Abstract
Let \(\gamma\) be a generator of a cyclic group G of order n. The least index of a self-mapping f of G is the index of the largest subgroup U of G such that \(f(x)x^{-r}\) is constant on each coset of U for some positive integer r. We determine the index of the univariate Diffie–Hellman mapping \(d(\gamma ^a)=\gamma ^{a^2}\), \(a=0,1,\ldots ,n-1\), and show that any mapping of small index coincides with d only on a small subset of G. Moreover, we prove similar results for the bivariate Diffie–Hellman mapping \(D(\gamma ^a,\gamma ^b)=\gamma ^{ab}\), \(a,b=0,1,\ldots ,n-1\). In the special case that G is a subgroup of the multiplicative group of a finite field we present improvements.
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References
Akbary, A., Ghioca, D., Wang, Q.: On permutation polynomials of prescribed shape. Finite Fields Appl. 15(2), 195–206 (2009)
Bach, E., Shallit, J.: Algorithmic Number Theory. MIT Press, Cambridge (1996)
Blake, I.F., Garefalakis, T.: On the complexity of the discrete logarithm and Diffie–Hellman problems. J. Complex. 20(2–3), 148–170 (2004)
Blake, I.F., Garefalakis, T.: Polynomial approximation of bilinear Diffie–Hellman maps. Finite Fields Appl. 14(2), 379–389 (2008)
Coppersmith, D., Shparlinski, I.: On polynomial approximation of the discrete logarithm and the Diffie–Hellman mapping. J. Cryptol. 13(3), 339–360 (2000)
El Mahassni, E., Shparlinski, I.: Polynomial representations of the Diffie–Hellman mapping. Bull. Aust. Math. Soc. 63(3), 467–473 (2001)
Işık, L., Winterhof, A.: Carlitz rank and index of permutation polynomials. Finite Fields Appl. 49, 156–165 (2018)
Kiltz, E., Winterhof, A.: On the interpolation of bivariate polynomials related to the Diffie–Hellman mapping. Bull. Aust. Math. Soc. 69(2), 305–315 (2004)
Kiltz, E., Winterhof, A.: Polynomial interpolation of cryptographic functions related to Diffie–Hellman and discrete logarithm problem. Discr. Appl. Math. 154(2), 326–336 (2006)
Konjagin, V.S.: The number of solutions of congruences of the nth degree with one unknown. Mat. Sb. (N.S.) 109(151)(2), 171–187, 327 (1979). (Russian)
Lange, T., Winterhof, A.: Polynomial interpolation of the elliptic curve and XTR discrete logarithm. In: Computing and combinatorics, Lecture Notes in Computer Science, vol. 2387, pp. 137–143. Springer, Berlin (2002)
Lange, T., Winterhof, A.: Interpolation of the elliptic curve Diffie–Hellman mapping. In: Applied algebra, algebraic algorithms and error-correcting codes (Toulouse, 2003), Lecture Notes in Computer Science, vol. 2643, pp. 51–60. Springer, Berlin (2003)
Mefenza, T., Vergnaud, D.: Polynomial interpolation of the generalized Diffie–Hellman and Naor–Reingold functions. Des. Codes Cryptogr. 87(1), 75–85 (2019)
Meidl, W., Winterhof, A.: A polynomial representation of the Diffie–Hellman mapping. Appl. Alg. Eng. Commun. Comput. 13, 313–318 (2002)
Mullen, G.L., Wan, D., Wang, Q.: Index Bounds for Value Sets of Polynomials Over Finite Fields, Applied Algebra and Number Theory, pp. 280–296. Cambridge University Press, Cambridge (2014)
Niederreiter, H., Winterhof, A.: Cyclotomic \({\cal{R}}\)-orthomorphisms of finite fields. Discr. Math. 295, 161–171 (2005)
Niederreiter, H., Winterhof, A.: Applied Number Theory. Springer, Cham (2015)
Shparlinski, I.: Cryptographic Applications of Analytic Number Theory. Complexity Lower Bounds and Pseudorandomness, Progr. Comput. Sc. Appl. Logic, vol. 22. Birkhäuser Verlag, Basel (2003)
Wang, Q.: Cyclotomic mapping permutation polynomials over finite fields. In: Sequences, Subsequences, and Consequences (International Workshop, SSC 2007, Los Angeles, CA, USA, May 31–June 2, 2007), Lecture Notes in Computer Sciene, vol. 4893, pp. 119–128. Springer, Berlin (2007)
Wang, Q.: Polynomials over finite fields: an index approach. In: Combinatorics and Finite Fields: Difference Sets, Polynomials, Pseudorandomness and Applications, Radon Series. Comput. Appl. Math. vol. 23, pp. 319–348. de Gruyter, Berlin/Boston (2019)
Winterhof, A.: A note on the interpolation of the Diffie–Hellman mapping. Bull. Math. Soc. 64, 475–477 (2001)
Acknowledgements
The second author is partially supported by the Austrian Science Fund FWF Project P 30405-N32. Parts of this paper were written during a visit of the first author to RICAM. She would like to express her sincere thanks for the hospitality during her visit. The authors would like to thank Steven Wang for useful discussions.
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Işık, L., Winterhof, A. On the index of the Diffie–Hellman mapping. AAECC 33, 587–595 (2022). https://doi.org/10.1007/s00200-020-00475-3
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DOI: https://doi.org/10.1007/s00200-020-00475-3