Discrete logarithm records#Elliptic curves

The current record ({{As of|2019|7|alt=as of July 2019}}) in a finite field of characteristic 2 was announced by Robert Granger, Thorsten Kleinjung, Arjen Lenstra, Benjamin Wesolowski, and Jens Zumbrägel on 10 July 2019.Jens Zumbrägel, "Discrete Logarithms in GF(2^30750)", 10 July 2019, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;62ab27f0.1907. This team was able to compute discrete logarithms in GF(2³⁰⁷⁵⁰) using 25,481,219 core hours on clusters based on the Intel Xeon architecture. This computation was the first large-scale example using the elimination step of the quasi-polynomial algorithm.R. Granger, T. Kleinjung, J. Zumbragel. [https://web.archive.org/web/20240428084403/https://www.ams.org/journals/tran/2018-370-05/S0002-9947-2017-07027-6/S0002-9947-2017-07027-6.pdf On the discrete logarithm problem in finite fields of fixed characteristic]. Trans. Amer. Math.

Soc. 370, no. 5 (2018), pp. 3129-3145.

Previous records in a finite field of characteristic 2 were announced by:

• Robert Granger, Thorsten Kleinjung, and Jens Zumbrägel on 31 January 2014. This team was able to compute discrete logarithms in GF(2⁹²³⁴) using about 400,000 core hours. New features of this computation include a modified method for obtaining the logarithms of degree two elements and a systematically optimized descent strategy.Jens Zumbrägel, "Discrete Logarithms in GF(2^9234)", 31 January 2014, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;9aa2b043.1401.
• Antoine Joux on 21 May 2013. His team was able to compute discrete logarithms in the field with 2⁶¹⁶⁸ = (2²⁵⁷)²⁴ elements using less than 550 CPU-hours. This computation was performed using the same index calculus algorithm as in the recent computation in the field with 2⁴⁰⁸⁰ elements.Antoine Joux, "Discrete logarithms in GF(2⁶¹⁶⁸) [=GF((2²⁵⁷)²⁴)]", May 21, 2013, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1305&L=NMBRTHRY&F=&S=&P=3034.
• Robert Granger, Faruk Göloğlu, Gary McGuire, and Jens Zumbrägel on 11 Apr 2013. The new computation concerned the field with 2⁶¹²⁰ elements and took 749.5 core-hours.
• Antoine Joux on Mar 22nd, 2013. This used the same algorithmAntoine Joux. A new index calculus algorithm with complexity $L(1/4+o(1))$ in very small characteristic, 2013, http://eprint.iacr.org/2013/095 for small characteristic fields as the previous computation in the field with 2¹⁷⁷⁸ elements. The new computation concerned the field with 2⁴⁰⁸⁰ elements, represented as a degree 255 extension of the field with 2¹⁶ elements. The computation took less than 14100 core hours.Antoine Joux, "Discrete logarithms in GF(2⁴⁰⁸⁰)", Mar 22, 2013, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1303&L=NMBRTHRY&F=&S=&P=13682.
• Robert Granger, Faruk Göloğlu, Gary McGuire, and Jens Zumbrägel on 19 Feb 2013. They used a new variant of the medium-sized base field function field sieve, for binary fields, to compute a discrete logarithm in a field of 2¹⁹⁷¹ elements. In order to use a medium-sized base field, they represented the field as a degree 73 extension of the field of 2²⁷ elements. The computation took 3132 core hours on an SGI Altix ICE 8200EX cluster using Intel (Westmere) Xeon E5650 hex-core processors.Faruk Gologlu et al., On the Function Field Sieve and the Impact of Higher Splitting Probabilities: Application to Discrete Logarithms in $\backslash mathbb\{F\}\_\{2^\{1971\}\}$, 2013, http://eprint.iacr.org/2013/074.
• Antoine Joux on 11 Feb 2013. This used a new algorithm for small characteristic fields. The computation concerned a field of 2¹⁷⁷⁸ elements, represented as a degree 127 extension of the field with 2¹⁴ elements. The computation took less than 220 core hours.Antoine Joux, "Discrete logarithms in GF(2¹⁷⁷⁸)", Feb. 11, 2013, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1302&L=NMBRTHRY&F=&S=&P=2317.

