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NOTES ON THE GENERALIZED GAUSS REDUCTION ALGORITHM.

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Titel: NOTES ON THE GENERALIZED GAUSS REDUCTION ALGORITHM.
Autoren: Baissalov, Y., Nauryzbayev, R.
Quelle: Eurasian Mathematical Journal; 2025, Vol. 16 Issue 2, p23-29, 7p
Schlagwörter: CRYPTOGRAPHY, LATTICE theory, QUANTUM computing, GAUSSIAN elimination, NP-hard problems, ALGORITHMS
Abstract: The hypothetical possibility of building a quantum computer in the near future has forced a revision of the foundations of modern cryptography. The fact is that many difficult algorithmic problems, such as the discrete logarithm, factoring a (large) natural number into prime factors, etc., on the complexity of which many cryptographic protocols are based these days, have turned out to be relatively easy to solve using quantum algorithms. Intensive research is currently underway to find problems that are difficult even for a quantum computer and have potential applications for cryptographic protocols. Our article contains notes related to the so-called generalized Gauss algorithm, which calculates the reduced basis of a two-dimensional lattice [8], [2]. Note that researchers are increasingly putting forward difficult algorithmic problems from lattice theory as candidates for the foundation of post-quantum cryptography. The majority of algorithmic problems related to lattice reduction become NP-hard as the lattice dimension increases [3], [1]. Fundamental problems such as the Shortest Vector Problem (SVP), the Closest Vector Problem (CVP), and Bounded Distance Decoding (BDD) are conjectured to remain hard even for quantum algorithms [4], [6]. Although the generalized Gauss reduction algorithm applies to two-dimensional lattices, where exact analysis is feasible (dimensions 3 and 4 are studied in [7], [5]), understanding such low-dimensional reductions provides important insights into the structure and complexity of lattice-based cryptographic constructions. [ABSTRACT FROM AUTHOR]
Copyright of Eurasian Mathematical Journal is the property of L.N. Gumilyov Eurasian National University and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: NOTES ON THE GENERALIZED GAUSS REDUCTION ALGORITHM.
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  Data: Eurasian Mathematical Journal; 2025, Vol. 16 Issue 2, p23-29, 7p
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  Data: <searchLink fieldCode="DE" term="%22CRYPTOGRAPHY%22">CRYPTOGRAPHY</searchLink><br /><searchLink fieldCode="DE" term="%22LATTICE+theory%22">LATTICE theory</searchLink><br /><searchLink fieldCode="DE" term="%22QUANTUM+computing%22">QUANTUM computing</searchLink><br /><searchLink fieldCode="DE" term="%22GAUSSIAN+elimination%22">GAUSSIAN elimination</searchLink><br /><searchLink fieldCode="DE" term="%22NP-hard+problems%22">NP-hard problems</searchLink><br /><searchLink fieldCode="DE" term="%22ALGORITHMS%22">ALGORITHMS</searchLink>
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  Label: Abstract
  Group: Ab
  Data: The hypothetical possibility of building a quantum computer in the near future has forced a revision of the foundations of modern cryptography. The fact is that many difficult algorithmic problems, such as the discrete logarithm, factoring a (large) natural number into prime factors, etc., on the complexity of which many cryptographic protocols are based these days, have turned out to be relatively easy to solve using quantum algorithms. Intensive research is currently underway to find problems that are difficult even for a quantum computer and have potential applications for cryptographic protocols. Our article contains notes related to the so-called generalized Gauss algorithm, which calculates the reduced basis of a two-dimensional lattice [8], [2]. Note that researchers are increasingly putting forward difficult algorithmic problems from lattice theory as candidates for the foundation of post-quantum cryptography. The majority of algorithmic problems related to lattice reduction become NP-hard as the lattice dimension increases [3], [1]. Fundamental problems such as the Shortest Vector Problem (SVP), the Closest Vector Problem (CVP), and Bounded Distance Decoding (BDD) are conjectured to remain hard even for quantum algorithms [4], [6]. Although the generalized Gauss reduction algorithm applies to two-dimensional lattices, where exact analysis is feasible (dimensions 3 and 4 are studied in [7], [5]), understanding such low-dimensional reductions provides important insights into the structure and complexity of lattice-based cryptographic constructions. [ABSTRACT FROM AUTHOR]
– Name: Abstract
  Label:
  Group: Ab
  Data: <i>Copyright of Eurasian Mathematical Journal is the property of L.N. Gumilyov Eurasian National University and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.32523/2077-9879-2025-16-2-23-29
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        Text: English
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      – SubjectFull: CRYPTOGRAPHY
        Type: general
      – SubjectFull: LATTICE theory
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      – SubjectFull: QUANTUM computing
        Type: general
      – SubjectFull: GAUSSIAN elimination
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      – SubjectFull: NP-hard problems
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      – SubjectFull: ALGORITHMS
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      – TitleFull: NOTES ON THE GENERALIZED GAUSS REDUCTION ALGORITHM.
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              Text: 2025
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