American Institute of Mathematical Sciences

Advanced
August  2017, 11(3): 453-469. doi: 10.3934/amc.2017038

Computing elliptic curve discrete logarithms with improved baby-step giant-step algorithm

Steven D. Galbraith 1,, , Ping Wang 2, and  Fangguo Zhang 3,

1. 

Mathematics Department, University of Auckland, New Zealand
2. 

College of Information Engineering, Shenzhen University, Shenzhen 518060, China
3. 

School of Data and Computer Science, Sun Yat-sen University, Guangzhou 510006, China

* Corresponding author

Received  June 2015 Revised  February 2016 Published  August 2017

Fund Project: This work was supported by the National Natural Science Foundation of China (No. 61672550,61402293).

The negation map can be used to speed up the computation of elliptic curve discrete logarithms using either the baby-step giant-step algorithm (BSGS) or Pollard rho. Montgomery's simultaneous modular inversion can also be used to speed up Pollard rho when running many walks in parallel. We generalize these ideas and exploit the fact that for any two elliptic curve points X and Y, we can efficiently get X-Y when we compute X+Y. We apply these ideas to speed up the baby-step giant-step algorithm. Compared to the previous methods, the new methods can achieve a significant speedup for computing elliptic curve discrete logarithms in small groups or small intervals.

Another contribution of our paper is to give an analysis of the average-case running time of Bernstein and Lange's "grumpy giants and a baby" algorithm, and also to consider this algorithm in the case of groups with efficient inversion.

Our conclusion is that, in the fully-optimised context, both the interleaved BSGS and grumpy-giants algorithms have superior average-case running time compared with Pollard rho. Furthermore, for the discrete logarithm problem in an interval, the interleaved BSGS algorithm is considerably faster than the Pollard kangaroo or Gaudry-Schost methods.

Keywords: Baby-step giant-step algorithm, elliptic curve discrete logarithm, negation map.
Mathematics Subject Classification: Primary: 11Y16; Secondary: 11T71.
Citation: Steven D. Galbraith, Ping Wang, Fangguo Zhang. Computing elliptic curve discrete logarithms with improved baby-step giant-step algorithm. Advances in Mathematics of Communications, 2017, 11 (3) : 453-469. doi: 10.3934/amc.2017038
References:
[1]

D. J. Bernstein and T. Lange, Computing small discrete logarithms faster, in INDOCRYPT 2012 (eds. S. D. Galbraith et al), Springer, 2012,317–338. doi: 10.1007/978-3-642-34931-7_19.  Google Scholar

[2]

D. J. Bernstein and T. Lange, Two grumpy giants and a baby, in Proc. 10th Algor. Number Theory Symp. (eds. E. W. Howe et al), 2013, 87–111. doi: 10.2140/obs.2013.1.87.  Google Scholar

[3]

D. J. Bernstein, T. Lange and P. Schwabe, On the correct use of the negation map in the Pollard rho method, in PKC 2011 (eds. D. Catalano et al), Springer, 2011,128–146. doi: 10.1007/978-3-642-19379-8_8.  Google Scholar

[4]

D. Boneh, E. Goh and K. Nissim, Evaluating 2-DNF formulas on ciphertexts in Theory of Cryptography-TCC 2005 (ed. J. Kilian), Springer, 2005,325-341. doi: 10.1007/978-3-540-30576-7_18.  Google Scholar

[5]

M. Chateauneuf, A. C. H. Ling and D. R. Stinson, Slope packings and coverings, and generic algorithms for the discrete logarithm problem, J. Combin. Des., 11 (2003), 36-50.  doi: 10.1002/jcd.10033.  Google Scholar

[6]

K. Fong, D. Hankerson, J. Lopez and A. Menezes, Field inversion and point halving revisited, IEEE Trans. Comp., 53 (2004), 1047-1059.   Google Scholar

[7]

S. D. Galbraith, J. M. Pollard and R. S. Ruprai, Computing discrete logarithms in an interval}, Math. Comp., 82 (2013), 1181-1195.  doi: 10.1090/S0025-5718-2012-02641-X.  Google Scholar

