Faster addition and doubling on elliptic curves

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Volumn 4833 LNCS, Issue , 2007, Pages 29-50

  Bernstein, Daniel J ^a   Lange, Tanja ^b  

^a UNIVERSITY OF ILLINOIS AT CHICAGO   (United States)

^b EINDHOVEN UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; BINARY SEQUENCES; CURVE FITTING; PROBLEM SOLVING;

ARBITRARY EDWARDS CURVES; ELLIPTIC CURVES; MULTI-SCALAR MULTIPLICATION;

COMPUTATIONAL METHODS;

EID: 38149061268     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/978-3-540-76900-2_3     Document Type: Conference Paper

References (45)

1. Antipa, A., Brown, D.R.L., Gallant, R.P., Lambert, R.J., Struik, R., Vanstone, S.A.: Accelerated verification of ECDSA signatures, in [43], pp. 307-318 (2006). MR 2007d:94044, www.cacr.math.uwaterloo.ca/techreports/2005/ tech_reports2005.html (Cited in §7)
2. Avanzi, R.M.: The complexity of certain multi-exponentiation techniques in cryptography. Journal of Cryptology 18, 357-373 (2005). MR 2007f:94027, www.eprint.iacr.org/2002/154 (Cited in §6, §7)
3. Barbosa, M., Page, D.: On the automatic construction of indistinguishable operations (2005), www.eprint.iacr.org/2005/174 (Cited in §8)
4. Bellare, M., Garay, J.A., Rabin, T.: Batch verification with applications to cryptography and checking, in [35], pp. 170-191 (1998). MR 99h:94043. (Cited in §7)
5. Bernstein, D.J.: A software implementation of NIST P-224 (2001), www.cr.yp.to/talks.html#2001.10.29 (Cited in §5)
6. Bernstein, D.J.: Differential addition chains (2006), www.cr.yp.to/papers.html#diffchain (Cited in §7, §8)
7. Bernstein, D.J.: Curve25519: new Diffie-Hellman speed records, in [45], pp. 207-228 (2006), www.cr.yp.to/papers.html#curve25519 (Cited in §1, §2,§4, §5, §8)
8. Bernstein, D.J., Lange, T.: Explicit-formulas database (2007), www. hyperelliptic.org/EFD (Cited in §2, §3, §3,§5)
9. Billet, O., Joye, M.: The Jacobi model of an elliptic curve and side-channel analysis, in [26], pp. 34-42 (2003). MR 2005c:94045, www.eprint.iacr.org/2002/125 (Cited in §1, §5)
10. Blake, I.F., Seroussi, G., Smart, N.P. (eds.): Advances in elliptic curve cryptography. London Mathematical Society Lecture Note Series, 317. Cambridge University Press, Cambridge (2005), MR 2007g:94001. See [27]
11. Bosma, W., Lenstra Jr., H.W.: Complete systems of two addition laws for elliptic curves. Journal of Number Theory 53, 229-240 (1995), MR 96f: 11079. (Cited in §3, §3)
12. Brier, É., Déchène, I., Joye, M.: Unified point addition formulae for elliptic curve cryptosystems, in [40], pp. 247-256 (2004) (Cited in §5)
13. Brier, É., Joye, M.: Weierstrass elliptic curves and side-channel attacks, in [39], pp. 335-345 (2002), www.geocities.com/MarcJoye/publications. html (Cited in §5, §8)
14. Brown, D.R.L.: Multi-dimensional Montgomery ladders for elliptic curves (2006), www.eprint.iacr.org/2006/220 (Cited in §7, §8)
15. Chevallier-Mames, B., Ciet, M., Joye, M.: Low-cost solutions for preventing simple side-channel analysis: side-channel atomicity. IEEE Transactions on Computers 53, 760-768 (2004), www.bcm.crypto.free.fr/pdf/CCJ04. pdf (Cited in §8)
16. Chudnovsky, D.V., Chudnovsky, G.V.: Sequences of numbers generated by addition in formal groups and new primality and factorization tests. Advances in Applied Mathematics 7, 385-434 (1986), MR 88h: 11094. (Cited in §5)
17. Cohen, H., Frey, G. (eds.): Handbook of elliptic and hyperelliptic curve cryptography. CRC Press, Boca Raton (2005), MR 2007f: 14020. See [22], [24], [33]
18. Cohen, H., Miyaji, A., Ono, T.: Efficient elliptic curve exponentiation using mixed coordinates, in [41], pp. 51-65 (1998), MR 1726152, www.math.u-bordeaux.fr/~cohen/asiacrypt98.dvi (Cited in §1, §5)
19. Coron, J.-S.: Resistance against differential power analysis for elliptic curve cryptosystems, in [32], pp. 292-302 (1999) (Cited in §8, §8, §8)
20. de Rooij, P.: Efficient exponentiation using precomputation and vector addition chains, in [21], pp. 389-399 (1995), MR 1479665. (Cited in §7)
