that is a maximal order in for all primes.
Of course grovers algorithm applies to any public key cryptosystem, but there is not a single system where we dont know a better algorithm than grovers. Elliptic curves in cryptography by ian blake, gadiel. Pdf since supersingular elliptic curve isogenies are one of the several. Are there any optimised implementations of the supersingular isogeny key exchange by defeo, jao, and plut. Supersingular abelian varieties in cryptology uci math. Stolbunov, constructing publickey cryptographic schemes based on class group action on a set of isogenous elliptic curves 2010 i.
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