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Open Problems for Polynomials over Finite Fields and Applications

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Abstract

We survey open problems for univariate polynomials over finite fields. We first comment in some detail on the existence and number of several classes of polynomials. The open problems in that part of the survey are of a more theoretical nature. Then, we center on classes of low-weight (irreducible) polynomials. The conjectures here are more practically oriented. Finally, we give brief descriptions of a selection of open problems from several areas including factorization of polynomials, special polynomials (APN functions, permutations), and relations between rational integers and polynomials.

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References

  1. J. Arney, E.A. Bender, Random mappings with constraints on coalescence and number of origins. Pac. J. Math. 103, 269–294 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  2. J.-P. Aumasson, M. Finiasz, W. Meier, S. Vaudenay, A hardware-oriented trapdoor cipher, in Information Security and Privacy. Lecture Notes in Computer Science, vol. 4586 (Springer, New York, 2007), pp. 184–199

    Google Scholar 

  3. R.C. Bose, On some connections between the design of experiments and information theory. Bull. Inst. Int. Stat. 38, 257–271 (1961)

    MATH  Google Scholar 

  4. K.A. Browning, J.F. Dillon, M.T. McQuistan, A.J. Wolfe, An APN permutation in dimension six, in Finite Fields: Theory and Applications. Contemp. Math., vol. 518 (The American Mathematical Society, Providence, 2010), pp. 33–42

    Google Scholar 

  5. L. Carlitz, Primitive roots in a finite field. Trans. Am. Math. Soc. 73, 373–382 (1952)

    Article  MATH  Google Scholar 

  6. W. Chou, I. Shparlinski, On the cycle structure of repeated exponentiation modulo a prime. J. Number Theory 107, 345–356 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. S.D. Cohen, Primitive polynomials with a prescribed coefficient. Finite Fields Appl. 12, 425–491 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. H. Cohen, G. Frey, R. Avanzi, C. Doche, T. Lange, K. Nguyen, F. Vercauteren, Handbook of Elliptic and Hyperelliptic Curve Cryptography, Series on Discrete Mathematics and Its Applications (CRC Press, Boca Raton, 2006)

    Google Scholar 

  9. S.D. Cohen, M. Prešern, The Hansen–Mullen primitive conjecture: completion of proof, in Number Theory and Polynomials. London Math. Soc. Lecture Note Series, vol. 352 (Cambridge University Press, Cambridge, 2008), pp. 89–120

    Google Scholar 

  10. S.D. Cohen, S. Huczynska, The primitive normal basis theorem—without a computer. J. Lond. Math. Soc. 2nd Ser. 67, 41–56 (2003)

    Google Scholar 

  11. C.J. Colbourn, J.H. Dinitz, Handbook of Combinatorial Designs, 2nd edn., Series on Discrete Mathematics and Its Applications (CRC Press, Boca Raton, 2007)

    Google Scholar 

  12. D. Coppersmith, Fast evaluation of logarithms in fields of characteristic two. IEEE Trans. Inf. Theory 30, 587–594 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  13. H. Davenport, Bases for finite fields. J. Lond. Math. Soc. 2nd Ser. 43, 21–39 (1968)

    Google Scholar 

  14. P. Dembowski, T.G. Ostrom, Planes of order n with collineation groups of order n 2. Math. Z. 103, 239–258 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  15. G. Effinger, Toward a complete twin primes theorem for polynomials over finite fields, in Finite Fields and Applications. Contemp. Math., vol. 461 (The American Mathematical Society, Providence, 2008), pp. 103–110

    Google Scholar 

  16. G. Effinger, D.R. Hayes, Additive Number Theory of Polynomials over a Finite Field, Oxford Mathematical Monographs (Oxford University Press, New York, 1991)

    Google Scholar 

  17. G. Effinger, K. Hicks, G.L. Mullen, Integers and polynomials: comparing the close cousins \(\mathbb{Z}\) and \(\mathbb{F}_{q}[x]\). Math. Intelligencer 27, 26–34 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. S. Fan, X. Wang, Primitive normal polynomials with a prescribed coefficient. Finite Fields Appl. 15, 682–730 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  19. P. Flajolet, A.M. Odlyzko, Random mapping statistics, in Advances in cryptology—EUROCRYPT ’89. Lecture Notes in Comput. Sci., vol. 434 (Springer, New York, 1990), pp. 329–354

