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On Multivariate Cryptosystems Based on Polynomially Compressed Maps with Invertible Decomposition

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Book cover Cryptography and Security Systems (CSS 2014)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 448))

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Abstract

Let K be a commutative ring and K n be an affine space over K of dimension n. We illustrate the concept of a family of polynomially compressed multivariate maps f(n) by relations of degree k with invertible decomposition via presentation of the explicit construction. Such a construction is based on the edge transitive family of graphs D(n, K). It uses the equations of connected component of the graph. The walk on the graph can be given by the sequence of its edges. It induces a cubical bijective transformation E(n) of the flag space isomorphic to K n + 1. The transformations related edges of the walk form invertible decomposition of E(n) into simple multipliers. Knowledge of such decomposition allows to find the pre-image of E(n)(x) fast. The map E(n) is not suitable for a public map because its inverse is also cubical. The restriction of the map \(\tilde{E(n)}\) on the chosen connected component is a multivariate transformation of unbounded degree (compressed map). The public user has some additional compression rules which allow for fast computation of the value \(\tilde{E(n)}\) on a given flag. The key holder (Alice) knows E(n), special decompression rules allow her to decrypt fast. To hide the graph Alice deformates E(n), the compression and decompression rules by two special affine transformations τ 1 and τ 2. The usage of τ 1 E(n)τ 2 allows to get a more densely compressed map.

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Authors and Affiliations

  1. Institute of Computer Science, Maria Curie-Sklodowska University, pl. M. Curie-Sklodowskiej 5, 20-031, Lublin, Poland

    Urszula Romańczuk-Polubiec & Vasyl Ustimenko

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  1. Urszula Romańczuk-Polubiec
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  2. Vasyl Ustimenko
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Editor information

Editors and Affiliations

  1. Insitute of Telecommunications, Warsaw University of Technology, ul. Nowowiejska 15/19, 00-665, Warsaw, Poland

    Zbigniew Kotulski

  2. Institute of Computer Science, Maria Curie-Skłodowska in Lublin, ul. Plac Marii Curie-Skłodowskiej 5, 20-031 Lublin and Polish-Japanese Institute of Information Technology, Koszykowa 8, Poland

    Bogdan Księżopolski

  3. Institute of Computer Science, Maria Curie-Skłodowska in Lublin, ul. Plac Marii Curie-Skłodowskiej 5, 20-031, Lublin, Poland

    Katarzyna Mazur

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Romańczuk-Polubiec, U., Ustimenko, V. (2014). On Multivariate Cryptosystems Based on Polynomially Compressed Maps with Invertible Decomposition. In: Kotulski, Z., Księżopolski, B., Mazur, K. (eds) Cryptography and Security Systems. CSS 2014. Communications in Computer and Information Science, vol 448. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44893-9_3

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  • DOI: https://doi.org/10.1007/978-3-662-44893-9_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-44892-2

  • Online ISBN: 978-3-662-44893-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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