Differential Addition in Edwards Coordinates Revisited and a Short Note on Doubling in Twisted Edwards Form

Srinivasa Rao Subramanya Rao

Abstract

Cryptographic algorithms in smart cards and other constrained environments increasingly rely on Elliptic Curves and thus it is desirable to have fast algorithms for elliptic curve arithmetic. In this paper, we provide (i) faster differential addition formulae for elliptic curve arithmetic on Generalized Edwards’ Curves improving upon the currently known formulae in the literature, proposed by Justus and Loebenberger at IWSEC 2010, (ii) more efficient affine differential addition formulae for a new model of Binary Edwards Curves proposed by Wu, Tang and Feng at INDOCRYPT 2012 and (iii) an algorithm for point doubling on Twisted Edwards Curves with a smaller footprint when the implementation is desired to work across Homogeneous Projective, Inverted and Extended Homogeneous Projective Coordinates.

References

  1. Bernstein, D. (2006a). Curve25519: New Diffie-Hellman speed records. In Public Key Cryptography - PKC 2006, LNCS 3958.
  2. Bernstein, D. (2006b). Differential Addition Chains. Technical report, http://cr.yp.to/ecdh/diffchain-2006 0219.pdf accessed on 30th Nov 2015.
  3. Bernstein, D. and Lange, T. (2007). Explicit Forms Database(EFD). Technical report, http://hyperellip tic.org/EFD/ accessed on 30th Nov 2015.
  4. Bernstein, D., Lange, T., and Farashahi, T. (2008a). Binary Edwards Curves. In Cryptographic Hardware and Embedded Systems - CHES 2008, LNCS 5154.
  5. Bernstein, D., P.Birkner, M.Joye, T.Lange, and C.Peters (2008b). Twisted Edwards Curves. In AFRICACRYPT 2008, LNCS 5023.
  6. Bernstein, D. and T.Lange (2007a). Faster addition and doubling on Elliptic curves. In ASIACRYPT 2007, LNCS 4833.
  7. Bernstein, D. and T.Lange (2007b). Inverted Edwards coordinates. In Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-17, LNCS 4851.
  8. D.Knuth (1998). The Art of Computer Programming Vol 2. Pearson Education.
  9. E.Brier and M.Joye (2002). Weierstrass Elliptic Curves and side channel attacks. In Public Key Cryptography - PKC 2002, LNCS 2274.
  10. Edwards, H. (2007). A normal form for elliptic curves. Bulletin of the AMS, 44(3):393422.
  11. Hisil, H. (2010). Elliptic Curves,Group Law, and Efficient Computation. PhD thesis, Queensland University of Technology.
  12. H.Wu, C.Tang, and R.Feng (2012). A new model of Binary Elliptic Curves. In INDOCRYPT 2012, LNCS 7668.
  13. J.Lopez and R.Dahab (1999). Fast multiplication on Elliptic Curves over GF (2m) without precomputation. In Cryptographic Hardware and Embedded Systems - CHES 1999, LNCS 1717.
  14. M.Joye and S.Yen (2002). The Montgomery Powering Ladder. In Cryptographic Hardware and Embedded Systems - CHES 2002, LNCS 2523.
  15. Montgomery, P. L. (1992). Evaluating recurrences of form Xm+n = f (Xm, Xn, Xm-n) via Lucas chains. Technical report, ftp://ftp.cwi.nl/pub/pmontgom/Lucas.ps.gz accessed on 30th Nov 2015.
  16. P.L.Montgomery (1987). Speeding the Pollard and Elliptic Curve methods of Factorization. In Mathematics of Computation Vol 48, Issue 177 Jan 1987.
  17. R.Justus and Loebenberger, D. (2010). Differential Addition in Generalized Edwards Coordinates. In 5th International Workshop on Security - IWSEC 2010,, LNCS 6434.
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Paper Citation


in Harvard Style

Subramanya Rao S. (2016). Differential Addition in Edwards Coordinates Revisited and a Short Note on Doubling in Twisted Edwards Form . In Proceedings of the 13th International Joint Conference on e-Business and Telecommunications - Volume 4: SECRYPT, (ICETE 2016) ISBN 978-989-758-196-0, pages 336-343. DOI: 10.5220/0005970603360343


in Bibtex Style

@conference{secrypt16,
author={Srinivasa Rao Subramanya Rao},
title={Differential Addition in Edwards Coordinates Revisited and a Short Note on Doubling in Twisted Edwards Form},
booktitle={Proceedings of the 13th International Joint Conference on e-Business and Telecommunications - Volume 4: SECRYPT, (ICETE 2016)},
year={2016},
pages={336-343},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0005970603360343},
isbn={978-989-758-196-0},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 13th International Joint Conference on e-Business and Telecommunications - Volume 4: SECRYPT, (ICETE 2016)
TI - Differential Addition in Edwards Coordinates Revisited and a Short Note on Doubling in Twisted Edwards Form
SN - 978-989-758-196-0
AU - Subramanya Rao S.
PY - 2016
SP - 336
EP - 343
DO - 10.5220/0005970603360343