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Gachkov, Igor

Open this publication in new window or tab >>The Complexity of Solving Low Degree Equations over Ring of Integers and Residue Rings### Gashkov, S. B.

### Gachkov, Igor

### Frolov, A. B.

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_some",{id:"formSmash:j_idt184:0:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_otherAuthors",{id:"formSmash:j_idt184:0:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: MOSCOW UNIVERSITY MATHEMATICS BULLETIN, ISSN 0027-1322, Vol. 74, no 1, p. 5-13Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Cham: Springer, 2019
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:kau:diva-72005 (URN)10.3103/S0027132219010029 (DOI)000465628800002 ()
#####

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Available from: 2019-05-09 Created: 2019-05-09 Last updated: 2019-05-10Bibliographically approved

Moscow State University, Russia.

Karlstad University, Faculty of Technology and Science, Department of Mathematics.

National Research University, Russia.

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.

Open this publication in new window or tab >>Fast Algorithm of Square Rooting in Some Finite Fields of Odd Characteristic### Gashkov, S. B.

### Gashkov, Igor

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_some",{id:"formSmash:j_idt184:1:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_otherAuthors",{id:"formSmash:j_idt184:1:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_otherAuthors",multiple:true}); 2018 (English)In: Moscow University Mathematics Bulletin, ISSN 0027-1322, Vol. 73, no 5, p. 176-181Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Cham: Springer, 2018
##### National Category

Geometry
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:kau:diva-70427 (URN)10.3103/S0027132218050029 (DOI)000450666000002 ()
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_j_idt359",{id:"formSmash:j_idt184:1:j_idt188:j_idt359",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_j_idt359",multiple:true});
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Available from: 2018-12-06 Created: 2018-12-06 Last updated: 2018-12-13Bibliographically approved

Moscow State University, Russia.

Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).

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).

Open this publication in new window or tab >>Bit parallel circuits for arithmetic operations in composite fields $ GF(2^{nm}) (CMMSE)### Gachkov, Igor

### Burtsev, A.A

### Khokhlov, R.A

### Gashkov, I

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_some",{id:"formSmash:j_idt184:2:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_otherAuthors",{id:"formSmash:j_idt184:2:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_otherAuthors",multiple:true}); 2010 (English)Conference paper, Published paper (Refereed)
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:kau:diva-10545 (URN)9788461355105 (ISBN)
#####

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Available from: 2012-02-08 Created: 2012-02-08 Last updated: 2013-06-12Bibliographically approved

Karlstad University, Faculty of Technology and Science, Department of Mathematics.

Open this publication in new window or tab >>Some Remarks on Testing Irreducibility of Polynomials and Normality of Bases in Finite Fields### Gachkov, Igor

### Gashkov, S.B

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_some",{id:"formSmash:j_idt184:3:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_otherAuthors",{id:"formSmash:j_idt184:3:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_otherAuthors",multiple:true}); 2010 (English)In: Fundamenta Informaticae, ISSN 0169-2968, E-ISSN 1875-8681, Vol. 104, no 3, p. 227-238Article in journal (Refereed) Published
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:kau:diva-10554 (URN)10.3233/FI-2010-346 (DOI)000285459900005 ()
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_j_idt359",{id:"formSmash:j_idt184:3:j_idt188:j_idt359",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_j_idt359",multiple:true});
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Available from: 2012-02-08 Created: 2012-02-08 Last updated: 2017-12-07Bibliographically approved

Karlstad University, Faculty of Technology and Science, Department of Mathematics.

Open this publication in new window or tab >>Probabilistic algorithm to find a normal basis in special finite fields### Gachkov, Igor

### Gashkov, S.B

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_some",{id:"formSmash:j_idt184:4:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_otherAuthors",{id:"formSmash:j_idt184:4:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_otherAuthors",multiple:true}); 2009 (English)Conference paper, Published paper (Refereed)
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:kau:diva-9865 (URN)9788461297276 (ISBN)
##### Conference

Proceedings of the International congerence of computational and mathematical Methods in Science and Engineering , CMMSE 2009 30 June, 1-3 July 2009.
#####

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Available from: 2012-02-08 Created: 2012-02-08 Last updated: 2013-06-12Bibliographically approved

Karlstad University, Faculty of Technology and Science, Department of Mathematics.

