New ECDLP record

A tweet today suggests the ECDLP computation started in the Faster discrete logarithms on FPGAs paper by Daniel J. Bernstein, Susanne Engels, Tanja Lange, Ruben Niederhagen, Christof Paar, Peter Schwabe and Ralf Zimmermann may have finally completed. See the below link for the tweet:

https://twitter.com/cryptocephaly/status/803542260256276481

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ECC 2016, İzmir

The 20th Elliptic Curve Cryptography workshop was held in İzmir, Turkey, from the 5th to the 7th of September this year. Many international colleagues unfortunately couldn’t be there this year: you all missed out. The conference was extremely well organized (special shout-out to the excellent Yaşar Geek Squad!), and almost all of the talks are online…  With video!  So rather than me writing about what was said and done, you can go and check it out for yourself.

It’s remarkable that this workshop has been running successfully for 20 years now: elliptic curve cryptography has come a long way.  It was great to be there to celebrate a milestone of sorts:

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Happy 20th birthday, ECC Workshop…  (Photo: Peter Stevenhagen)

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The famous Geek Squad of Izmir, who made everything happen. (Photo: Gamze Orhon)

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Did I mention how well-organised it all was?  Here’s one of the lecture theatres during a break.

–Ben Smith

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Twelfth Algorithmic Number Theory Symposium, Kaiserslautern, 2016

The Algorithmic Number Theory Symposium (ANTS-XII) took place at the Technical University of Kaiserslautern from August 29 to September 2, 2016.

Apart from the excellent invited lectures, the most memorable event of the conference was the late-night walk through the forest, illuminated by hand-held flaming torches, from the conference dinner at Bremerhof.

The Selfridge Prize was presented to J. Steffen Müller (Oldenburg) for the paper “Computing canonical heights on elliptic curves in quasi-linear time” by J. Steffen Müller and Michael Stoll.

The published papers are available in the LMS Journal of Computational Mathematics. Sadly this will be the final year that the proceedings appear in this journal, since the journal is being closed down.

There were relatively few papers with major relevance to ECC, but the following papers may be of some interest to readers of this blog:

  • Chris Peikert “Finding Short Generators of Ideals, and Implications for Cryptography“. This was an overview of the work presented in his paper with Cramer, Ducas and Regev.
  • Gary McGuire, Henriette Heer and Oisin Robinson “JKL-ECM: An implementation of ECM using Hessian curves”. The paper was about choosing elliptic curves in Hessian form with large torsion groups for the elliptic curve factoring method.
  • Jung Hee Cheon, Jinhyuck Jeong and Changmin Lee “An algorithm for NTRU problems and cryptanalysis of the GGH multilinear map without an encoding of zero”. This paper is another nail in the coffin of multilinear maps.
  • Francois Morain, Charlotte Scribot and Benjamin Smith “Computing cardinalities of Q-curve reductions over finite fields”. This was about a variant of the SEA method that is suitable for counting points on special curves with an endomorphism of a special type. Such curves are suitable for fast implementations of ECC, and so the method in this paper helps to speed up parameter generation when using such curves.
  • Luca Defeo, Jerome Plût, Eric Schost and Cyril Hugounenq “Explicit isogenies in quadratic time in any characteristic”. The paper is about a Couveignes-type method for computing an explicit isogeny between two curves. This is a useful ingredient in point counting algorithms. The new method is appropriate when working in characteristic p that is neither “large” nor “very small”.
  • Jean-François Biasse, Claus Fieker and Michael Jacobson “Fast heuristic algorithms for computing relations in the class group of a quadratic order with applications to isogeny evaluation”. This paper is about the problem of “smoothing” an isogeny by reducing the ideal corresponding to it in the ideal class group of the order. It introduces some nice techniques that had not been used in this context previously.

The rump session contained a number of jokes about Australia and New Zealand. Aurore Guillevic mentioned some recent DLP records (mostly already mentioned on this blog). Rump session slides will be available eventually here.

The 2018 edition of the ANTS conference is expected to take place in Madison, Wisconsin.

— Steven Galbraith

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CRYPTO and CHES 2016, Santa Barbara, CA, USA

Like every three years since 2010, CRYPTO and CHES this year were jointly held at the University of California Santa Barbara on the third week of August, and featured the usual chocolate strawberries, beach barbecue and India pale ale.