    Google Scholar 

  20. Ph. Flajolet, R. Sedgewick, Analytic Combinatorics (Cambridge University Press, Cambridge, 2009)

    Book  MATH  Google Scholar 

  21. P. Flajolet, X. Gourdon, D. Panario, The complete analysis of a polynomial factorization algorithm over finite fields. J. Algorithms 40, 37–81 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  22. K. Ford, The distribution of integers with a divisor in a given interval. Ann. Math. 168, 367–433 (2008)

    Article  MATH  Google Scholar 

  23. J. von zur Gathen, J. Gerhard, Modern Computer Algebra, 2nd edn. (Cambridge University Press, Cambridge/New York/Melbourne, 2003)

    Google Scholar 

  24. J. von zur Gathen, D. Panario, Factoring polynomials over finite fields: a survey. J. Symb. Comput. 31, 3–17 (2001)

    Google Scholar 

  25. J. von zur Gathen, D. Panario, B. Richmond, Interval partitions and polynomial factorization. Algorithmica 63, 363–397 (2012)

    Google Scholar 

  26. J. von zur Gathen, V. Shoup, Computing Frobenius maps and factoring polynomials. Comput. Complex. 2, 187–224 (1992)

    Google Scholar 

  27. S. Gao, J. Howell, D. Panario, Irreducible polynomials of given forms, in Finite Fields: Theory, Applications, and Algorithms. Contemp. Math., vol. 225 (The American Mathematical Society, Providence, 1999), pp. 43–54 s

    Google Scholar 

  28. T.A. Gassert, Chebyshev action on finite fields. arXiv:1209.4396v3

    Google Scholar 

  29. D.A. Goldston, J. Pintz, C.Y. Yldrm, Primes in tuples I. Ann. of Math. 170, 819–862 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  30. D. Hachenberger, Primitive complete normal bases for regular extensions. Glasgow Math. J. 43, 383–398 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  31. D. Hachenberger, Primitive complete normal bases: existence in certain 2-power extensions and lower bounds. Discrete Math. 310, 3246–3250 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  32. T. Hansen, G.L. Mullen, Primitive polynomials over finite fields. Math. Comput. 59, 639–643, S47–S50 (1992)

    Article  MathSciNet  Google Scholar 

  33. A.S. Hedayat, N.J.A. Sloane, J. Stufken, Orthogonal Arrays, Theory and Applications, Springer Series in Statistics (Springer, New York, 1999)

    Google Scholar 

  34. M. Herrmann, G. Leander, A practical key recovery attack on basic \(\mathcal{T}\mathcal{C}\mathcal{H}o\), in Public Key Cryptography—PKC 2009. Lecture Notes in Comput. Sci., vol. 5443 (Springer, New York, 2009), pp. 411–424

    Google Scholar 

  35. C.-N. Hsu, The distribution of irreducible polynomials in \(\mathbf{F}_{q}[t]\). J. Number Theory 61, 85–96 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  36. IEEE, Standard specifications for public key cryptography, Standard P1363-2000, Institute of Electrical and Electronics Engineering, 2000, Draft D13 available at http://grouper.ieee.org/groups/1363/P1363/draft.html

  37. G. Ivanyos, M. Karpinski, L. Rónyai, N. Saxena, Trading GRH for algebra: algorithms for factoring polynomials and related structures. Math. Comput. 81, 493–531 (2012)

    Article  MATH  Google Scholar 

  38. E. Jensen, M.R. Murty, Artin’s conjecture for polynomials over finite fields, in Number Theory, Trends in Mathematics (Birkhauser, Basel, 2000), pp. 167–181

    Google Scholar 

  39. K.S. Kedlaya, C. Umans, Fast modular composition in any characteristic, in 49th Annual IEEE Symposium on Foundations of Computer Science (2008), pp. 146–155

    Google Scholar 

  40. D. Koukoulopoulos, Divisors of shifted primes. Int. Math. Res. Not. IMRN 2010, 4585–4627 (2010)

    MATH  MathSciNet  Google Scholar 

  41. H.W. Lenstra, Jr., R.J. Schoof, Primitive normal bases for finite fields. Math. Comp. 48, 217–231 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  42. R. Lidl, G.L. Mullen, When does a polynomial over a finite field permute the elements of the field? Am. Math. Mon. 95, 243–246 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  43. R. Lidl, G.L. Mullen, When does a polynomial over a finite field permute the elements of the field? II. Am. Math. Mon. 100, 71–74 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  44. R. Lidl, H. Niederreiter, Finite Fields, vol. 20, 2nd edn., Encyclopedia of Mathematics and its Applications (Cambridge University Press, Cambridge, 1997)