Open this publication in new window or tab >>Using the package Coding Theory for a search technique for Quasi-perfect Codes### Gachkov, Igor

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_some",{id:"formSmash:j_idt184:5:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_otherAuthors",{id:"formSmash:j_idt184:5:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_otherAuthors",multiple:true}); 2008 (English)Conference paper, Published paper (Refereed)
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:kau:diva-25300 (URN)
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_j_idt359",{id:"formSmash:j_idt184:5:j_idt188:j_idt359",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_j_idt359",multiple:true});
#####

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Available from: 2013-01-22 Created: 2013-01-22 Last updated: 2013-01-22

Karlstad University, Faculty of Technology and Science, Department of Mathematics.

Open this publication in new window or tab >>A geometric approach to finding new lower bounds of A ( n , d , w )### Gachkov, Igor

### Ekberg, A.O.

### Taub, D.

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_some",{id:"formSmash:j_idt184:6:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_otherAuthors",{id:"formSmash:j_idt184:6:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_otherAuthors",multiple:true}); 2007 (English)In: Designs, Codes and Cryptography v. 43, N 2-3 / June, 2007 pp. 85-91Article in journal (Refereed)
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:kau:diva-16549 (URN)
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_j_idt359",{id:"formSmash:j_idt184:6:j_idt188:j_idt359",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_j_idt359",multiple:true});
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Available from: 2013-01-21 Created: 2013-01-21 Last updated: 2013-01-21

Karlstad University, Faculty of Technology and Science, Department of Mathematics.

Open this publication in new window or tab >>Improvements on the Juxtaposing theorem### Gachkov, Igor

### Larsson, H

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_some",{id:"formSmash:j_idt184:7:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_otherAuthors",{id:"formSmash:j_idt184:7:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_otherAuthors",multiple:true}); 2007 (English)In: Serdica J. Computing 1 Volume 1, Number 2, pp. 207-212Article in journal (Refereed)
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:kau:diva-20236 (URN)
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_j_idt359",{id:"formSmash:j_idt184:7:j_idt188:j_idt359",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_j_idt359",multiple:true});
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Available from: 2013-01-21 Created: 2013-01-21 Last updated: 2013-01-21

Karlstad University, Faculty of Technology and Science, Department of Mathematics.

Open this publication in new window or tab >>New Optimal Constant Weight Codes### Gachkov, Igor

### Taub, D

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_some",{id:"formSmash:j_idt184:8:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_otherAuthors",{id:"formSmash:j_idt184:8:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_otherAuthors",multiple:true}); 2007 (English)In: The electronic journal of combinatorics 14 , no. 1, Note 13, pp.1-6Article in journal (Refereed)
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:kau:diva-21781 (URN)
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_j_idt359",{id:"formSmash:j_idt184:8:j_idt188:j_idt359",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_j_idt359",multiple:true});
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Available from: 2013-01-21 Created: 2013-01-21 Last updated: 2013-01-21

Karlstad University, Faculty of Technology and Science, Department of Mathematics.

Open this publication in new window or tab >>Visualisation of the Mathematical Process: Boolean Algebra and Graph Theory with TI-83/89### Gachkov, Igor

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_some",{id:"formSmash:j_idt184:9:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_otherAuthors",{id:"formSmash:j_idt184:9:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_otherAuthors",multiple:true}); 2007 (English)In: Journal of the Korea Society of Mathematical Education Vol.11, No.2, pp. 143-151Article in journal (Refereed)
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:kau:diva-25569 (URN)
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_j_idt359",{id:"formSmash:j_idt184:9:j_idt188:j_idt359",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_j_idt359",multiple:true});
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Available from: 2013-01-22 Created: 2013-01-22 Last updated: 2013-01-22

Karlstad University, Faculty of Technology and Science, Department of Mathematics.