The scientific programs of both conferences overlapped somewhat (the CRYPTO program ran from Monday through Thursday morning, while CHES ran from Wednesday to Friday with optional tutorials on Tuesday), and CRYPTO now has parallel sessions, so attendees to both conferences effectively had to choose between three parallel tracks. Yet you could probably attend all the talks related to elliptic curve cryptography, as there just weren’t that many.

At CRYPTO, the two most relevant presentations were given in the Algorithmic Number Theory session on Wednesday morning:

  • Taechan Kim presented his paper with Razvan Barbulescu, “Extended Tower Number Field Sieve: A New Complexity for the Medium Prime Case”, about which Aurore Guillevic wrote an extensive survey on this blog a few months ago. The paper obtains a better complexity for the discrete logarithm problem in some composite degree extensions of finite fields, and although Taechan spent a good part of his talk trying to downplay the concrete impact, it actually translates to a significant reduction in the security of the most popular pairing-friendly elliptic curves. In particular, after this attack, 256-bit Barreto-Naehrig curves no longer offer 128 bits of security, but perhaps closer to 96 or so.
  • Craig Costello presented his paper with Patrick Longa and Michael Naehrig, “Efficient Algorithms for Supersingular Isogeny Diffie-Hellman”, which uses a number of clever tricks to implement the postquantum-secure isogeny-based key exchange protocol of De Feo, Jao and Plût significantly more efficiently than what previously thought possible. Although SIDH still lags behind other popular postquantum constructions based e.g. on lattices by several orders of magnitude in terms of performance, it uses comparatively short keys, can be combined with classical ECDH very cheaply, and in any case is based on a very different type of security assumption that may look more appealing to the algebraic geometrically inclined.

Other papers related to elliptic curves include:

  • “Design in Type-I, Run in Type-III: Fast and Scalable Bilinear-Type Conversion using Integer Programming” by Masayuki Abe, Fumitaka Hoshino and Miyako Ohkubo, which explains how to algorithmically convert pairing-based protocols using symmetric pairings to the asymmetric setting at a minimal overhead using integer linear programming techniques;
  • the CRYPTO best paper, “Breaking the Circuit Size Barrier for Secure Computation Under DDH“ by Elette Boyle, Niv Gilboa and Yuval Ishai, which is not elliptic curve crypto per se, but relies on an interesting observation regarding discrete logarithms. The idea is that if two parties hold a secret sharing of a small value z in the exponent, i.e. g^{z_1} and g^{z_2} with z_2-z_1=z, they can derive from that an additive secret sharing of z itself without any interaction. To do so, they agree on a polynomially dense subset X of distinguished points in the group, and count how many steps it takes to reach an element of X from their respective share. If z is small enough compared to the relative density of X, they should reach the same element of X with good probability, and in that case, if it took x_1 (resp. x_2) steps for the first (resp. second) party, we have g^{z_1+x_1} = g^{z_2+x_2}, hence x_2-x_1 = z_2-z_1 = z: x_1,x_2 is a secret sharing of z obtained without interaction!

At CHES, on the other hand, there were several interesting papers about the implementation of elliptic and hyperelliptic curve cryptography on various platforms.

  • On desktop CPUs: the paper by Thomaz Oliveira, Julio López and Francisco Rodríguez-Henríquez, “Software Implementation of Koblitz Curves over Quadratic Fields”. Usual Koblitz curves are defined over \mathbb{F}_2 and use the fast Frobenius endomorphism (x,y)\mapsto (x^2,y^2) instead of doublings to speed up scalar multiplication. This paper instead investigates curves defined over \mathbb{F}_4 together with the corresponding, slightly less fast Frobenius (x,y)\mapsto (x^4,y^4), and shows that the quadratic extension structure of the corresponding fields yields interesting performance benefits. The authors obtain a constant time scalar multiplication in under 70k cycles on Haswell and 52k cycles on Skylake at the 128-bit security level, which is quite respectable, even though they have to rely on a suboptimal field size of close to 300 bits to find a curve with a sufficiently large prime subgroup.
  • On embedded CPUs: the paper by Leijla Batina, Joost Renes, Peter Schwabe and Benjamin Smith, “µKummer: Efficient Hyperelliptic Signatures and Key Exchange on Microcontrollers”. It is well-known that Kummer surfaces support a notion of scalar multiplication, but not point addition directly because it is not compatible with quotienting by [-1]. As a result, they would only be used for protocols like Diffie-Hellman, and not for e.g. signatures, which require point additions. However, Chung, Costello and Smith recently observed that you can simply lift back to the actual Jacobian after carrying out your fast variable base point multiplication on the Kummer, and doing so is likely to be faster than doing everything in a Jacobian, especially if you want constant time arithmetic. This CHES paper is a concrete demonstration of that idea on constrained software platforms (AVR ATmega and ARM Cortex M0), where the authors break earlier speed records for (H)ECC by wide margins.
  • In hardware: the paper by Kimmo Järvinen, Andrea Miele, Reza Azarderakjsh and Patrick Longa, “FourQ on FPGA: New Hardware Speed Records for Elliptic Curve Cryptography over Large Prime Characteristic Fields”. FourQ is a very nice curve introduced by Costello and Longa at last year’s ASIACRYPT, which currently holds essentially all of the speed records on desktop CPUs for constant-time scalar multiplication (both fixed and variable base) by a comfortable margin. This CHES paper implements it on FPGA, and finds that it also performs faster than other implementations over large characteristic fields (although not nearly as fast as comparable binary field designs).