    Google Scholar 

  45. A. MacFie, D. Panario, Random mappings with restricted preimages, in Proceedings of LatinCrypt 2012. Lecture Notes in Comput. Sci., vol.7533 (Springer, Berlin, 2012), pp. 254–270

    Google Scholar 

  46. C. Mauduit, J. Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers. Ann. Math. 171, 1591–1646 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  47. J. Maynard, Small gaps between primes. arXiv:1311.4600 (2013)

    Google Scholar 

  48. A.J. Menezes, P.C. van Oorschot, S.A. Vanstone, Handbook of Applied Cryptography, Series on Discrete Mathematics and its Applications (CRC Press, Boca Raton, 1997)

    Google Scholar 

  49. I.H. Morgan, G.L. Mullen, Completely normal primitive basis generators of finite fields. Utilitas Math. 49, 21–43 (1996)

    MATH  MathSciNet  Google Scholar 

  50. G.L. Mullen, D. Panario, Handbook of Finite Fields, Series on Discrete Mathematics and Its Applications (CRC Press, Boca Raton, 2013)

    Google Scholar 

  51. A. Munemasa, Orthogonal arrays, primitive trinomials, and shift-register sequences. Finite Fields Appl. 4, 252–260 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  52. D. Panario, What do random polynomials over finite fields look like? in Finite Fields and Applications. Lecture Notes in Comput. Sci., vol. 2948 (Springer, Berlin, 2004), pp. 89–108

    Google Scholar 

  53. V.S. Pless, W.C. Huffman, R.A. Brualdi, Handbook of Coding Theory (North-Holland, Amsterdam, 1998)

    MATH  Google Scholar 

  54. P. Pollack, A polynomial analogue of the twin primes conjecture. Proc. Am. Math. Soc. 136, 3775–3784 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  55. J.M. Pollard, A Monte Carlo method for factorization, Nordisk Tidskr. Informationsbehandling (BIT) 15, 331–334 (1975)

    MATH  MathSciNet  Google Scholar 

  56. M. Scott, Optimal irreducible polynomials for GF(2m) arithmetic, in Software Performance Enhancement for Encryption and Decryption (SPEED 2007), Cryptology ePrint Archive (2007)

    Google Scholar 

  57. M. Sha, S. Hu, Monomial dynamical systems of dimension over finite fields. Acta Arith. 148, 309–331 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  58. R.G. Swan, Factorization of polynomials over finite fields. Pac. J. Math. 12, 1099–1106 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  59. S. Ugolini, Graphs associated with the map \(X\mapsto X + X^{-1}\) in finite fields of characteristic three and five. J. Number Theory 133, 1207–1228 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  60. T. Vasiga, J. Shallit, On the iteration of certain quadratic maps over GF(p). Discrete Math. 277, 219–240 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  61. D. Wan, Generators and irreducible polynomials over finite fields. Math. Comput. 66, 1195–1212 (1997)

    Article  MATH  Google Scholar 

  62. A. Weil, Sur les Courbes Algébriques et les Variétés qui s’en dÉduisent, Actualités Sci. Ind., no. 1041; Publ. Inst. Math. Univ. Strasbourg, vol. 7 (Hermann et Cie., Paris, 1945/1948)

    Google Scholar 

  63. Y. Zhang, Bounded gaps between primes. Ann. Math. 170, 1121–1174 (2014)

    Article  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Carleton University, Ottawa, ON, Canada, K1S 5B6

    Daniel Panario

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  1. Daniel Panario
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Correspondence to Daniel Panario .

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  1. Dept. of Computer Science, University of California, Santa Barbara, Santa Barbara, California, USA

    Çetin Kaya Koç

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Panario, D. (2014). Open Problems for Polynomials over Finite Fields and Applications. In: Koç, Ç. (eds) Open Problems in Mathematics and Computational Science. Springer, Cham. https://doi.org/10.1007/978-3-319-10683-0_6

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  • DOI: https://doi.org/10.1007/978-3-319-10683-0_6

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