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Structured ramp secret sharing schemata over rings of real polynomials.

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Title: Structured ramp secret sharing schemata over rings of real polynomials.
Authors: Meletiou, Gerasimos C.1 (AUTHOR) gmelet@uoi.gr, Papadakis, Nikolaos K.2 (AUTHOR) npapadakis@sse.gr, Triantafyllou, Dimitrios S.1,2 (AUTHOR) dtriant@sse.gr, Vrahatis, Michael N.3 (AUTHOR) vrahatis@upatras.gr
Source: Applied Numerical Mathematics. Feb2025:Part A, Vol. 208, p317-339. 23p.
Subjects: Numerical solutions for linear algebra, Triangularization (Mathematics), Matrix decomposition, Polynomial rings, Polynomials
Abstract: Two new ramp secret sharing schemata based on polynomials are proposed. For both schemata, the secret is considered to be a polynomial created by the dealer. The participants are separated into ℓ ⩾ 2 , groups, that are specified by the dealer's levels L i for i = 1 , 2 , ... , ℓ and each level L i , i ⩾ 2 , is separated into subsets. The shares of the secret are given to participants in the form of polynomials. For the first proposed scheme, the dealer creates ℓ polynomials one for each level. Specific participants from every subset of each level have to cooperate all together in order to construct the polynomial of their level. Next all the authorized participants cooperate for computing the greatest common divisor of the polynomials in order to retrieve the secret. In the second scheme, the authorized participants cooperate per two levels using a bottom-up procedure. In both schemata the greatest common divisor can be evaluated by implementing numerical linear algebra methods, and precisely factorization of matrices of special form such as Sylvester matrices. The triangularization of these matrices can be obtained by exploiting their special structure for the reduction of the required floating point operations. The innovative idea of the paper at hand is the use of real polynomials in secret sharing schemata. This is particularly useful since the greatest common divisor can always be computed with efficient accuracy using effective numerical methods. New theoretical results are proved and provided that support the error analysis of our approach. [ABSTRACT FROM AUTHOR]
Copyright of Applied Numerical Mathematics is the property of Elsevier B.V. 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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DbLabel: Engineering Source
An: 181539869
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PubTypeId: academicJournal
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  Data: Structured ramp secret sharing schemata over rings of real polynomials.
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  Data: <searchLink fieldCode="AR" term="%22Meletiou%2C+Gerasimos+C%2E%22">Meletiou, Gerasimos C.</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> gmelet@uoi.gr</i><br /><searchLink fieldCode="AR" term="%22Papadakis%2C+Nikolaos+K%2E%22">Papadakis, Nikolaos K.</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> npapadakis@sse.gr</i><br /><searchLink fieldCode="AR" term="%22Triantafyllou%2C+Dimitrios+S%2E%22">Triantafyllou, Dimitrios S.</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> dtriant@sse.gr</i><br /><searchLink fieldCode="AR" term="%22Vrahatis%2C+Michael+N%2E%22">Vrahatis, Michael N.</searchLink><relatesTo>3</relatesTo> (AUTHOR)<i> vrahatis@upatras.gr</i>
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  Data: <searchLink fieldCode="JN" term="%22Applied+Numerical+Mathematics%22">Applied Numerical Mathematics</searchLink>. Feb2025:Part A, Vol. 208, p317-339. 23p.
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  Data: <searchLink fieldCode="DE" term="%22Numerical+solutions+for+linear+algebra%22">Numerical solutions for linear algebra</searchLink><br /><searchLink fieldCode="DE" term="%22Triangularization+%28Mathematics%29%22">Triangularization (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Matrix+decomposition%22">Matrix decomposition</searchLink><br /><searchLink fieldCode="DE" term="%22Polynomial+rings%22">Polynomial rings</searchLink><br /><searchLink fieldCode="DE" term="%22Polynomials%22">Polynomials</searchLink>
– Name: Abstract
  Label: Abstract
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  Data: Two new ramp secret sharing schemata based on polynomials are proposed. For both schemata, the secret is considered to be a polynomial created by the dealer. The participants are separated into ℓ ⩾ 2 , groups, that are specified by the dealer's levels L i for i = 1 , 2 , ... , ℓ and each level L i , i ⩾ 2 , is separated into subsets. The shares of the secret are given to participants in the form of polynomials. For the first proposed scheme, the dealer creates ℓ polynomials one for each level. Specific participants from every subset of each level have to cooperate all together in order to construct the polynomial of their level. Next all the authorized participants cooperate for computing the greatest common divisor of the polynomials in order to retrieve the secret. In the second scheme, the authorized participants cooperate per two levels using a bottom-up procedure. In both schemata the greatest common divisor can be evaluated by implementing numerical linear algebra methods, and precisely factorization of matrices of special form such as Sylvester matrices. The triangularization of these matrices can be obtained by exploiting their special structure for the reduction of the required floating point operations. The innovative idea of the paper at hand is the use of real polynomials in secret sharing schemata. This is particularly useful since the greatest common divisor can always be computed with efficient accuracy using effective numerical methods. New theoretical results are proved and provided that support the error analysis of our approach. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Applied Numerical Mathematics is the property of Elsevier B.V. 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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RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1016/j.apnum.2024.06.003
    Languages:
      – Code: eng
        Text: English
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      Pagination:
        PageCount: 23
        StartPage: 317
    Subjects:
      – SubjectFull: Numerical solutions for linear algebra
        Type: general
      – SubjectFull: Triangularization (Mathematics)
        Type: general
      – SubjectFull: Matrix decomposition
        Type: general
      – SubjectFull: Polynomial rings
        Type: general
      – SubjectFull: Polynomials
        Type: general
    Titles:
      – TitleFull: Structured ramp secret sharing schemata over rings of real polynomials.
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            NameFull: Meletiou, Gerasimos C.
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            NameFull: Papadakis, Nikolaos K.
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            NameFull: Triantafyllou, Dimitrios S.
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            NameFull: Vrahatis, Michael N.
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          Dates:
            – D: 01
              M: 02
              Text: Feb2025:Part A
              Type: published
              Y: 2025
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            – TitleFull: Applied Numerical Mathematics
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