The CHES rump session also featured some annoucements of note, including a concrete complexity estimate by Francisco Rodríguez-Henríquez and his colleagues of the quasilinear attack on discrete logs in the formerly 128-bit secure field \mathbb{F}_{3^{6\cdot 509}} that used to be recommended for pairings (answer: if everybody in the world was working on it 8 hours per day, 1000 \mathbb{F}_{3^6} multiplications per hour, it would only take about 10 months!). The annoucement that personally got me the most excited is an improved implementation result for the binary curve GLS254 on desktop CPUs due to Thomaz Oliveira, Diego Aranha, Julio López and Francisco Rodríguez-Henríquez, who adapted techniques from the quadratic Koblitz paper above to blow up the competition again with that curve: at 48k cycles on Haswell and 38k cycles on Skylake for 128-bit secure scalar multiplication, it is even faster than Kummers and FourQ!

— Mehdi Tibouchi

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eprint 2016/704

Nicolas Courtois has recently uploaded to eprint the paper High Saturation Complete Graph Approach for EC Point Decomposition and ECDL Problem of over 80 pages about ECDLP.

The paper is written in an unusual style. It is a bit like a research notebook, containing sketches of ideas, rather than a polished mathematics paper.

The paper is mainly about the point decomposition problem, which is the fundamental problem behind all recent work on index calculus algorithms (see: these blog
posts). Precisely this problem is: Given a point R and a factor base F write R = P_1 + ... + P_M for P_i \in F.

The standard approach these days is to use summation polynomials: We find solutions x_1,\dots, x_M to the Semaev summation polynomial S_{M+1}( x_1, \dots, x_M, x(R)) = 0 and then compute the corresponding points. Currently these methods have not had any practical impact on the security of elliptic curve cryptography.

The preprint contains several ideas, whose relevance and impact are yet to be fully determined.

One idea is a way to generate a lot of equations without adding too many new variables. Courtois chooses K random elliptic curve points S_1,...,S_K (where K is any natural number) and defines new variables
Z_{i,j} = x( P_i + S_j )
for 1 \le i \le M, 1 \le j \le K.
Courtois then notes that if j_1,...,j_M \in \{1,...,K\} and R = P_1 + ... + P_M then
R + \sum_{i=1}^M S_{j_i} = (P_1 + S_{j_1}) + ... + (P_M + S_{j_M})
Hence we have
S_{M+1} \left( R + \sum_{i=1}^M S_{j_i}, Z_{1,j_1}, ..., Z_{M, j_M} \right) = 0.
There are K^M choices for (j_1,...,j_M), so we get a system of K^M equations in the KM variables Z_{i,j}. On the one hand, we now have a greatly over-determined system, and so it should be easier to solve than traditional systems. On the other hand, the system has too many solutions as the variables Z_{i,j} are unconstrained.

Hence the next problem is to add constraints to the system to reduce the number of solutions. If one can find suitable constraints then one should be able to define a corresponding notion of factor base. Some ideas are sketched in the paper, in particular in Section 18.2, but I have not yet formed an opinion about well these ideas will work. The paper only considers elliptic curves over prime fields \mathbb{F}_p, but similar ideas might be used for curves over other fields.