[8]

S. D. Galbraith and R. S. Ruprai, Using equivalence classes to speed up the discrete logarithm problem in a short interval, in PKC 2010 (eds. P. Nguyen et al), Springer, 2010,368–383. doi: 10.1007/978-3-642-13013-7_22.  Google Scholar

[9]

R. Gallant, R. Lambert and S. Vanstone, Improving the parallelized Pollard lambda search on binary anomalous curves, Math. Comp., 69 (1999), 1699-1705.  doi: 10.1090/S0025-5718-99-01119-9.  Google Scholar

[10]

P. Gaudry and E. Schost, A low-memory parallel version of Matsuo, Chao and Tsujii's algorithm, in ANTS Ⅵ (ed. D. A. Buell), Springer, 2004,208–222. doi: 10.1007/978-3-540-24847-7_15.  Google Scholar

[11]

R. Granger, D. Page and M. Stam, On small characteristic algebraic tori in pairing-based cryptography, LMS J. Comp. Math., 9 (2006), 64-85.  doi: 10.1112/S1461157000001194.  Google Scholar

[12]

R. Henry, K. Henry and I. Goldberg, Making a nymbler Nymble using VERBS, in PETS 2010 (eds. M. J. Atallah), Springer, 2010,111–129. Google Scholar

[13]

N. Koblitz, Elliptic curve cryptosystems, Math. Comp., 48 (1987), 203-209.  doi: 10.2307/2007884.  Google Scholar

[14]

V. Miller, Use of elliptic curves in cryptography, in Crypto '85 (ed. H. C. Williams), Springer, 1986,417–426. doi: 10.1007/3-540-39799-X_31.  Google Scholar

[15]

P. L. Montgomery, Speeding the Pollard and elliptic curve methods of factorization, Math. Comp., 48 (1987), 243-264.  doi: 10.2307/2007888.  Google Scholar

[16]

V. I. Nechaev, Complexity of a determinate algorithm for the discrete logarithm, Math. Notes, 55 (1994), 165-172.  doi: 10.1007/BF02113297.  Google Scholar

[17]

J. M. Pollard, Monte Carlo methods for index computation (mod p), Math. Comp., 32 (1978), 918-924.  doi: 10.2307/2006496.  Google Scholar

[18]

J. Pollard, Kangaroos, Monopoly and discrete logarithms, J. Crypt., 13 (2000), 437-447.  doi: 10.1007/s001450010010.  Google Scholar

[19]

D. Shanks, Five number-theoretic algorithms, in Proc. 2nd Manitoba Conf. Numer. Math. , Winnipeg, 1973, 51–70.  Google Scholar

[20]

P. van Oorschot and M. Wiener, Parallel collision search with cryptanalytic applications, J. Crypt., 12 (1999), 1-28.  doi: 10.1007/PL00003816.  Google Scholar

[21]

P. Wang and F. Zhang, Computing elliptic curve discrete logarithms with the negation map, Inf. Sci., 195 (2012), 277-286.  doi: 10.1016/j.ins.2012.01.044.  Google Scholar

[22]

P. Wang and F. Zhang, Improving the parallelized Pollard rho method for computing elliptic curve discrete logarithms in 4th Int. Conf. Emerging Intell. Data Web Techn. (EIDWT-2013) 2013. doi: 10.1109/EIDWT.2013.55.  Google Scholar

[23]

M. Wiener and R. Zuccherato, Faster attacks on elliptic curve cryptosystems, in Selected Areas in Cryptography '98 (eds. S. E. Tavares et al), Springer, 1998,190–120. doi: 10.1007/3-540-48892-8_15.  Google Scholar

show all references

References:
[1]

D. J. Bernstein and T. Lange, Computing small discrete logarithms faster, in INDOCRYPT 2012 (eds. S. D. Galbraith et al), Springer, 2012,317–338. doi: 10.1007/978-3-642-34931-7_19.  Google Scholar