The current record ({{As of|2014|alt=as of 2014}}) in a finite field of characteristic 2 of prime degree was announced by Thorsten Kleinjung on 17 October 2014. The calculation was done in a field of 2¹²⁷⁹ elements and followed essentially the path sketched for $\backslash mathrm\{GF\}(2^\{4\; \backslash cdot\; 1223\})$ in Granger, Robert, Thorsten Kleinjung, and Jens Zumbrägel. “Breaking `128-Bit Secure’ Supersingular Binary Curves (or How to Solve Discrete Logarithms in $\{\backslash mathbb\; F\}\_\{2^\{4\; \backslash cdot\; 1223\}\}$ and $\{\backslash mathbb\; F\}\_\{2^\{12\; \backslash cdot\; 367\}\}$).” arXiv:1402.3668 [cs, Math], February 15, 2014. https://arxiv.org/abs/1402.3668. with two main exceptions in the linear algebra computation and the descent phase. The total running time was less than four core years.Thorsten Kleinjung, 2014 October 17, "Discrete Logarithms in GF(2^1279)", https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;256db68e.1410. The previous record in a finite field of characteristic 2 of prime degree was announced by the CARAMEL group on April 6, 2013. They used the function field sieve to compute a discrete logarithm in a field of 2⁸⁰⁹ elements.The CARAMEL group: Razvan Barbulescu and Cyril Bouvier and Jérémie Detrey and Pierrick Gaudry and Hamza Jeljeli and Emmanuel Thomé and Marion Videau and Paul Zimmermann, “Discrete logarithm in GF(2⁸⁰⁹) with FFS”, April 6, 2013, http://eprint.iacr.org/2013/197.

The current record ({{As of|2016|7|alt=as of July 2016}}) for a field of characteristic 3 was announced by

Gora Adj, Isaac Canales-Martinez, Nareli Cruz-Cortés, Alfred Menezes, Thomaz Oliveira, Francisco Rodriguez-Henriquez, and Luis Rivera-Zamarripa on 18 July 2016. The calculation was done in the 4841-bit finite field with 3^6 · 509 elements and was performed on several computers at CINVESTAV and

the University of Waterloo. In total, about 200 core years of computing time was expended on the computation.Francisco Rodriguez-Henriquez, 18 July 2016, "Discrete Logarithms in GF(3^{6*509})", https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;65bedfc8.1607.

Previous records in a finite field of characteristic 3 were announced:

• in the full version of the Asiacrypt 2014 paper of Joux and Pierrot (December 2014).{{Cite web|last1=Joux|first1=Antoine|last2=Pierrot|first2=Cécile|title=Improving the Polynomial time Precomputation of Frobenius Representation Discrete Logarithm Algorithms|url=http://www-polsys.lip6.fr/~pierrot/papers/Simplified%20Frob%20Rep.pdf|url-status=dead|archive-url=https://web.archive.org/web/20141211051755/http://www-polsys.lip6.fr/~pierrot/papers/Simplified%20Frob%20Rep.pdf|archive-date=2014-12-11|access-date=2014-12-11}} The DLP is solved in the field GF(3^5 · 479), which is a 3796-bit field. This work did not exploit any "special" aspects of the field such as Kummer or twisted-Kummer properties. The total computation took less than 8600 CPU-hours.
• by Gora Adj, Alfred Menezes, Thomaz Oliveira, and Francisco Rodríguez-Henríquez on 26 February 2014, updating a previous announcement on 27 January 2014. The computation solve DLP in the 1551-bit field GF(3^6 · 163), taking 1201 CPU hours.Francisco Rodríguez-Henríquez, “Announcement,” 27 January 2014, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;763a9e76.1401.Gora Adj and Alfred Menezes and Thomaz Oliveira and Francisco Rodríguez-Henríquez, "Computing Discrete Logarithms in F_{3^{6*137}} and F_{3^{6*163}} using Magma", 26 Feb 2014, http://eprint.iacr.org/2014/057.
• in 2012 by a joint Fujitsu, NICT, and Kyushu University team, that computed a discrete logarithm in the field of 3^6 · 97 elements and a size of 923 bits,Kyushu University, NICT and Fujitsu Laboratories Achieve World Record Cryptanalysis of Next-Generation Cryptography, 2012, http://www.nict.go.jp/en/press/2012/06/PDF-att/20120618en.pdf. using a variation on the function field sieve and beating the previous record in a field of 3^6 · 71 elements and size of 676 bits by a wide margin.Takuya Hayashi et al., Solving a 676-bit Discrete Logarithm Problem in GF(3⁶ⁿ), 2010, http://eprint.iacr.org/2010/090.