There are several other ideas in the paper, including some new polynomial equations that might be used to play a similar role to the summation polynomials.

Overall, the paper contains some interesting ideas that are not yet fully developed. Currently the paper does not describe a complete index calculus algorithm and it is difficult for me to determine whether or not the methods are likely to lead to an improvement over existing techniques. No precise complexity statements are made in the paper.

I hope that other researchers will investigate these ideas. I look forward to following the development of work on this topic.

— Steven Galbraith

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Upcoming conferences and recent announcements

  • The list of speakers for the ECC 2016 conference is at
    http://ecc2016.yasar.edu.tr/invited.html.

    Here are the speakers:

    • Benjamin Smith, “Fast, uniform, and compact scalar multiplication for elliptic curves and genus 2 Jacobians with applications to signature schemes” and “μKummer: efficient hyperelliptic signatures and key exchange on microcontrollers”
    • Cyril Hugounenq, “Explicit isogenies in quadratic time in any characteristic”
    • Daniel Genkin, “ECDH Key-Extraction via Low-Bandwidth Electromagnetic Attacks on PCs” and “ECDSA Key Extraction from Mobile Devices via Nonintrusive Physical Side Channels” and “CacheBleed: A Timing Attack on OpenSSL Constant Time RSA”
    • Jens Groth, “On the Size of Pairing-based Non-interactive Arguments”
    • Maike Massierer, “Computing L-series of geometrically hyperelliptic curves of genus three”
    • Mehmet Sabır Kiraz, “Pairings and Cloud Security”
    • Pascal Sasdrich, “Implementing Curve25519 for Side-Channel–Protected Elliptic Curve Cryptography”
    • Patrick Longa, “Efficient algorithms for supersingular isogeny Diffie-Hellman”
    • Razvan Barbulescu, “Extended Tower Number Field Sieve: A New Complexity for Medium Prime Case”
    • Sebastian Kochinke, “Computing discrete logarithms with special linear systems”
    • Shashank Singh, “A General Polynomial Selection Method and New Asymptotic Complexities for the Tower Number Field Sieve Algorithm”
    • Shoukat Ali, “A new algorithm for residue multiplication modulo $2^{521}-1$”
    • Tung Chou, “The Simplest Protocol for Oblivious Transfer” and “Sandy2x: new Curve25519 speed records”

    The conference organisers wish to reassure conference attendees that it is safe to come to Turkey for the conference: “The life in Izmir is just as usual: sunny and slow-going. We are preparing for ECC and we would like to serve our guests in the best way we can.”

  • The schedule for the Algorithmic Number Theory conference (ANTS) is at
    http://www.mathematik.uni-kl.de/~thofmann/ants/schedule.html.

  • Aurore Guillevic, François Morain and Emmanuel Thomé recently announced a solution to an ECDLP instance on a 170-bit pairing-friendly curve with embedding degree 3.
    In other words, they solved a DLP in a finite field \mathbb{F}_{p^3}^* of size around 510 bits. You can read further details here and here.

  • Thorsten Kleinjung, Claus Diem, Arjen K. Lenstra, Christine Priplata and Colin Stahlke have reported a new DLP computation using the number field sieve in a 768-bit field \mathbb{F}_{p}^*. The details are here.

— Steven Galbraith

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Complete group laws for prime-order elliptic curves: a step towards safer implementations of standard curves

Eurocrypt 2016 didn’t feature many papers directly relevant to curve-based cryptography, but one presentation stood out: Joost Renes presented his work with Craig Costello and Lejla Batina on complete addition laws for prime-order elliptic curves.

From a cryptographic point of view, complete addition laws are important for designing and implementing uniform and constant-time algorithms on elliptic curves. The most well-known and successful example of this is the (twisted) Edwards model for elliptic curves, where a single formula can be used for doubling and adding, without any exceptions or special cases. In contrast, consider the traditional chord-and-tangent group law on the Weierstrass model of an elliptic curve.  This group law can’t be applied to compute P + P, for example; instead, we have a separate doubling formula using the tangent. And until now, nobody has written down a single efficient complete addition law for the points on prime-order elliptic curves – including the prime-order curves that were standardized by NIST, and their international counterparts like Brainpool and ANSSI. This means that implementing standardized curves in a safe way is a much more complicated business than it should be!

