Abstract
Modular exponentiation is one of the core operations in most of the public-key cryptosystems. It consists of a sequence of modular multiplications. The performance of public-key cryptographic transformations is strongly influenced by the competent implementation of modular exponentiation and modular multiplication. This paper presents the hardware implementation of modular exponentiation on two processor cores. Montgomery multiplication method is modified according to the needs of dual-core implementation to improve the core utilization. It is implemented with different radices ranging from \(2^2\) to \(2^{32}\). The performance of the proposed design is analyzed and compared with the existing techniques in terms of number of clock cycles, throughput, power, and area. The proposed design has been developed using Verilog and synthesized using Xilinx-14.6 ISE for usage in FPGA, and the same has been synthesized using Cadence for ASIC. But here the results are presented based on FPGA.
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References
Diffie, W., Hellman, M.E.: New directions in cryptography. IEEE Trans. Inf. Theory 22(6), 644–654 (1976)
Rivest, R.L., Shamir, A., Adleman, L.: A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM 21(2), 120–126 (1978)
de Dormale, G.M., Quisquater, J.-J.: High-speed hardware implementations of elliptic curve cryptography: a survey. 53(2), 72–84 (2007)
Tibouchi, M., Kim, T.: Improved elliptic curve hashing and point representation. Des. Codes Cryptogr. 1–17 (2016)
Bos, J.W., Costello, C., Longa, P., Naehrig, M.: Selecting elliptic curves for cryptography: an efficiency and security analysis. J. Cryptogr. Eng. 1–28 (2015)
ElGamal, T.: A public key cryptosystem and a signature scheme based on discrete logarithms. In: Advances in Cryptology, pp. 10–18. Springer (1985)
Kaminaga, M., Yoshikawa, H., Suzuki, T.: Double counting in-ary RSA precomputation reveals the secret exponent. IEEE Trans. Inf. Forensics Secur. 10(7), 1394–1401 (2015)
Huang, X., Wang, W.: A novel and efficient design for an RSA cryptosystem with a very large key size. IEEE Trans. Circuits Syst. II Express Briefs 62(10), 972–976 (2015)
Garg, H.K., Xiao, H.: New residue arithmetic based Barrett algorithms: modular integer computations. IEEE Access 4, 4882–4890 (2016)
Montgomery, P.L.: Modular multiplication without trial division. Math. Comput. 44(170), 519–521 (1985)
Meng, Q., Chen, T., Dai, Z., Chen, Q.: A scalable hybrid modular multiplication algorithm. J. Electron. (China) 25(3), 378–383 (2008)
Shieh, M.-D., Chen, J.-H., Wu, H.-H., Lin, W.-C.: A new modular exponentiation architecture for efficient design of RSA cryptosystem. IEEE Trans. Very Large Scale Integr. (VLSI) Syst. 16(9), 1151–1161 (2008)
Kong, Y., Asif, S., Khan, M.A.U.: Modular multiplication using the core function in the residue number system. Appl. Algebra Eng. Commun. Comput. 1–16 (2015)
Wu, T., Li, S.G., Liu, L.T.: Fast RSA decryption through high-radix scalable montgomery modular multipliers. Sci. China Inf. Sci. 58(6), 1–16 (2015)
Miyamoto, A., Homma, N., Aoki, T., Satoh, A.: Systematic design of RSA processors based on high-radix montgomery multipliers. IEEE Trans. Very Large Scale Integr. (VLSI) Syst. 19(7):1136–1146 (2011)
Yao, G.X., Fan, J., Cheung, R.C.C., Verbauwhede, I.: Novel RNS parameter selection for fast modular multiplication. IEEE Trans. Comput. 63(8), 2099–2105 (2014)
Kuang, S.-R., Wang, J.-P., Chang, K.-C., Hsu, H.-W.: Energy-efficient high-throughput montgomery modular multipliers for RSA cryptosystems. IEEE Trans. Very Large Scale Integr. (VLSI) Syst. 21(11), 1999–2009 (2013)
Kuang, S.-R., Wu, K.-Y., Lu, R.-Y.: Low-cost high-performance VLSI architecture for montgomery modular multiplication
Koç, C.K., Acar, T., Kaliski, B.S. Jr.: Analyzing and comparing montgomery multiplication algorithms. IEEE Micro 16(3), 26–33 (1996)
Chen, X., Li, J., Ma, J., Tang, Q., Lou, W.: New algorithms for secure outsourcing of modular exponentiations. IEEE Trans. Parallel Distrib. Syst. 25(9), 2386–2396 (2014)
Néto, J.C., Tenca, A.F., Ruggiero, W.V.: A parallel and uniform-partition method for montgomery multiplication. IEEE Trans. Comput. 63(9), 2122–2133 (2014)
Schinianakis, D., Stouraitis, T.: Multifunction residue architectures for cryptography. IEEE Trans. Circuits Syst. I Regul. Pap. 61(4), 1156–1169 (2014)
Rezai, A., Keshavarzi, P.: High-throughput modular multiplication and exponentiation algorithms using multibit-scan-multibit-shift technique. IEEE Trans. Very Large Scale Integr. (VLSI) Syst. 23(9), 1710–1719 (2015)
Paillier, P.: Public-key cryptosystems based on composite degree residuosity classes. In: Advances in cryptology (UROCRYPT’99), pp. 223–238. Springer (1999)
Gordon, D.M.: A survey of fast exponentiation methods. J. Algorithm. 27(1), 129–146 (1998)
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Vollala, S., Ramasubramanian, N., Shameedha Begum, B., Joshi, A.D. (2019). Dual-Core Implementation of Right-to-Left Modular Exponentiation. In: Sa, P., Bakshi, S., Hatzilygeroudis, I., Sahoo, M. (eds) Recent Findings in Intelligent Computing Techniques . Advances in Intelligent Systems and Computing, vol 707. Springer, Singapore. https://doi.org/10.1007/978-981-10-8639-7_5
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