21. De Santis, A. (ed.): Advances in cryptology: EUROCRYPT 1994. LNCS, vol. 950. Springer, Heidelberg (1995), MR 98h:94001. See [20]
22. Doche, C.: Exponentiation, in [17], pp. 145-168 (2005) MR 2162725. (Cited in §6, §7)
23. Doche, C., Icart, T., Kohel, D.R.: Efficient scalar multiplication by isogeny decompositions, in [45], pp. 191-206 (2006) (Cited in §1)
24. Doche, C., Lange, T.: Arithmetic of elliptic curves, in [17], pp. 267-302 (2005), MR 2162729. (Cited in §5)
25. Edwards, H.M.: A normal form for elliptic curves. Bulletin of the American Mathematical Society 44, 393-422 (2007), www.ams.org/bull/2007-44-03/ S0273-0979-07-01153-6/home.html (Cited in §1, §3)
26. Fossorier, M.P.C., Høholdt, T., Poli, A. (eds.): Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. LNCS, vol. 2643. Springer, Heidelberg (2003). ISBN 3-540-40111-3. MR 2004j:94001. (Sec [9])
27. Joye, M.: Defences against side-channel analysis, in [10], pp. 87-100 (2005) (Cited in §8)
28. Joye, M., Quisquater, J.-J.: Hessian elliptic curves and side-channel attacks, in [31], pp. 402-410 (2001). MR 2003k:94032, www.geocities.com/ MarcJoye/publications.html (Cited in §1, §5)
29. Joye, M., Yen, S.-M.: The Montgomery powering ladder, in [30], pp. 291-302 (2003), www.gemplus.com/smart/rd/publications/pdf/JY03mont.pdf (Cited in §8)
30. Kaliski Jr., B.S., Koç, Ç.K., Paar, C. (eds.): Cryptographic hardware and embedded systems-CHES 2002. LNCS, vol. 2523. Springer, Heidelberg (2003). ISBN 3-540-42521-7. See [29]
31. Koç, Ç.K., Naccache, D., Paar, C. (eds.): Cryptographic hardware and embedded systems-CHES 2001. LNCS, vol. 2162. Springer, Heidelberg (2001). ISBN 3-540-42521-7. MR 2003g:94002. See [28], [34], [42]
32. Koç, Ç.K., Paar, C. (eds.): Cryptographic hardware and embedded systems. In: first international workshop CHES 1999. LNCS, vol. 1717. Springer, Heidelberg (1999). ISBN 3-540-66646-X. See [19]
33. Lange, T.: Mathematical countermeasures against side-channel attacks, in [17], pp. 687-714 (2005), MR 2163785. (Cited in §8, §8)
34. Liardet, P.-Y., Smart, N.P.: Preventing SPA/DPA in ECC systems using the Jacobi form, in [31], pp. 391-401 (2001), MR 2003k:94033. (Cited in §1, §5, §8)
35. Lucchesi, C.L., Moura, A.V. (eds.): LATIN 1998: theoretical informatic. LNCS, vol. 1380. Springer, Heidelberg (1998). ISBN 3-540-64275-7. MR 99d:68007. See [4]
36. Mangard, S., Oswald, E., Popp, T.: Power analysis attacks: revealing the secrets of smart cards. Springer, Heidelberg (2007) (Cited in §8, §8)
37. Miller, V.S.: Use of elliptic curves in cryptography, in [44], pp. 417-426 (1986) MR 88b:68040. (Cited in §1)
38. Montgomery, P.L.: Speeding the Pollard and elliptic curve methods of factorization. Mathematics of Computation 48, 243-264 (1987) MR 88e:11130, www.links.jstor.org/sici?sici=0025-5718(198701)48:177〈243:STPAEC〉2.0. CO; 2-3 (Cited in §1, §6, §7)
39. Naccache, D., Paillier, P. (eds.): Public key cryptography. In: PKC 2002. LNCS, vol. 2274. Springer, Heidelberg (2002). ISBN 3-540-43168-3. MR 2005b:94044. See [13]
40. Nedjah, N., de Macedo Mourelle, L. (eds.): Embedded Cryptographic Hardware: Methodologies & Architectures, Nova Science Publishers (2004) ISBN 1-59454-012-8. See [12]
41. Ohta, K., Pei, D. (eds.): Advances in cryptology-ASIACRYPT 1998. LNCS, vol. 1514. Springer, Berlin (1998). ISBN 3-540-65109-8. MR 2000h:94002. See [18]
42. Oswald, E., Aigner, M.: Randomized addition-subtraction chains as a countermeasure against power attack, in [31], pp. 39-50 (2001) MR 2003m:94068. (Cited in §8)
43. Preneel, B., Tavares, S.E. (eds.): Selected Areas in Cryptography. In: SAC 2005. LNCS, vol. 3897, Springer, Heidelberg (2006). ISBN3-540-33108-5. MR 2007b:94002. See [1]
44. Williams, H.C. (ed.): CRYPTO 1985. LNCS, vol. 218. Springer, Berlin (1986). ISBN 3-540-16463-4. MR 87d:94002. See [37]
45. Yung, M., Dodis, Y., Kiayias, A., Malkin, T. (eds.): 9th international conference on theory and practice in public-key cryptography. LNCS, vol. 3958. Springer, Heidelberg (2006). ISBN 978-3-540-33851-2. See [7], [23]