Over fields of "moderate"-sized characteristic, notable computations as of 2005 included those a field of 65537²⁵ elements (401 bits) announced on 24 Oct 2005, and in a field of 370801³⁰ elements (556 bits) announced on 9 Nov 2005.A. Durand, “New records in computations over large numbers,” The Security Newsletter, January 2005, http://eric-diehl.com/letter/Newsletter1_Final.pdf {{Webarchive|url=https://web.archive.org/web/20110710202518/http://eric-diehl.com/letter/Newsletter1_Final.pdf |date=2011-07-10 }}. The current record (as of 2013) for a Kummer extension finite field of "moderate" characteristic was announced on 6 January 2013. The team used a new variation of the function field sieve for the medium prime case to compute a discrete logarithm in a Kummer extension field of 33341353⁵⁷ elements (a 1425-bit finite field).Antoine Joux, “Discrete Logarithms in a 1425-bit Finite Field,” January 6, 2013, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1301&L=NMBRTHRY&F=&S=&P=2214.Faster index calculus for the medium prime case. Application to 1175-bit and 1425-bit finite fields, Eprint Archive, http://eprint.iacr.org/2012/720 The same technique had been used a few weeks earlier to compute a discrete logarithm in a Kummer extension field of 33553771⁴⁷ elements (an 1175-bit finite field).Antoine Joux, “Discrete Logarithms in a 1175-bit Finite Field,” December 24, 2012, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1212&L=NMBRTHRY&F=&S=&P=13902.{{Dead link|date=January 2022}} The current record (as of 2022) for a finite field of "moderate" characteristic (which is not necessarily a Kummer extension) is the computation of discrete logarithm in a field of 2111023⁵⁰ elements (a 1051-bit finite field);{{cite journal|last1=Mukhopadhyay|first1=Madhurima|last2=Sarkar|first2=Palash|last3=Singh|first3=Shashank|last4=Thomé|first4=Emmanuel|title=New discrete logarithm computation for the medium prime case using the function field sieve|journal=Advances in Mathematics of Communications|year=2022|volume=16 |issue=3 |page=449 |doi=10.3934/amc.2020119|doi-access=free}} previous record{{cite journal|last1=Sarkar|first1=Palash|last2=Singh|first2=Shashank|title=Fine Tuning the Function Field Sieve Algorithm for the Medium Prime Case|journal=IEEE Transactions on Information Theory|year=2016|volume=62 |issue=4 |pages=2233–2253 |url=https://ieeexplore.ieee.org/document/7405328|doi=10.1109/TIT.2016.2528996|url-access=subscription}} of discrete logarithm computations over such fields was over fields having 297079⁴⁰ elements (a 728-bit

finite field) and 64373³⁷ elements (a 592-bit finite field). These computations were done using new ideas to speed up the function field sieve.

On 25 June 2014, Razvan Barbulescu, Pierrick Gaudry, Aurore Guillevic, and François Morain announced a new computation of a discrete logarithm in a finite field whose order has 160 digits and is a degree 2 extension of a prime field.Razvan Barbulescu, “Discrete logarithms in GF(p^2) --- 160 digits,” June 24, 2014, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;2ddabd4c.1406. The algorithm used was the number field sieve (NFS), with various modifications. The total computing time was equivalent to 68 days on one core of CPU (sieving) and 30 hours on a GPU (linear algebra).

Certicom Corp. has issued a series of Elliptic Curve Cryptography challenges. Level I involves fields of 109-bit and 131-bit sizes. Level II includes 163, 191, 239, 359-bit sizes. All Level II challenges are currently believed to be computationally infeasible.Certicom Corp., “The Certicom ECC Challenge,” https://www.certicom.com/content/certicom/en/the-certicom-ecc-challenge.html

The Level I challenges which have been met are:Certicom Research, Certicom ECC Challenge (Certicom Research, November 10, 2009), {{cite web |url=http://www.certicom.com/images/pdfs/challenge-2009.pdf |title=Archived copy |accessdate=2010-12-30 |url-status=dead |archiveurl=https://web.archive.org/web/20151022135946/https://www.certicom.com/images/pdfs/challenge-2009.pdf |archivedate=2015-10-22 }} .

• ECC2K-108, involving taking a discrete logarithm on a Koblitz curve over a field of 2¹⁰⁸ elements. The prize was awarded on 4 April 2000 to a group of about 1300 people represented by Robert Harley. They used a parallelized Pollard rho method with speedup.
• ECC2-109, involving taking a discrete logarithm on a curve over a field of 2¹⁰⁹ elements. The prize was awarded on 8 April 2004 to a group of about 2600 people represented by Chris Monico. They also used a version of a parallelized Pollard rho method, taking 17 months of calendar time.
• ECCp-109, involving taking a discrete logarithm on a curve modulo a 109-bit prime. The prize was awarded on 15 Apr 2002 to a group of about 10308 people represented by Chris Monico. Once again, they used a version of a parallelized Pollard rho method, taking 549 days of calendar time.

None of the 131-bit (or larger) challenges have been met {{As of|2019|alt=as of 2019}}.

