Abstract
This paper describes a design of an elliptic curve scalar multiplication and finite field arithmetic.
The scalar multiplication design resists to Simple Power Analysis(SPA) and solves performance problem induced by SPA countermeasure. Izu and Takagi[9] proposed a parallel multiplication method resistant against SPA. When it is implemented in parallel with two processors, the computing time for n-bit scalar multiplication is 1·(point doubling) + (n-1)·(point addition). Although our design uses one multiplier and one inverter for finite field operation, it takes 2n·(inversion) to compute n-bit scalar multiplication. If our algorithm utilizes two processors, the computation time is n(point addition) which is almost same as Izu and Takagi’s result.
The proposed inverter is resistant to Timing Analysis(TA) and Differential Power Analysis(DPA). López and Dahab[13] argued that for GF(2n), projective coordinates perform better than the affine coordinates do when inversion operation is more than 7 times slower than the multiplication operation. Speed ratio of the proposed inverter to the proposed multiplier is 6. Thus, the proposed architecture is efficient on the affine coordinates.
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References
Boneh, D., DeMillo, R., Lipton, R.: On the Importance of Checking Cryptographic Protocols for Faults. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 37–51. Springer, Heidelberg (1997)
Coron, J.-S.: Resistance Against Differential Power Analysis for Elliptic Curve Cryptosystems. In: Koç, Ç.K., Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 292–302. Springer, Heidelberg (1999)
Fischer, W., Giraud, C., Knudsen, E.W., Seifert, J.-P.: Parallel scalar multiplication on general elliptic curves over F p hedged against Non-Differential Side-Channel Attacks (2002), Available from http://eprint.iacr.org
Gutub, A.A.-A., Tenca, A.F., Savaş, E., Koç, Ç.K.: Scalable and unified hardware to compute Montgomery inverse in GF(p) and GF(2n). In: Kaliski Jr., B.S., Koç, Ç.K., Paar, C. (eds.) CHES 2002. LNCS, vol. 2523, pp. 484–499. Springer, Heidelberg (2003)
Ha, J.C., Moon, S.J.: Randomized Signed-Scalar Multiplication of ECC to Resist Power Attacks. In: Kaliski Jr., B.S., Koç, Ç.K., Paar, C. (eds.) CHES 2002. LNCS, vol. 2523, pp. 553–565. Springer, Heidelberg (2003)
Hankerson, D., Hernandez, J., Menezes, A.: Software Implementation of Elliptic Curve Cryptography Over Binary Fields. In: Paar, C., Koç, Ç.K. (eds.) CHES 2000. LNCS, vol. 1965, pp. 1–24. Springer, Heidelberg (2000)
Hasan, M.A.: Power Analysis Attacks and Algorithmic Approaches to their Countermeasures for Koblitz Curve Cryptosystems. In: Paar, C., Koç, Ç.K. (eds.) CHES 2000. LNCS, vol. 1965, pp. 93–108. Springer, Heidelberg (2000)
Institute of Electrical and Electronics Engineers (IEEE), IEEE standard specifications for public-key cryptography, IEEE Std 1363-2000 (2000)
Izu, T., Takagi, T.: A Fast Parallel Elliptic Curve Multiplication Resistant against Side Channel Attacks. In: Naccache, D., Paillier, P. (eds.) PKC 2002. LNCS, vol. 2274, pp. 280–296. Springer, Heidelberg (2002)
Joye, M., Quisquater, J.: Hessian ellitpic curves and side-channel attacks. In: Koç, Ç.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 402–410. Springer, Heidelberg (2001)
Kocher, P.: Timing Attacks on Implementations of Diffie-Hellman, RSA, DSS and Other Systems. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 104–113. Springer, Heidelberg (1996)
Kocher, P., Jaffe, J., Jun, B.: Differential power analysis. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 388–397. Springer, Heidelberg (1999)
López, J., Dahab, R.: Fast Multiplication on Elliptic Curves over GF(2m) without Precomputation. In: Koç, Ç.K., Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 316–327. Springer, Heidelberg (1999)
National Institute of Standards and Technology(NIST), Digital Signature Standard(DSS) FIPS PUB 186-2 (2000)
Okeya, K., Sakurai, K.: Power analysis breaks elliptic curve cryptosystems even secure against the timing attack. In: Roy, B., Okamoto, E. (eds.) INDOCRYPT 2000. LNCS, vol. 1977, pp. 178–190. Springer, Heidelberg (2000)
Okeya, K., Sakurai, K.: On Insecurity of the Side Channel Attack Countermeasure Using Addition-Subtraction Chains under Distinguishability between Addition and Doubling. In: Batten, L.M., Seberry, J. (eds.) ACISP 2002. LNCS, vol. 2384, pp. 420–435. Springer, Heidelberg (2002)
Oswald, E., Aigner, M.: Randomized Addition-Subtraction Chains as a Countermeasure against Power Attacks. In: Koç, Ç.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 39–50. Springer, Heidelberg (2001)
Certicom Research, Standards for efficient cryptography-SEC2 : Recommended elliptic curve cryptography domain parameters (2000), Available from http://www.secg.org
Savaş, E., Koç, Ç.K.: Architectures for unified field inversion with applications in elliptic curve cryptography. In: The 9th IEEE International Conference on Electronics, Circuits and Systems - ICECS 2002, pp. 1155–1158. IEEE, Los Alamitos (2002)
Savaş, E., Tenca, A.F., Koç, Ç.K.: A Scalable and Unified Multiplier Architecture for Finite Fields GF(p) and GF(2n). In: Paar, C., Koç, Ç.K. (eds.) CHES 2000. LNCS, vol. 1965, pp. 281–296. Springer, Heidelberg (2000)
Wolkerstorfer, J.: Dual-Field Arithmetic Unit for GF(p) and GF(2m). In: Kaliski Jr., B.S., Koç, Ç.K., Paar, C. (eds.) CHES 2002. LNCS, vol. 2523, pp. 501–515. Springer, Heidelberg (2003)
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Yoon, J.C., Jung, S.W., Lee, S. (2004). Architecture for an Elliptic Curve Scalar Multiplication Resistant to Some Side-Channel Attacks. In: Lim, JI., Lee, DH. (eds) Information Security and Cryptology - ICISC 2003. ICISC 2003. Lecture Notes in Computer Science, vol 2971. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24691-6_12
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DOI: https://doi.org/10.1007/978-3-540-24691-6_12
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