Abstract
Elliptic curve cryptography (ECC) is an ideal choice for low-resource applications because it provides the same level of security with smaller key sizes than other existing public key encryption schemes. For low-resource applications, designing efficient functional units for elliptic curve computations over binary fields results in an effective platform for an embedded co-processor. This paper proposes such a co-processor designed for area-constrained devices by utilizing state of the art binary Edwards curve equations over mixed point addition and doubling. The binary Edwards curve offers the security advantage that it is complete and is, therefore, immune to the exceptional points attack. In conjunction with Montgomery Ladder, such a curve is naturally immune to most types of simple power and timing attacks. The recently presented formulas for mixed point addition in [1] were found to be invalid, but were corrected such that the speed and register usage were maintained. We utilize corrected mixed point addition and doubling formulas to achieve a secure, but still fast implementation of a point multiplication on binary Edwards curves. Our synthesis results over NIST recommended fields for ECC indicate that the proposed co-processor requires about 50 % fewer clock cycles for point multiplication and occupies a similar silicon area when compared to the most recent in literature.
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Acknowledgements
The authors would like to thank the reviewers for their constructive comments. This material is based upon work supported by the National Science Foundation under Award No. CNS-1464118 to Reza Azarderakhsh.
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A Appendix
A Appendix
1.1 A.1 Subroutines
This contains a code listing of the program in assembly.
Algorithm 3 shows the Itoh-Tsujii [19] inversion subroutine for \(\mathbb {F}_{2^{283}}\). This follows the addition chain (1,2,4,8,16,17,34,35,70,140,141,282). Eleven multiplications are required for this binary field. A similar approach was done for \(\mathbb {F}_{2^{163}}\) and \(\mathbb {F}_{2^{233}}\).
Algorithm 4 shows the half-trace subroutine. This is a simple double square and add routine that produces the result after \(\frac{m-1}{2}\) iterations.
Algorithm 5 shows the beginning of the main program that was used. This includes the initialization of the point and the repeated step of the Montgomery ladder [11].
Algorithm 6 shows the end of the main program that was used. This includes the recovery of \(w_{2},w_{3},x_{2},y_{2}\).
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Koziel, B., Azarderakhsh, R., Mozaffari-Kermani, M. (2015). Low-Resource and Fast Binary Edwards Curves Cryptography. In: Biryukov, A., Goyal, V. (eds) Progress in Cryptology -- INDOCRYPT 2015. INDOCRYPT 2015. Lecture Notes in Computer Science(), vol 9462. Springer, Cham. https://doi.org/10.1007/978-3-319-26617-6_19
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