Advertisement

Detection of Primes in the Set of Residues of Divisors of a Given Number

Conference paper
  • 440 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10737)

Abstract

We consider the following problem: given the set of residues modulo p of all divisors of some squarefree number n, can we find efficiently a small set of residues containing all residues coming from prime factors? We present an algorithm which solves this problem for p and n satisfying some natural conditions and show that there are plenty of such numbers. One interesting feature of the proof is that it relies on additive combinatorics. We also give some application of this result to algorithmic number theory.

Keywords

Small Doubling Probabilistic Polynomial Time Reduction Large Additional Energy Hensel Lifting Deterministic Reduction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgement

I would like to thank Doctor Bartosz Źrałek for introducing me to the topic of deterministic reductions of integer factorization to computing values of number-theoretic functions and encouraging me to work further on the problem considered here. I would like to thank Professor Tomasz Schoen for fruitful discussions on additive combinatorics and its connection to the problem. I would also like to thank anonymous referee for suggesting a simplification of the proof of Lemma 20 which led to a better estimate and the other anonymous referee for suggestions concerning the notation, which increased the readability of the paper.

References

  1. 1.
    Adleman, L.M., McCurley, K.S.: Open problems in number theoretic complexity, II. In: Adleman, L.M., Huang, M.-D. (eds.) ANTS 1994. LNCS, vol. 877, pp. 291–322. Springer, Heidelberg (1994).  https://doi.org/10.1007/3-540-58691-1_70 CrossRefGoogle Scholar
  2. 2.
    Bach, E., Shallit, J.: Algorithmic Number Theory. MIT Press, Cambridge (1996)zbMATHGoogle Scholar
  3. 3.
    Bach, E., Miller, G., Shallit, J.: Sums of divisors, perfect numbers and factoring. SIAM J. Comput. 15(4), 1143–1154 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bluestein, L.I.: A linear filtering approach to the computation of the discrete Fourier transform. Northeast Electron. Res. Eng. Meet. Rec. 10, 218–219 (1968)Google Scholar
  5. 5.
    Burthe Jr., R.J.: The average least witness is 2. Acta Arithmetica 80, 327–341 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Erdős, P., Kac, M.: The Gaussian law of errors in the theory of additive number theoretic functions. Am. J. Math. 62, 738–742 (1940)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Green, B.J., Ruzsa, I.Z.: Sets with small sumsets and rectification. Bull. London Math. Soc. 38(1), 43–52 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Katz, N.H., Koester, P.: On additive doubling and energy. SIAM J. Discrete Math. 24(4), 1684–1693 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Mertens, F.: Ein Beitrag zur analytischen Zahlentheorie. J. Reine Angew. Math. 78, 46–62 (1874)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Mead, D.G.: Newton’s identities. Am. Math. Mon. 99(8), 749–751 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Miller, G.: Riemann’s hypothesis and tests for primality. J. Comput. Syst. Sci. 13, 300–317 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Plünnecke, H.: Eine zahlentheorische anwendung der graphtheorie. J. Reine Angew. Math. 243, 171–183 (1970)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Pollard, J.M.: Monte Carlo methods for index computation (mod p). Math. Comput. 32(143), 918–924 (1978)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Ruzsa, I.: Sumsets and structure. In: Combinatorial Number Theory and Additive Group Theory, pp. 87–210 (2009)Google Scholar
  15. 15.
    Schoen, T.: New bounds in Balog-Szemer’edi-Gowers theorem. Combinatorica 34(5), 1–7 (2014)MathSciNetGoogle Scholar
  16. 16.
    Schönhage, A., Strassen, V.: Schnelle Multiplikation großer Zahlen. Computing 7, 281–292 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Shoup, V.: On the deterministic complexity of factoring polynomials over finite fields. Inf. Process. Lett. 33, 261–267 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Tao, T.C., Vu, H.V.: Additive Combinatorics. Cambridge Studies in Advanced Mathematics, vol. 105. Cambridge University Press, Cambridge (2006)CrossRefzbMATHGoogle Scholar
  19. 19.
    Vinogradov, A.I.: The density hypothesis for Dirichlet L-series. Izv. Akad. Nauk SSSR Ser. Mat. 29(4), 903–934 (1965). (in Russian)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Źrałek, B.: A deterministic version of Pollard’s p-1 algorithm. Math. Comput. 79, 513–533 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Discrete MathematicsAdam Mickiewicz University in PoznańPoznańPoland

Personalised recommendations