Products with variables from low-dimensional affine spaces and shifted power identity testing in finite fields

Abstract Families of stable cyclic groups of nonlinear polynomial transformations of affine spaces Kn over general commutative ring K of with n increasing order can be used in the key exchange protocols and El Gamal multivariate cryptosystems related to them. We suggest to use high degree of noncommutativity of affine Cremona group and modify multivariate El Gamal algorithm via conjugations of two polynomials of kind gk and g−1 given by key holder (Alice) or giving them as elements of different transformation groups. Recent results on the existence of families of stable transformations of prescribed degree and density and exponential order over finite fields can be used for the implementation of schemes as above with feasible computational complexity.

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Integral automorphisms of affine spaces over finite fields

Designs Codes and Cryptography ◽

10.1007/s10623-016-0246-z ◽

2016 ◽

Vol 84 (1-2) ◽

pp. 181-188

Author(s):

István Kovács ◽

Klavdija Kutnar ◽

János Ruff ◽

Tamás Szőnyi

Keyword(s):

Finite Fields ◽

Affine Spaces

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Integral point sets in higher dimensional affine spaces over finite fields

Journal of Combinatorial Theory Series A ◽

10.1016/j.jcta.2009.03.001 ◽

2009 ◽

Vol 116 (6) ◽

pp. 1120-1139 ◽

Cited By ~ 2

Author(s):

Sascha Kurz ◽

Harald Meyer

Keyword(s):

Finite Fields ◽

Integral Point ◽

Point Sets ◽

Affine Spaces ◽

Higher Dimensional

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Values of rational functions in small subgroups of finite fields and the identity testing problem from powers

International Journal of Number Theory ◽

10.1142/s1793042120500128 ◽

2019 ◽

Vol 16 (02) ◽

pp. 219-231

Author(s):

László Mérai

Keyword(s):

Finite Field ◽

Rational Functions ◽

Algorithmic Problem ◽

Identity Testing ◽

Affine Subspaces ◽

Algorithmic Problems ◽

Low Dimensional ◽

High Degree ◽

Multiplicative Groups ◽

Oracle Access

Motivated by some algorithmic problems, we give lower bounds on the size of the multiplicative groups containing rational function images of low-dimensional affine subspaces of a finite field [Formula: see text] considered as a linear space over a subfield [Formula: see text]. We apply this to the recently introduced algorithmic problem of identity testing of “hidden” polynomials [Formula: see text] and [Formula: see text] over a high degree extension of a finite field, given oracle access to [Formula: see text] and [Formula: see text].

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Non-Hurwitz Classical Groups

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157000001303 ◽

2007 ◽

Vol 10 ◽

pp. 21-82 ◽

Cited By ~ 6

Author(s):

R. Vincent ◽

A.E. Zalesski

Keyword(s):

Finite Fields ◽

Unitary Groups ◽

Classical Groups ◽

Special Linear ◽

Low Dimensional

AbstractIn previous work by Di Martino, Tamburini and Zalesski [Comm. Algebra28 (2000) 5383–5404] it is shown that certain low-dimensional classical groups over finite fields are not Hurwitz. In this paper the list is extended by adding the special linear and special unitary groups in dimensions 8.9,11.13. We also show that all groups Sp(n, q) are not Hurwitz forqeven andn= 6,8,12,16. In the range 11 <n< 32 many of these groups are shown to be non-Hurwitz. In addition, we observe that PSp(6, 3),PΩ±(8, 3k),PΩ±10k), Ω(11,3k), Ω±(14,3k), Ω±(16,7k), Ω(n, 7k) forn= 9,11,13, PSp(8, 7k) are not Hurwitz.

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