Skip to main content

Main navigation

  • About ITEFI
  • Research
  • Formación y empleo
  • OpenLab
  • Servicios científico técnicos
  • Staff Directory

A Group Law on the Projective Plane with Applications in Public Key Cryptography

abelian group law
discrete logarithm problem
norm of an extension
projective cubic curve
Raúl Durán Díaz, Luis Hernández Encinas and Jaime Muñoz Masqué
Mathematics, 8(5), 734 (2020), 20pp., Special Issue "Mathematics Cryptography and Information Security (Q1, Mathematics, F.I. 1.105)
https://doi.org/10.3390/math8050734

In the context of new threats to Public Key Cryptography arising from a growing computational power both in classic and in quantum worlds, we present a new group law defined on a subset of the projective plane $F$$P$2 over an arbitrary field $F$ , which lends itself to applications in Public Key Cryptography and turns out to be more efficient in terms of computational resources. In particular, we give explicitly the number of base field operations needed to perform the mentioned group law. Based on it, we present a Diffie-Hellman-like key agreement protocol. We analyze the computational difficulty of solving the mathematical problem underlying the proposed Abelian group law and we prove that the security of our proposal is equivalent to the discrete logarithm problem in the multiplicative group of the cubic extension of the finite field considered. We present an experimental setup in order to show real computation times along a comparison with the group operation in the group of points of an elliptic curve. Based on current state-of-the-art algorithms, we provide parameter ranges suitable for real world applications. Finally, we present a promising variant of the proposed group law, by moving from the base field $F$ to the ring $Z/pqZ$ , and we explain how the security becomes enhanced, though at the cost of a longer key length.

Funding

This research has been partially supported by Ministerio de Economía, Industria y Competitividad (MINECO), Agencia Estatal de Investigación (AEI), and European Regional Development Fund (ERDF, EU), through project COPCIS, Grant No. TIN2017-84844-C2-1-R, and by Comunidad de Madrid (Spain) through project CYNAMON, Grant No. P2018/TCS-4566-CM, co-funded along with ERDF.

GiCSI
Acoustics and Non Destructive Evaluation (DAEND)
  • Environmental Acoustics (GAA)
  • G Carma: Materials Characterization by Non Destructive Evaluation
  • ULAB, Ultrasounds for Liquid Analysis and Bioengineering
Information and Communication Technologies (TIC)
  • Cybersecurity and Privacy Protection Research Group (GiCP)
  • Research group on Cryptology and Information Security (GiCSI)
    • Quantum Communications Laboratory (LCQE)
  • Multichannel Ultrasonic Signal Processing Group (MUSP)
Sensors and Ultrasonic Systems (DSSU)
  • Ultrasonic Systems and Technologies (USTG)
  • Nanosensors and Smart Systems (NoySi)
  • Ultrasonic Resonators for cavitation and micromanipulation (RESULT)
  • Advanced Sensor Technology (SENSAVAN)
  • Quantum Electronics (QE)
Laboratorios
  • Laboratorio de Acústica
  • Laboratorio de Metrología Ultrasónica Médica (LMUM)
  • Laboratorio de Comunicaciones Cuánticas
  • Laboratory for International Collaboration in Advanced Biophotonics Imaging

Instituto de Tecnologías Físicas y de la Información Leonardo Torres Quevedo  - ITEFI
C/ Serrano, 144. 28006 - Madrid • Tel.: (+34) 91 561 88 06  Contacto  •  Intranet
EDIFICIO PARCIALMENTE ACCESIBLE POR PERSONAS CON MOVILIDAD REDUCIDA