[2]

D. J. Bernstein and T. Lange, Two grumpy giants and a baby, in Proc. 10th Algor. Number Theory Symp. (eds. E. W. Howe et al), 2013, 87–111. doi: 10.2140/obs.2013.1.87.  Google Scholar

[3]

D. J. Bernstein, T. Lange and P. Schwabe, On the correct use of the negation map in the Pollard rho method, in PKC 2011 (eds. D. Catalano et al), Springer, 2011,128–146. doi: 10.1007/978-3-642-19379-8_8.  Google Scholar

[4]

D. Boneh, E. Goh and K. Nissim, Evaluating 2-DNF formulas on ciphertexts in Theory of Cryptography-TCC 2005 (ed. J. Kilian), Springer, 2005,325-341. doi: 10.1007/978-3-540-30576-7_18.  Google Scholar

[5]

M. Chateauneuf, A. C. H. Ling and D. R. Stinson, Slope packings and coverings, and generic algorithms for the discrete logarithm problem, J. Combin. Des., 11 (2003), 36-50.  doi: 10.1002/jcd.10033.  Google Scholar

[6]

K. Fong, D. Hankerson, J. Lopez and A. Menezes, Field inversion and point halving revisited, IEEE Trans. Comp., 53 (2004), 1047-1059.   Google Scholar

[7]

S. D. Galbraith, J. M. Pollard and R. S. Ruprai, Computing discrete logarithms in an interval}, Math. Comp., 82 (2013), 1181-1195.  doi: 10.1090/S0025-5718-2012-02641-X.  Google Scholar

[8]

S. D. Galbraith and R. S. Ruprai, Using equivalence classes to speed up the discrete logarithm problem in a short interval, in PKC 2010 (eds. P. Nguyen et al), Springer, 2010,368–383. doi: 10.1007/978-3-642-13013-7_22.  Google Scholar

[9]

R. Gallant, R. Lambert and S. Vanstone, Improving the parallelized Pollard lambda search on binary anomalous curves, Math. Comp., 69 (1999), 1699-1705.  doi: 10.1090/S0025-5718-99-01119-9.  Google Scholar

[10]

P. Gaudry and E. Schost, A low-memory parallel version of Matsuo, Chao and Tsujii's algorithm, in ANTS Ⅵ (ed. D. A. Buell), Springer, 2004,208–222. doi: 10.1007/978-3-540-24847-7_15.  Google Scholar

[11]

R. Granger, D. Page and M. Stam, On small characteristic algebraic tori in pairing-based cryptography, LMS J. Comp. Math., 9 (2006), 64-85.  doi: 10.1112/S1461157000001194.  Google Scholar

[12]

R. Henry, K. Henry and I. Goldberg, Making a nymbler Nymble using VERBS, in PETS 2010 (eds. M. J. Atallah), Springer, 2010,111–129. Google Scholar

[13]

N. Koblitz, Elliptic curve cryptosystems, Math. Comp., 48 (1987), 203-209.  doi: 10.2307/2007884.  Google Scholar

[14]

V. Miller, Use of elliptic curves in cryptography, in Crypto '85 (ed. H. C. Williams), Springer, 1986,417–426. doi: 10.1007/3-540-39799-X_31.  Google Scholar

[15]

P. L. Montgomery, Speeding the Pollard and elliptic curve methods of factorization, Math. Comp., 48 (1987), 243-264.  doi: 10.2307/2007888.  Google Scholar

[16]

V. I. Nechaev, Complexity of a determinate algorithm for the discrete logarithm, Math. Notes, 55 (1994), 165-172.  doi: 10.1007/BF02113297.  Google Scholar

[17]

J. M. Pollard, Monte Carlo methods for index computation (mod p), Math. Comp., 32 (1978), 918-924.  doi: 10.2307/2006496.  Google Scholar

[18]

J. Pollard, Kangaroos, Monopoly and discrete logarithms, J. Crypt., 13 (2000), 437-447.  doi: 10.1007/s001450010010.  Google Scholar

