Constructing Permutation Rational Functions from Isogenies

A permutation rational function $f\in\mathbb{F}\/_q(x)$ is a rational function that induces a bijection on $\mathbb{F}\/_q$, that is, for all $y\in\mathbb{F}\/_q$ there exists exactly one $x\in\mathbb{F}\/_q$ such that $f(x)=y$. Permutation rational functions are intimately related to exceptional rational functions, and, more generally, exceptional covers of the projective line, of which they form the first important example. In this paper, we show how to efficiently generate many permutation rational functions over large finite fields using isogenies of elliptic curves, and discuss some cryptographic applications. Our algorithm is based on Fried's modular interpretation of certain dihedral exceptional covers of the projective line [Finite Fields: Theory, Applications, and Algorithms, Contemp. Math. 168, 1994, pp. 69--100].

  • 1.  T. M. Apostol , Introduction to Analytic Number Theory , Springer , New York , 1976 . Google Scholar

  • 2.  D. Boneh , Twenty years of attacks on the RSA cryptosystem , Notices Amer. Math. Soc. , 46 ( 1999 ), pp. 203 -- 213 . Google Scholar

  • 3.  R. Bröker K. Lauter and  A. V. Sutherland , Modular polynomials via isogeny volcanoes , Math. Comp. , 81 ( 2012 ), pp. 1201 -- 1231 , https://doi.org/10.1090/S0025-5718-2011-02508-1. CrossrefISIGoogle Scholar

  • 4.  J.-M. Couveignes and  R. Lercier , The geometry of some parameterizations and encodings , Adv. Math. Commun. , 8 ( 2014 ), pp. 437 -- 458 . CrossrefISIGoogle Scholar

  • 5.  P.-A. Fouque and  M. Tibouchi , Deterministic encoding and hashing to odd hyperelliptic curves , in Pairing-Based Cryptography , Lecture Notes in Comput. Sci. 6487 , Springer , Berlin , 2010 , pp. 265 -- 277 . Google Scholar

  • 6.  M. Fouquet and  F. Morain , Isogeny volcanoes and the SEA algorithm , in Algorithmic Number Theory, C. Fieker and D. R. Kohel, eds., Lecture Notes in Comput. Sci. 2369 , Springer , 2002 , pp. 276 -- 291 , https://doi.org/10.1007/3-540-45455-1_23. CrossrefGoogle Scholar

  • 7.  M. D. Fried , Global construction of general exceptional covers, in Finite Fields: Theory, Applications, and Algorithms, Contemp. Math. 168, G. L. Mullen and P. J. Shiue, eds ., AMS , Providence, RI , 1994 , pp. 69 -- 100 . Google Scholar

  • 8.  M. D. Fried , The place of exceptional covers among all Diophantine relations , Finite Fields Appl. , 11 ( 2005 ), pp. 367 -- 433 . CrossrefISIGoogle Scholar

  • 9.  R. M. Guralnick P. Müller and  J. Saxl , The Rational Function Analogue of a Question of Schur and Exceptionality of Permutation Representations, Mem. Amer. Math. Soc. 773 , AMS , Providence, RI , 2003 . Google Scholar

  • 10.  R. M. Guralnick T. J. Tucker and  M. E. Zieve , Exceptional covers and bijections on rational points , Int. Math. Res. Not. IMRN , 2007 ( 2007 ), m m004 . ISIGoogle Scholar

  • 11.  J.-G. Kammerer R. Lercier and  G. Renault , Encoding points on hyperelliptic curves over finite fields in deterministic polynomial time , in Pairing-Based Cryptography , Lecture Notes in Comput. Sci. 6487 , Springer , Berlin , 2010 , pp. 278 -- 297 . Google Scholar

  • 12.  D. R. Kohel , Endomorphism Rings of Elliptic Curves over Finite Fields , Ph.D. thesis, University of California at Berkeley, Berkeley, CA, 1996 , http://iml.univ-mrs.fr/~kohel/pub/thesis.pdf. , http://iml.univ-mrs.fr/~kohel/pub/thesis.pdf. Google Scholar

  • 13.  R. T. Moenck and  A. B. Borodin , Fast modular transforms via division , in IEEE 13th Annual Symposium on Switching and Automata Theory, IEEE Press , New York , 1972 , pp. 90 -- 96 . Google Scholar

  • 14.  The PARI Group , PARI/GP , http://pari.math.u-bordeaux.fr/ ( 2016 ). , http://pari.math.u-bordeaux.fr/. Google Scholar

  • 15.  M. Tibouchi , Hachage vers les courbes elliptiques et cryptanalyse de schémas RSA , Ph.D. thesis, University of Paris 7 and University of Luxembourg , Paris and Luxembourg, 2011 . Google Scholar

  • 16.  M. Tibouchi , Indifferentiable deterministic hashing to elliptic and hyperelliptic curves , in ECC 2013 , IEEE, Piscataway, NJ , 2013 . Google Scholar

  • 17.  M. Tibouchi , Impossibility of surjective Icart-like encodings , in ProvSec 2014 , S. S. M. Chow, J. K. Liu, L. C. K. Hui, and S. Yiu, eds., Lecture Notes in Comput . Sci. 8782, Springer , Cham, Switzerland, 2014, pp. 29 -- 39 , https://doi.org/10.1007/978-3-319-12475-9_3. Google Scholar

  • 18.  H. Weber , Elliptische Funktionen und Algebraische Zahlen , Lehrbuch der Algebra 3, Friedrich Vieweg und Sohn , Braunschweig , Germany , 1891 . Google Scholar