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El fundamento matemático de la criptografía post-cuántica

Chuanming Zong1

  • 1Center for Applied Mathematics, Tianjin University, Tianjin 300072, China.

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La computación cuántica amenaza el cifrado actual. Este artículo explora los fundamentos matemáticos de la criptografía post-cuántica, vinculando problemas de celosía como SVP y CVP a las formas de embalaje de bolas y cuadráticas.

Área de la Ciencia:

  • Criptografía
  • La computación cuántica
  • Teoría de los números

Sus antecedentes:

  • Los algoritmos cuánticos desarrollados por P. Shor en 1994 y el advenimiento de las computadoras cuánticas representan una amenaza significativa para los métodos de comunicación secretos actuales como RSA y ElGamal.
  • El Instituto Nacional de Estándares y Tecnología (NIST) está estandarizando la criptografía post-cuántica (PQC) para abordar esta crisis, con candidatos basados en la teoría de la red y las funciones hash.

Objetivo del estudio:

  • Proporcionar un artículo de revisión sobre los fundamentos matemáticos de la teoría de la complejidad de la criptografía post-cuántica.
  • Demostrar las raíces matemáticas de PQC en problemas fundamentales como el embalaje de bolas, la cubierta de bolas y las formas cuadráticas definidas positivas.

Principales métodos:

  • Introducción a la criptografía postcuántica (PQC).

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  • Demostración de las conexiones matemáticas entre los criptosistemas basados en celosía y los problemas computacionales.
  • Explicación de la relación entre el problema del vector más corto (SVP), el problema del vector más cercano (CVP) y las formas cuadráticas definidas positivas.

    • Principales resultados:

    • El NIST ha anunciado los estándares iniciales de PQC (FIPS 203, 204, 205) basados en CRYSTALS-Kyber, CRYSTALS-Dilithium y Sphincs+.
    • La seguridad de los criptosistemas basados en celosía está fundamentalmente vinculada a la dureza de SVP y CVP.
    • SVP y CVP pueden entenderse como problemas de embalaje de bolas y cubrimiento de bolas, respectivamente, y son equivalentes a problemas aritméticos que involucran formas cuadráticas definidas positivas.

    Conclusiones:

    • Comprender los fundamentos matemáticos de los problemas de celosía es crucial para desarrollar y analizar sistemas criptográficos post-cuánticos.
    • Este trabajo cierra la brecha entre los conceptos matemáticos abstractos y su aplicación práctica para asegurar las comunicaciones contra las amenazas cuánticas.