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  • 1. Burtsev, A.
    et al.
    Gashkov, S.
    Gachkov, Igor
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    On complexity Boolean circuits for arithmetics operations in some composite binary fields2006In: Vestnik, Mosk. Universitet. Ser1. Math. N 5. pp.10-16Article in journal (Refereed)
  • 2.
    Gachkov, Igor
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Error Correcting codes with Mathematica2003In: Lecture note in Computer science LNCS 2657Article in journal (Refereed)
  • 3.
    Gachkov, Igor
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Non binary codes and Mathematica calculations: Reed-Solomon Codes over GF( 2n )2006In: Lecture notes in Computer science LNCS 3516 pp.663-666Article in journal (Refereed)
  • 4.
    Gachkov, Igor
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Optimal Constant Weight Codes2006In: Lecture note in Computer science LNCS 3991 pp. 912 915Article in journal (Refereed)
  • 5.
    Gachkov, Igor
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Using the package Coding Theory for a search technique for Quasi-perfect Codes2008Conference paper (Refereed)
  • 6.
    Gachkov, Igor
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Visualisation of the Mathematical Process: Boolean Algebra and Graph Theory with TI-83/892007In: Journal of the Korea Society of Mathematical Education Vol.11, No.2, pp. 143-151Article in journal (Refereed)
  • 7.
    Gachkov, Igor
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    , Kazarin
    , Sidelnikov
    Relative duality in MacWilliams Identity2003In: Lecture note in Computer science LNCS 2643Article in journal (Refereed)
  • 8.
    Gachkov, Igor
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Burtsev, A.A
    Khokhlov, R.A
    Gashkov, I
    Bit parallel circuits for arithmetic operations in composite fields $ GF(2^{nm}) (CMMSE)2010Conference paper (Refereed)
  • 9.
    Gachkov, Igor
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Ekberg, A.O.
    Taub, D.
    A geometric approach to finding new lower bounds of A ( n , d , w )2007In: Designs, Codes and Cryptography v. 43, N 2-3 / June, 2007 pp. 85-91Article in journal (Refereed)
  • 10.
    Gachkov, Igor
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Gashkov, S.B
    Probabilistic algorithm to find a normal basis in special finite fields2009Conference paper (Refereed)
  • 11.
    Gachkov, Igor
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Gashkov, S.B
    Some Remarks on Testing Irreducibility of Polynomials and Normality of Bases in Finite Fields2010In: Fundamenta Informaticae, ISSN 0169-2968, E-ISSN 1875-8681, Vol. 104, no 3, p. 227-238Article in journal (Refereed)
  • 12.
    Gachkov, Igor
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Larsson, H
    Improvements on the Juxtaposing theorem2007In: Serdica J. Computing 1 Volume 1, Number 2, pp. 207-212Article in journal (Refereed)
  • 13.
    Gachkov, Igor
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Taub, D
    New Optimal Constant Weight Codes2007In: The electronic journal of combinatorics 14 , no. 1, Note 13, pp.1-6Article in journal (Refereed)
  • 14.
    Gashkov, S. B.
    et al.
    Moscow State University, Russia.
    Gachkov, Igor
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Frolov, A. B.
    National Research University, Russia.
    The Complexity of Solving Low Degree Equations over Ring of Integers and Residue Rings2019In: MOSCOW UNIVERSITY MATHEMATICS BULLETIN, ISSN 0027-1322, Vol. 74, no 1, p. 5-13Article in journal (Refereed)
    Abstract [en]

    It is proved that for an arbitrary polynomial f(x)Zpn[X] of degree d the Boolean complexity of calculation of one its root (if it exists) equals O(dM(n(p))) for a fixed prime p and growing n, where (p) = remvoelog(2)p, and M(n) is the Boolean complexity of multiplication of two binary n-bit numbers. Given the known decomposition of this number into prime factors n = m(1)...m(k), mi=pini, i = 1,..., k, with fixed k and primes p(i), i = 1,..., k, and growing n, the Boolean complexity of calculation of one of solutions to the comparison f(x) = 0 mod n equals O(dM((n))). In particular, the same estimate is obtained for calculation of one root of any given degree in the residue ring Z(m). As a corollary, it is proved that the Boolean complexity of calculation of integer roots of a polynomial f(x) is equal to O-d(M(n)), where f(x)=adxd+ad-1xd-1+...+a0,aiZ , |a(i)| < 2(n), i = 0,..., d.

  • 15.
    Gashkov, S. B.
    et al.
    Moscow State University, Russia.
    Gashkov, Igor
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).
    Fast Algorithm of Square Rooting in Some Finite Fields of Odd Characteristic2018In: Moscow University Mathematics Bulletin, ISSN 0027-1322, Vol. 73, no 5, p. 176-181Article in journal (Refereed)
    Abstract [en]

    It was proved that the complexity of square root computation in the Galois field GF(3 (s) ), s = 2 (k) r, is equal to O(M(2 (k) )M(r)k + M(r) log(2) r) + 2 (k) kr (1+o(1)), where M (n) is the complexity of multiplication of polynomials of degree n over fields of characteristics 3. The complexity of multiplication and division in the field GF(3 (s) ) is equal to O(M(2 (k) )M(r)) and O(M(2 (k) )M(r)) + r (1+o(1)), respectively. If the basis in the field GF(3 (r) ) is determined by an irreducible binomial over GF(3) or is an optimal normal basis, then the summands 2 (k) kr (1+o(1)) and r (1+o(1)) can be omitted. For M(n) one may take n log(2) n psi(n) where psi(n) grows slower than any iteration of the logarithm. If k grow and r is fixed, than all the estimates presented here have the form O (r) (M (s) log (2) s) = s (log (2) s)(2) psi(s).

  • 16. Gashkov, S.
    et al.
    Gachkov, Igor
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    BerlekampMassey Algorithm, Continued Fractions, Pade Approximations, and Orthogonal Polynomials2006In: Mathematical Notes, vol. 79, no. 1, 2006, pp. 4154Article in journal (Refereed)
  • 17. Gashkov, S.
    et al.
    Gachkov, Igor
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    On the complexity of calculation of differentials and gradients2006In: Discrete Mathematics and Applications Volume 15, No. 4, pp. 327--440Article in journal (Refereed)
  • 18. Gashkov, S
    et al.
    Gachkov, Igor
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Remark on testing Irreducibility of polynomials over Finite Fields2005In: Lecture Series on Computer and Computational Sciences Volume 4 2005 pp.204-206Article in journal (Refereed)
1 - 18 of 18
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