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Combining Montgomery Multiplication with Tag Tracing for the Pollard Rho Algorithm in Prime Order Fields

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Security, Privacy, and Applied Cryptography Engineering (SPACE 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13783))

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Abstract

In this short paper we show how to apply Montgomery multiplication to the tag tracing variant of the Pollard rho algorithm applied to prime order fields. This combines the advantages of tag tracing with those of Montgomery multiplication. In particular, compared to the previous version of tag tracing, the use of Montgomery multiplication entirely eliminates costly modular reductions and replaces these with much more efficient divisions by a suitable power of two.

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References

  1. Bos, J.W., Montgomery, P.L.: Montgomery arithmetic from a software perspective. In: Bos, J.W., Lenstra, A.K. (eds.) Topics in Computational Number Theory Inspired by Peter L. Montgomery, pp. 10–39. Cambridge University Press (2017)

    Google Scholar 

  2. Cheon, J.H., Hong, J., Kim, M.: Accelerating Pollard’s rho algorithm on finite fields. J. Cryptol. 25(2), 195–242 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Montgomery, P.L.: Modular multiplication without trial division. Math. Comput. 44(170), 519–521 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  4. van Oorschot, P., Wiener, M.: Parallel collision search with cryptanalytic applications. J. Cryptol. 12, 1–28 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Pollard, J.M.: A Monte Carlo method for index computation \((\text{ mod } \, p)\). Math. Comput. 32(143), 918–924 (1978)

    MATH  Google Scholar 

  6. Schnorr, C., Lenstra, H.W.: A Monte Carlo factoring algorithm with linear storage. Math. Comput. 43(167), 289–311 (1984)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank Ruben Niederhagen and the reviewers for helpful comments and suggestions on how to improve the presentation of the paper.

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Authors and Affiliations

  1. Indian Institute of Technology, Kanpur, India

    Madhurima Mukhopadhyay

  2. Indian Statistical Institute, Kolkata, India

    Palash Sarkar

Authors
  1. Madhurima Mukhopadhyay
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  2. Palash Sarkar
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Correspondence to Madhurima Mukhopadhyay .

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Editors and Affiliations

  1. Radboud University, Nijmegen, The Netherlands

    Lejla Batina

  2. Radboud University, Nijmegen, The Netherlands

    Stjepan Picek

  3. Indian Institute of Technology Kharagpur, Kharagpur, India

    Mainack Mondal

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Mukhopadhyay, M., Sarkar, P. (2022). Combining Montgomery Multiplication with Tag Tracing for the Pollard Rho Algorithm in Prime Order Fields. In: Batina, L., Picek, S., Mondal, M. (eds) Security, Privacy, and Applied Cryptography Engineering. SPACE 2022. Lecture Notes in Computer Science, vol 13783. Springer, Cham. https://doi.org/10.1007/978-3-031-22829-2_8

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  • DOI: https://doi.org/10.1007/978-3-031-22829-2_8

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-22828-5

  • Online ISBN: 978-3-031-22829-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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