[19]

D. Shanks, Five number-theoretic algorithms, in Proc. 2nd Manitoba Conf. Numer. Math. , Winnipeg, 1973, 51–70.  Google Scholar

[20]

P. van Oorschot and M. Wiener, Parallel collision search with cryptanalytic applications, J. Crypt., 12 (1999), 1-28.  doi: 10.1007/PL00003816.  Google Scholar

[21]

P. Wang and F. Zhang, Computing elliptic curve discrete logarithms with the negation map, Inf. Sci., 195 (2012), 277-286.  doi: 10.1016/j.ins.2012.01.044.  Google Scholar

[22]

P. Wang and F. Zhang, Improving the parallelized Pollard rho method for computing elliptic curve discrete logarithms in 4th Int. Conf. Emerging Intell. Data Web Techn. (EIDWT-2013) 2013. doi: 10.1109/EIDWT.2013.55.  Google Scholar

[23]

M. Wiener and R. Zuccherato, Faster attacks on elliptic curve cryptosystems, in Selected Areas in Cryptography '98 (eds. S. E. Tavares et al), Springer, 1998,190–120. doi: 10.1007/3-540-48892-8_15.  Google Scholar

Figure 1.  Graph of $c(t) t^2$ for $t \in [0,1]$ obtained by simulations for the three values $M = \sqrt{N/2}, M = \sqrt{N}$ and $M = \tfrac{1}{2}\sqrt{N}$. The multiple lines denote the results for different experiments coming from different choices of $N$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Graph of $c(t) t^2$ for $t \in [0,1]$ obtained by simulations, for grumpy giants method using $x$-coordinates, with $M = \sqrt{N/2}$ and $ M = \sqrt{N}$
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Size of set $\mathcal{L}_L$ written as $c(t) L^2$ where $t = L/\sqrt{N}$.
$t$ 0 0.12 0.24 0.37 0.49 0.61 0.73 0.85 0.97
$c(t)$ 3.00 3.00 3.00 2.99 2.79 2.30 1.77 1.35 1.06
Table Options

Download as excel

$t$ 0 0.12 0.24 0.37 0.49 0.61 0.73 0.85 0.97
$c(t)$ 3.00 3.00 3.00 2.99 2.79 2.30 1.77 1.35 1.06
Table 2.  Results of experiments with the grumpy-giants algorithm without negation
Bits #Elliptic Curves #DLPs per Curve average value for c standard deviation
28 100 10000 1.2579 0.0083
29 100 10000 1.2533 0.0064
30 100 10000 1.2484 0.0062
31 100 10000 1.2517 0.0067
32 100 10000 1.2736 0.0054
Table Options

Download as excel

Bits #Elliptic Curves #DLPs per Curve average value for c standard deviation
28 100 10000 1.2579 0.0083
29 100 10000 1.2533 0.0064
30 100 10000 1.2484 0.0062
31 100 10000 1.2517 0.0067
32 100 10000 1.2736 0.0054
Table 3.  The table lists constants $c$ such that the named algorithm requires $(c + o(1)) \sqrt{N}$ group operations for large enough groups of size $N$. The first block lists algorithms for general groups, and all these results are known (see Section 2). The values for the grumpy-giant algorithm (marked by an asterisk) are conjectural and the values for the rho and Gaudry-Schost algorithm are heuristic. The second block lists algorithms for groups having an efficiently computable inversion (see Section 3). Some of these results are new (the first one appears as an exercise in the first author's textbook). The third block lists algorithms that exploit efficient inversion as well as our main observation, and these results are all new (see Section 5)
Algorithm Average-case Worst-case
Textbook BSGS [19] $1.5$ $2.0$
Textbook BSGS optimised for average-case [18] $1.414$ $2.121$
Pollard interleaving BSGS [17] $1.333$ $2.0$
Grumpy giants [2] $1.25^* $ $ \le 3$
Pollard rho using distinguished points [20] $1.253$ $\infty$
Gaudry-Schost [7] $1.661$ $\infty$
BSGS with negation $1.0$ $1.5$
Pollard interleaving BSGS with negation $0.943$ $1.414$
Grumpy giants with negation $0.9^*$ $\le 2.7$
Pollard rho using negation [3,21] $0.886(1 + o(1))$ $\infty$
Gaudry-Schost using negation [8] $1.36$ $\infty$
Interleaved BSGS with block computation $0.38$ $0.57$
Grumpy giants with block computation $0.36^*$ $\le 1.08$
Pollard rho with Montgomery trick $0.47 $ $\infty$
Gaudry-Schost with Montgomery trick $0.72 $ $\infty$
Table Options

