Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Solving the "Isomorphism of Polynomials with Two Secrets" Problem for
  all Pairs of Quadratic Forms   We study the Isomorphism of Polynomial (IP2S) problem with m=2 homogeneous
quadratic polynomials of n variables over a finite field of odd characteristic:
given two quadratic polynomials (a, b) on n variables, we find two bijective
linear maps (s,t) such that b=t . a . s. We give an algorithm computing s and t
in time complexity O~(n^4) for all instances, and O~(n^3) in a dominant set of
instances.
  The IP2S problem was introduced in cryptography by Patarin back in 1996. The
special case of this problem when t is the identity is called the isomorphism
with one secret (IP1S) problem. Generic algebraic equation solvers (for example
using Gr\"obner bases) solve quite well random instances of the IP1S problem.
For the particular cyclic instances of IP1S, a cubic-time algorithm was later
given and explained in terms of pencils of quadratic forms over all finite
fields; in particular, the cyclic IP1S problem in odd characteristic reduces to
the computation of the square root of a matrix.
  We give here an algorithm solving all cases of the IP1S problem in odd
characteristic using two new tools, the Kronecker form for a singular quadratic
pencil, and the reduction of bilinear forms over a non-commutative algebra.
Finally, we show that the second secret in the IP2S problem may be recovered in
cubic time.

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The categories of this paper are: q-fin
29
The categories of this paper are: funct-an
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The categories of this paper are: solv-int
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The categories of this paper are: hep-lat
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The categories of this paper are: mtrl-th
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The categories of this paper are: alg-geom
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The categories of this paper are: bayes-an
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The categories of this paper are: cond-mat
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The categories of this paper are: econ
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The categories of this paper are: stat
0
The categories of this paper are: hep-ph
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The categories of this paper are: physics
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The categories of this paper are: acc-phys
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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-th
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The categories of this paper are: dg-ga