Over twenty years ago, Bosma and Lenstra studied the group laws on elliptic curves. They concentrated on the case where an elliptic curve E is embedded in the projective plane as a Weierstrass model, but Arene, Kohel, and Ritzenthaler have since generalized their results to any projective embedding of any elliptic curve, and also to abelian varieties of any dimensions. To make things more precise, suppose E is an elliptic curve in projective Weierstrass form, with coordinates X, Y, and Z.  A group law is a triple of polynomials (F_X,F_Y,F_Z) such that P + Q = (F_X(P,Q):F_Y(P,Q):F_Z(P,Q)) for the pairs (P,Q) of points in some open subset of E\times E (the cartesian product of E with itself). This means that the group law is allowed to “fail” on some subset of the pairs of points on E, provided that subset is defined by a nontrivial algebraic relation. Basically, the group law is allowed to fail on what I will call a failure curve of points in the surface E\times E (this curve may be reducible; strictly speaking, it’s an effective divisor on E\times E). For any fixed P, there is a bounded number of Q such that the group law fails to add Q to P, and that bound depends only on the group law, not P.

To make this notion of failure more precise, we can add another requirement to the definition of a group law: for any pair (P,Q) of input points on E\times E, either (F_X(P,Q):F_Y(P,Q):F_Z(P,Q)) = P + Q (ie, the group law holds) or (F_X(P,Q),F_Y(P,Q),F_Z(P,Q)) = (0,0,0). From a computational point of view this is very nice, because (0,0,0) does not correspond to a projective point; so if we evaluate the group law at a pair of points then either the result is correct, or it is not a projective point – and this case is extremely easy to identify. If the formula is ever wrong, it’s so obviously wrong that you see it immediately.

A complete system of group laws is a collection of group laws that covers all of the pairs of points on E: that is, the common intersection of all of the failure curves is empty. Bosma and Lenstra showed that for an elliptic curve, any complete system must contain at least 2 group laws. However, this result only holds over the algebraic closure; and in cryptography, we don’t work over the algebraic closure, we work over a fixed finite field \mathbb{F}_q. And this is the crucial point: we can get away with just one group law, so long as none of the pairs of points on its failure curve are defined over \mathbb{F}_q! If they’re all defined over some extension, then they will never be inputs to our cryptographic algorithm, and we can simply ignore their existence.  Arene, Kohel, and Ritzenthaler take this to its logical conclusion, showing that this can always be done for any elliptic curve or abelian variety (apart from some pathological cases over extremely small fields, which are irrelevant to cryptography).

The particularly nice thing about Bosma and Lenstra’s paper is that they give a clear description of what the failure curves look like for the simplest nontrivial class of group laws, which is where all of the polynomials are biquadratic (that is, each F_X, F_Y, and F_Z is homogeneous of degree 2 in the coordinates of P, and also in the coordinates of Q). In this case, the failure curves all correspond to lines: for each group law (F_X,F_Y,F_Z), there is a line L in the projective plane such that the group law fails on (P,Q) if and only if P-Q is in the intersection of E with L.

At Eurocrypt, Joost Renes explained that there is an obvious way to apply this to prime-order elliptic curves: for a complete group law on such a curve E/\mathbb{F}_p, we can take the biquadratic group law whose failure line is the x-axis. This works because the intersection with E consists of the nonzero 2-torsion points – and since E(\mathbb{F}_p) has prime order, none of those are defined over \mathbb{F}_p, so they can’t be the difference of any pair of \mathbb{F}_p-points, and therefore the group law can’t fail on any pair of points in E(\mathbb{F}_p).  So far so good.  But Bosma and Lenstra actually wrote down that group law as an example in their paper, and the formulae take up a whole page and a half!  I’m sure that plenty of cryptographers had seen it, and thought “that’s all very nice in theory, but…”

So now we come to the main contribution of the paper: Joost, Craig, and Lejla show that this “it’ll never work” intuition is completely wrong.  They simplify the formulae (following Arene, Kohel, and Ritzenthaler); they derive efficient straight-line algorithms to evaluate the polynomials; and they show that not only can this group law be computed efficiently, it’s actually competitive with the best known (non-complete) formulae for projective Weierstrass curves.  So if you find yourself implementing the group law on a prime-order curve (for whatever reason, be it scientific or political), then you should definitely consider doing it the way their paper suggests.  You won’t lose much speed, and you’ll gain a lot of safety and simplicity.

–Ben Smith

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