Private Key Extension of Polly Cracker Cryptosystems

In 1993 Koblitz and Fellows proposed a public key cryptosystem, Polly Cracker, based on the problem of solving multivariate systems of polynomial equations, which was soon generalized to a Dröbner basis formulation. Since then a handful of improvements of this construction has been proposed. In this paper it is suggested that security, and possibly efficiency, of any Polly Cracker-type cryptosystem could be increased by altering the premises regarding private- and public information

A Survey on Homomorphic Encryption Schemes: Theory and Implementation

Legacy encryption systems depend on sharing a key (public or private) among the peers involved in exchanging an encrypted message. However, this approach poses privacy concerns. Especially with popular cloud services, the control over the privacy of the sensitive data is lost. Even when the keys are not shared, the encrypted material is shared with a third party that does not necessarily need to access the content. Moreover, untrusted servers, providers, and cloud operators can keep identifying elements of users long after users end the relationship with the services. Indeed, Homomorphic Encryption (HE), a special kind of encryption scheme, can address these concerns as it allows any third party to operate on the encrypted data without decrypting it in advance. Although this extremely useful feature of the HE scheme has been known for over 30 years, the first plausible and achievable Fully Homomorphic Encryption (FHE) scheme, which allows any computable function to perform on the encrypted data, was introduced by Craig Gentry in 2009. Even though this was a major achievement, different implementations so far demonstrated that FHE still needs to be improved significantly to be practical on every platform. First, we present the basics of HE and the details of the well-known Partially Homomorphic Encryption (PHE) and Somewhat Homomorphic Encryption (SWHE), which are important pillars of achieving FHE. Then, the main FHE families, which have become the base for the other follow-up FHE schemes are presented. Furthermore, the implementations and recent improvements in Gentry-type FHE schemes are also surveyed. Finally, further research directions are discussed. This survey is intended to give a clear knowledge and foundation to researchers and practitioners interested in knowing, applying, as well as extending the state of the art HE, PHE, SWHE, and FHE systems.Comment: - Updated. (October 6, 2017) - This paper is an early draft of the survey that is being submitted to ACM CSUR and has been uploaded to arXiv for feedback from stakeholder

Why you cannot even hope to use Gr\uf6bner bases in cryptography: an eternal golden braid of failures

In 1994, Moss Sweedler\u2019s dog proposed a cryptosystem, known as Barkee\u2019s Cryptosystem, and the related cryptanalysis. Its explicit aim was to dispel the proposal of using the urban legend that \u201cGr\uf6bner bases are hard to compute\u201d, in order to devise a public key cryptography scheme. Therefore he claimed that \u201cno scheme using Gr\uf6bner bases will ever work\u201d. Later, further variations of Barkee\u2019s Cryptosystem were proposed on the basis of another urban legend, related to the infiniteness (and consequent uncomputability) of non-commutative Gr\uf6bner bases; unfortunately Pritchard\u2019s algorithm for computing (finite) non-commutative Gr\uf6bner bases was already available at that time and was sufficient to crash the system proposed by Ackermann and Kreuzer. The proposal by Rai, where the private key is a principal ideal and the public key is a bunch of polynomials within this principal ideal, is surely immune to Pritchard\u2019s attack but not to Davenport\u2019s factorization algorithm. It was recently adapted specializing and extending Stickel\u2019s Diffie\u2013Hellman protocols in the setting of Ore extension. We here propose a further generalization and show that such protocols can be broken simply via polynomial division and Buchberger reduction

Criptografa multvariable, bases de Groebner y ataques

En el presente trabajo se analizan los fundamentos teóricos que avalan la teoría de bases de Groebner, desarrollada por Bruno Buchberger en su tesis doctoral en 1965. Posteriormente, se estudian dos tipos de criptosistemas bajo los que subyace dicha teoría, llamados criptosistemas Barkee y criptosistemas Polly Cracker, y se comprueba su seguridad ante ciertos ataques algebraicos.<br /

Bounded Fully Homomorphic Encryption from Monoid Algebras

We present a new method that produces bounded FHE schemes (see Definition 3), starting with encryption schemes that support one algebraic operation. We use this technique to construct examples of encryption schemes that, theoretically can handle any algebraic function on encrypted data

Weyl Gröbner Basis Cryptosystems

In this thesis, we shall consider a certain class of algebraic cryptosystems called Gröbner Basis Cryptosystems. In 1994, Koblitz introduced the Polly Cracker cryptosystem that is based on the theory of Gröbner basis in commutative polynomials rings. The security of this cryptosystem relies on the fact that the computation of Gröbner basis is, in general, EXPSPACE-hard. Cryptanalysis of these commutative Polly Cracker type cryptosystems is possible by using attacks that do not require the computation of Gröbner basis for breaking the system, for example, the attacks based on linear algebra. To secure these (commutative) Gröbner basis cryptosystems against various attacks, among others, Ackermann and Kreuzer introduced a general class of Gröbner Basis Cryptosystems that are based on the difficulty of computing module Gröbner bases over general non-commutative rings. The objective of this research is to describe a special class of such cryptosystems by introducing the Weyl Gröbner Basis Cryptosystems. We divide this class of cryptosystems in two parts namely the (left) Weyl Gröbner Basis Cryptosystems and Two-Sided Weyl Gröbner Basis Cryptosystems. We suggest to use Gröbner bases for left and two-sided ideals in Weyl algebras to construct specific instances of such cryptosystems. We analyse the resistance of these cryptosystems to the standard attacks and provide computational evidence that secure Weyl Gröbner Basis Cryptosystems can be built using left (resp. two-sided) Gröbner bases in Weyl algebras

Bases de Groebner y aplicaciones a sistemas criptográficos

En este trabajo, propondremos una alternativa a los métodos criptográficos existentes puesto que la inminente llegada de los ordenadores cuánticos nos hace buscar nuevos algoritmos. Para ello con la utilización de las bases de Groebner describiremos los sistemas de Polly-Cracker y el Inverso Polinomico y veremos por medio de un análisis de datos que el criptosistema de Polly-Cracker es difícil de romper