In July 2009, Joppe W. Bos, Marcelo E. Kaihara, Thorsten Kleinjung, Arjen K. Lenstra and Peter L. Montgomery announced that they had carried out a discrete logarithm computation on an elliptic curve (known as secp112r1Certicom Research, "SEC 2: Recommended Elliptic Curve Domain Parameters" https://www.secg.org/SEC2-Ver-1.0.pdf) modulo a 112-bit prime. The computation was done on a cluster of over 200 PlayStation 3 game consoles over about 6 months. They used the common parallelized version of Pollard rho method.Joppe W. Bos and Marcelo E. Kaihara, “PlayStation 3 computing breaks 2^60 barrier: 112-bit prime ECDLP solved,” EPFL Laboratory for cryptologic algorithms - LACAL, http://lacal.epfl.ch/112bit_prime

In April 2014, Erich Wenger and Paul Wolfger from Graz University of Technology solved the discrete logarithm of a 113-bit Koblitz curve in extrapolated{{Refn|The computation ran for 47 days, but not all of the FPGAs used were active all the time, which meant that it was equivalent to an extrapolated time of 24 days.|group=note|name=extrapolated47to24}} 24 days using an 18-core Virtex-6 FPGA cluster.Erich Wenger and Paul Wolfger, “Solving the Discrete Logarithm of a 113-bit Koblitz Curve with an FPGA Cluster” http://eprint.iacr.org/2014/368 In January 2015, the same researchers solved the discrete logarithm of an elliptic curve defined over a 113-bit binary field. The average runtime is around 82 days using a 10-core Kintex-7 FPGA cluster.Erich Wenger and Paul Wolfger, “Harder, Better, Faster, Stronger - Elliptic Curve Discrete Logarithm Computations on FPGAs” http://eprint.iacr.org/2015/143/

On 2 December 2016, Daniel J. Bernstein, Susanne Engels, Tanja Lange, Ruben Niederhagen, Christof Paar, Peter Schwabe, and Ralf Zimmermann announced the solution of a generic 117.35-bit elliptic curve discrete logarithm problem on a binary curve, using an optimized FPGA implementation of a parallel version of Pollard's rho method. The attack ran for about six months on 64 to 576 FPGAs in parallel.Ruben Niederhagen, “117.35-Bit ECDLP on Binary Curve,” https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;628a3b51.1612

On 23 August 2017, Takuya Kusaka, Sho Joichi, Ken Ikuta, Md. Al-Amin Khandaker, Yasuyuki Nogami, Satoshi Uehara, Nariyoshi Yamai, and Sylvain Duquesne announced that they had solved a discrete logarithm problem on a 114-bit "pairing-friendly" Barreto–Naehrig (BN) curve,{{cite web|date=23 August 2017|title=114-bit ECDLP on a BN curve has been solved|url=http://isec.ec.okayama-u.ac.jp/whats-new/114-bit-ecdlp-has-been-solved|url-status=dead|archive-url=https://web.archive.org/web/20180527201520/http://isec.ec.okayama-u.ac.jp/whats-new/114-bit-ecdlp-has-been-solved/|archive-date=27 May 2018|website=isec.ec.okayama-u.ac.jp|accessdate=2018-05-03}} using the special sextic twist property of the BN curve to efficiently carry out the random walk of Pollard's rho method. The implementation used 2000 CPU cores and took about 6 months to solve the problem.{{cite conference |title= Solving 114-Bit ECDLP for a Barreto–Naehrig Curve|last1=Kusaka |first1=Takuya |last2=Joichi |first2= Sho |first3=Ken|last3= Ikuta|first4= Md. Al-Amin |last4=Khandaker|first5= Yasuyuki|last5= Nogami| first6=Satoshi|last6= Uehara|first7= Nariyoshi |last7=Yamai| first8= Sylvain|last8= Duquesne|series=Lecture Notes in Computer Science |date=2018 |volume=10779 |publisher= Springer|book-title=Information Security and Cryptology – ICISC 2017 |pages= 231–244|doi= 10.1007/978-3-319-78556-1_13|isbn=978-3-319-78555-4 |url= https://hal.archives-ouvertes.fr/hal-01633653/file/article.pdf}}

On 16 June 2020, Aleksander Zieniewicz (zielar) and Jean Luc Pons ([https://github.com/JeanLucPons JeanLucPons]) announced the solution of a 114-bit interval elliptic curve discrete logarithm problem on the secp256k1 curve by solving a 114-bit private key in Bitcoin Puzzle Transactions Challenge. To set a new record, they used their own software{{cite web | title=Pollard's kangaroo for SECPK1 | first1=Jean-Luc |last1=Pons | first2=Aleksander | last2=Zieniewicz | website=GitHub | date=17 January 2022 | url=https://github.com/JeanLucPons/Kangaroo}} based on the Pollard Kangaroo on 256x [https://www.nvidia.com/en-us/data-center/v100/ NVIDIA Tesla V100] GPU processor and it took them 13 days. Two weeks earlier - They used the same number of graphics cards to solve a 109-bit interval ECDLP in just 3 days.

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• [https://dldb.loria.fr/ Computations of discrete logarithms sorted by date]

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