Solving systems of diagonal polynomial equations over finite fields

  title={Solving systems of diagonal polynomial equations over finite fields},
  author={G{\'a}bor Ivanyos and Miklos Santha},
  journal={Theor. Comput. Sci.},
3 Citations

Zero sum subsequences and hidden subgroups

We propose a method for solving the hidden subgroup problem in nilpotent groups. The main idea is iteratively transforming the hidden subgroup to its images in the quotient groups by the members of a

Transformation Method for Solving System of Boolean Algebraic Equations

A new transformation method for Solving Systems of Boolean Algebraic Equations (SBAE) written with logical operations are transformed (approximated) in a system of harmonic-polynomial equations in the unit n-dimensional cube Kn with the usual operations of addition and multiplication of numbers.

Fast Quantum Algorithm for Solving Multivariate Quadratic Equations

This paper considers the quantum security of the problem of solving a system of {\it $m$ Boolean multivariate quadratic equations in $n$ variables} (\MQb); a central problem in post-quantum cryptography and presents a Las-Vegas quantum algorithm solving this problem that is the fastest algorithm for solving it.

On Solving Systems of Diagonal Polynomial Equations Over Finite Fields

This work designs polynomial time quantum algorithms for two algebraic hidden structure problems: for the hidden subgroup problem in certain semidirect product \(p\)-groups of constant nilpotency class, and for the multi-dimensional univariate hidden polynometric graph problem when the degree of the polynomials is constant.

Efficient quantum algorithm for identifying hidden polynomials

A aquantum algorithm is presented that correctly identifies hidden m-variate polynomials for all but a finitenumber of values of d with constant probability and that has a running time that is onlypolylogarithmic in d.

Polynomial time quantum algorithms for certain bivariate hidden polynomial problems

The new approach yields an efficient quantum algorithm for the bivariate HPGP even when the input consists of several level set superpositions, a more difficult version of the problem than the one where the input is given by an oracle.

Deterministic equation solving over finite fields

It is shown how to compute a field generator that is an nth power, and how to write elements as sums of nth powers, for a given integer n, which takes polynomial time in n and in the logarithm of the field size.

The Hidden Subgroup Problem and Quantum Computation Using Group Representations

A natural generalization of the algorithm for the abelian case to the nonabelian case is analyzed and it is shown that the algorithm determines the normal core of a hidden subgroup: in particular, normal subgroups can be determined.

On taking roots in finite fields

The main result is shown that finding the least x such that x2 = a MOD(m) is NP-complete (even if m is factored).

Normal subgroup reconstruction and quantum computation using group representations

It is shown that an immediate generalization of the Abelian case solution to the non-Abelian case does not efficiently solve Graph Isomorphism.

Quantum measurements and the Abelian Stabilizer Problem

  • A. Kitaev
  • Mathematics
    Electron. Colloquium Comput. Complex.
  • 1996
A polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm is presented, based on a procedure for measuring an eigenvalue of a unitary operator.

Quantum measurements and the Abelian Stabilizer Problem

We present a polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm. Thus we extend famous Shor’s results [7]. Our method is based on

An Efficient Quantum Algorithm for the Hidden Subgroup Problem in Nil-2 Groups

It is shown that the hidden subgroup problem in nil-2 groups, that is in groups of nilpotency class at most 2, can be solved efficiently by a quantum procedure, and the existence of a solution is guaranteed by the Chevalley-Warning theorem.