Download as excel

Algorithm Average-case Worst-case
Textbook BSGS [19] $1.5$ $2.0$
Textbook BSGS optimised for average-case [18] $1.414$ $2.121$
Pollard interleaving BSGS [17] $1.333$ $2.0$
Grumpy giants [2] $1.25^* $ $ \le 3$
Pollard rho using distinguished points [20] $1.253$ $\infty$
Gaudry-Schost [7] $1.661$ $\infty$
BSGS with negation $1.0$ $1.5$
Pollard interleaving BSGS with negation $0.943$ $1.414$
Grumpy giants with negation $0.9^*$ $\le 2.7$
Pollard rho using negation [3,21] $0.886(1 + o(1))$ $\infty$
Gaudry-Schost using negation [8] $1.36$ $\infty$
Interleaved BSGS with block computation $0.38$ $0.57$
Grumpy giants with block computation $0.36^*$ $\le 1.08$
Pollard rho with Montgomery trick $0.47 $ $\infty$
Gaudry-Schost with Montgomery trick $0.72 $ $\infty$
Table 4.  Size of set $\mathcal{L}_L$ written as $c(t) L^2$ where $t = L/\sqrt{N}$
$t$ 0 0.15 0.30 0.46 0.61 0.76 0.91
$c(t)$ 6.00 5.76 5.47 4.10 2.56 1.72 1.20
Table Options

Download as excel

$t$ 0 0.15 0.30 0.46 0.61 0.76 0.91
$c(t)$ 6.00 5.76 5.47 4.10 2.56 1.72 1.20
Table 5.  Results of experiments with the grumpy-giants algorithm exploiting efficient inversion
Bits #Elliptic Curves #DLPs per Curve average value for c standard deviation
28 100 10000 0.8926 0.0077
29 100 10000 0.9053 0.0061
30 100 10000 0.8961 0.0073
31 100 10000 0.9048 0.0068
32 100 10000 0.9207 0.0065
Table Options

Download as excel

Bits #Elliptic Curves #DLPs per Curve average value for c standard deviation
28 100 10000 0.8926 0.0077
29 100 10000 0.9053 0.0061
30 100 10000 0.8961 0.0073
31 100 10000 0.9048 0.0068
32 100 10000 0.9207 0.0065
Table 6.  Results of experiments with the 4-giants algorithm without negation
Bits #Elliptic Curves #DLPs per Curve average value for $c$
28 100 10000 1.2867
29 100 10000 1.3002
30 100 10000 1.2926
31 100 10000 1.2944
32 100 10000 1.3150
Table Options

Download as excel

Bits #Elliptic Curves #DLPs per Curve average value for $c$
28 100 10000 1.2867
29 100 10000 1.3002
30 100 10000 1.2926
31 100 10000 1.2944
32 100 10000 1.3150
[1]

Lianshuan Shi, Enmin Feng, Huanchun Sun, Zhaosheng Feng. A two-step algorithm for layout optimization of structures with discrete variables. Journal of Industrial & Management Optimization, 2007, 3 (3) : 543-552. doi: 10.3934/jimo.2007.3.543
[2]

Kazeem Olalekan Aremu, Chinedu Izuchukwu, Grace Nnenanya Ogwo, Oluwatosin Temitope Mewomo. Multi-step iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2161-2180. doi: 10.3934/jimo.2020063
[3]

