RSA Cryptosystem Based on Early Word Based Montgomery Modular Multiplication

  • Rupali VermaEmail author
  • Maitreyee Dutta
  • Renu Vig
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10975)


RSA is a public key cryptosystem in which encryption and decryption are modular exponentiation functions. Modular exponentiation is achieved by repeated modular multiplications. Montgomery modular multiplication is an efficient algorithm, hence is widely used for RSA public key cryptosystem. Performance of RSA depends on throughput of Montgomery modular multiplication. This paper presents RSA with Early Word based Montgomery modular multiplication. Early word based approach is scalable and Radix 4 Early Word Based Common Multiplicand Montgomery is proposed. RSA cryptosystem is implemented on virtex 5 FPGAs. The processing elements in Early Word based Montgomery use target device resources DSP48E for addition of operands. Two factors: algorithmic approach and use of target device resources have improved the performance of RSA on FPGAs.


Early word based FPGA Montgomery RSA 


  1. 1.
    Rivest, R.L., Shamir, A., Adleman, L.: A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM 21(2), 120–126 (1978)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Koc, C.K.: High-Speed RSA Implementation, RSA Labs, Redwood City, CA, Technical report (1994)Google Scholar
  3. 3.
    Montgomery, P.L.: Modular multiplication without trial division. Math. Comput. 44(170), 519–521 (1985)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Tenca, A.F., Koc, C.K.: A Scalable architecture for modular multiplication based on montgomery’s algorithm. IEEE Trans. Comput. 52(9), 1215–1221 (2003)CrossRefGoogle Scholar
  5. 5.
    Harris, D., Krishnamurthy, R., Anders, M., Mathew, S., Hsu, S.: An improved unified scalable radix 2 Montgomery multiplier. In: Proceedings of the 17th IEEE Symposium on Computer Arithmetic (ARITH), pp. 172–178 (2005)Google Scholar
  6. 6.
    Shieh, M.D., Lin, W.C.: Word based Montgomery modular multiplication algorithm for low latency scalable architectures. IEEE Trans. Comput. 59(8), 1145–1151 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Lin, W.C., Ye, J.H., Shieh, M.D.: Scalable Montgomery modular multiplication architecture with low-latency and low-memory bandwidth requirement. IEEE Trans. Comput. 63(2), 475–483 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Huang, M., Gaj, K., Ghazawi, T.E.: New hardware architectures for Montgomery modular multiplication algorithm. IEEE Trans. Comput. 60(7), 923–936 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chen, D.S., Li, H.T., Wang, Y.W.: A prediction-based scalable design for Montgomery modular multiplication. In: Proceedings of 3rd International Conference on Electric and Electronics, pp. 46–50 (2013)Google Scholar
  10. 10.
    McIvor, C., McLoone, M., McCanny, J.V.: Modified Montgomery modular multiplication and RSA exponentiation techniques. IEE Proc. Comput. Digit. Techniques 151(6), 402–408 (2004)Google Scholar
  11. 11.
    Verma, R., Dutta, M., Vig, R.: Early-word-based Montgomery modular multiplication algorithm. In: Proceedings of IEEE 2nd International conference on SPIN, pp. 595–600 (2015)Google Scholar
  12. 12.
    Sutter, G.D., Deschamps, J.-P., Imana, J.L.: Modular multiplication and exponentiation architectures for fast RSA cryptosystem based on digit serial computation. IEEE Trans. Ind. Electron. 58(7), 3101–3109 (2011)CrossRefGoogle Scholar
  13. 13.
    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. VLSI Syst. 16(9), 1151–1161 (2008)CrossRefGoogle Scholar
  14. 14.
    Joye, M., Yen, S-M.: The Montgomery powering ladder. In: Kaliski, B.S., Koç, Ç.K., Paar, C. (eds.) CHES 2002. LNCS, vol. 2523, pp. 291–302. Springer, Heidelberg (2003). Scholar
  15. 15.
    Wu, T., Li, S., Liu, L.: Fast, compact and symmetric modular exponentiation architecture by common multiplicand Montgomery modular multiplication. Integr. VLSI J. 46, 323–332 (2013)CrossRefGoogle Scholar
  16. 16.
    Verma, R., Dutta, M., Vig, R.: Carry save common multiplicand Montgomery for RSA cryptosystem. Am. J. Appl. Sci. 11(5), 851–856 (2014)CrossRefGoogle Scholar
  17. 17.
    Walter, C.D.: Montgomery exponentiation needs no final subtractions. Electron. Lett. 32(21), 1831–1832 (1999)CrossRefGoogle Scholar
  18. 18.
    Elridge, S.E., Walter, C.D.: Hardware implementation of Montgomery’s modular multiplication algorithm. IEEE Trans. Comput. 42(6), 693–699 (1993)CrossRefGoogle Scholar
  19. 19.
    Mesquita, D., Perin, G., Herrmann, F.L., Martins, J.B.: An efficient implementation of montgomery powering ladder in reconfigurable hardware. In: Proceedings of 23rd Annual Symposium on Integrated Circuits and Systems Design SBCCI, pp. 121–126 (2010)Google Scholar
  20. 20.
    Perin, G., Mesquita, D.G., Herrmann, F.L., Martins, J.B.: Montgomery modular multiplication on reconfigurable hardware: fully systolic array vs parallel implementation. In: Proceedings of IEEE VI Southern Programmable Logic Conference, pp. 61–66 (2010)Google Scholar
  21. 21.
    Wang, P., Liu, Z., Wang, L., Gao, N.: High radix montgomery modular multiplier on modern FPGA. In: Proceedings of 12th IEEE International Conference on Trust, Security and Privacy in Computing and Communication, pp. 1484–1489 (2013)Google Scholar
  22. 22.
    Fournaris, A.P.: Fault and simple power attack resistant RSA using Montgomery modular multiplication. In: Proceedings of 2010 IEEE International Symposium on Circuits and Systems, pp. 1875–1878 (2010)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Punjab Engineering CollegeChandigarhIndia
  2. 2.NITTTRChandigarhIndia
  3. 3.UIET, Panjab UniversityChandigarhIndia

Personalised recommendations