Behrouz Kheirfam. A full Nesterov-Todd step infeasible interior-point algorithm for symmetric optimization based on a specific kernel function. Numerical Algebra, Control & Optimization, 2013, 3 (4) : 601-614. doi: 10.3934/naco.2013.3.601
[4]

Yanqin Bai, Lipu Zhang. A full-Newton step interior-point algorithm for symmetric cone convex quadratic optimization. Journal of Industrial & Management Optimization, 2011, 7 (4) : 891-906. doi: 10.3934/jimo.2011.7.891
[5]

Yinghong Xu, Lipu Zhang, Jing Zhang. A full-modified-Newton step infeasible interior-point algorithm for linear optimization. Journal of Industrial & Management Optimization, 2016, 12 (1) : 103-116. doi: 10.3934/jimo.2016.12.103
[6]

Leonid Faybusovich, Cunlu Zhou. Long-step path-following algorithm for quantum information theory: Some numerical aspects and applications. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021017
[7]

Xiaoyu Xing, Hailiang Yang. American type geometric step options. Journal of Industrial & Management Optimization, 2013, 9 (3) : 549-560. doi: 10.3934/jimo.2013.9.549
[8]

Cuixia Miao, Yuzhong Zhang. Scheduling with step-deteriorating jobs to minimize the makespan. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1955-1964. doi: 10.3934/jimo.2018131
[9]

Delphine Boucher. A first step towards the skew duadic codes. Advances in Mathematics of Communications, 2018, 12 (3) : 553-577. doi: 10.3934/amc.2018033
[10]

Jon Aaronson, Michael Bromberg, Nishant Chandgotia. Rational ergodicity of step function skew products. Journal of Modern Dynamics, 2018, 13: 1-42. doi: 10.3934/jmd.2018012
[11]

Santos González, Llorenç Huguet, Consuelo Martínez, Hugo Villafañe. Discrete logarithm like problems and linear recurring sequences. Advances in Mathematics of Communications, 2013, 7 (2) : 187-195. doi: 10.3934/amc.2013.7.187
[12]

Andrew J. Steyer, Erik S. Van Vleck. Underlying one-step methods and nonautonomous stability of general linear methods. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2859-2877. doi: 10.3934/dcdsb.2018108
[13]

Peter E. Kloeden, Björn Schmalfuss. Lyapunov functions and attractors under variable time-step discretization. Discrete & Continuous Dynamical Systems, 1996, 2 (2) : 163-172. doi: 10.3934/dcds.1996.2.163
[14]

Michael A. Saum, Tim Schulze. The role of processing speed in determining step patterns during directional epitaxy. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 443-457. doi: 10.3934/dcdsb.2009.11.443
[15]

Lifeng Chen, Jifa Jiang. Stochastic epidemic models driven by stochastic algorithms with constant step. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 721-736. doi: 10.3934/dcdsb.2016.21.721
[16]

Yoonsang Lee, Bjorn Engquist. Variable step size multiscale methods for stiff and highly oscillatory dynamical systems. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 1079-1097. doi: 10.3934/dcds.2014.34.1079
[17]

Behrouz Kheirfam, Guoqiang Wang. An infeasible full NT-step interior point method for circular optimization. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 171-184. doi: 10.3934/naco.2017011
[18]

Tingting Wu, Yufei Yang, Huichao Jing. Two-step methods for image zooming using duality strategies. Numerical Algebra, Control & Optimization, 2014, 4 (3) : 209-225. doi: 10.3934/naco.2014.4.209
[19]

Van Hieu Dang. An extension of hybrid method without extrapolation step to equilibrium problems. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1723-1741. doi: 10.3934/jimo.2017015
[20]

Angelamaria Cardone, Dajana Conte, Beatrice Paternoster. Two-step collocation methods for fractional differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2709-2725. doi: 10.3934/dcdsb.2018088

2019 Impact Factor: 0.734

Tools

Metrics

  • PDF downloads (454)
  • HTML views (81)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences