In this paper we introduce an algebra embedding $\iota:K \to S$ from the free associative algebra $K $ generated by a finite or countable set $X$ into the skew monoid ring $S = P * \Sigma$ defined by the commutative polynomial ring $P = K[X\times N^*]$ and by the monoid $\Sigma = $ generated by a suitable endomorphism $\sigma:P\to P$. If $P = K[X]$ is any ring of polynomials in a countable set of commuting variables, we present also a general Gr\"obner bases theory for graded two-sided ideals of the graded algebra $S = \bigoplus_i S_i$ with $S_i = P \sigma^i$ and $\sigma:P \to P$ an abstract endomorphism satisfying compatibility conditions with ordering and divisibility of the monomials of $P$. Moreover, using a suitable grading for the algebra $P$ compatible with the action of $\Sigma$, we obtain a bijective correspondence, preserving Gr\"obner bases, between graded $\Sigma$-invariant ideals of $P$ and a class of graded two-sided ideals of $S$. By means of the embedding $\iota$ this results in the unification, in the graded case, of the Gr\"obner bases theories for commutative and non-commutative polynomial rings. Finally, since the ring of ordinary difference polynomials $P = K[X\times N]$ fits the proposed theory one obtains that, with respect to a suitable grading, the Gr\"obner bases of finitely generated graded ordinary difference ideals can be computed also in the operators ring $S$ and in a finite number of steps up to some fixed degree.

Skew polynomial rings, Groebner bases and the letterplace embedding of the free associative algebra

Zacharias representation of effective associative rings

In 1994, Moss Sweedler’s dog proposed a cryptosystem, known as Barkee’s Cryptosystem, and the related cryptanalysis. Its explicit aim was to dispel the proposal of using the urban legend that “Grobner bases are hard to compute”, in order to devise a public key cryptography scheme. Therefore he claimed that “no scheme using Grobner bases will ever work”. Later, further variations of Barkee’s Cryptosystem were proposed on the basis of another urban legend, related to the infiniteness (and consequent uncomputability) of non-commutative Grobner bases; unfortunately Pritchard’s algorithm for computing (finite) non-commutative Grobner 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’s attack but not to Davenport’s factorization algorithm. It was recently adapted specializing and extending Stickel’s Diffie–Hellman 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.

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

We adapt tag-variables and Buchberger reduction in order, given two elements, $$pg\in R$$
 into an effective ring R, to express g as the evaluation of a polynomial $$f(X)\in R[X]$$
 at p, $$g=f(p)$$
  As a by-product, we present also an attack to a couple of Cryptographical protocols

Why you cannot even hope to use Ore algebras in Cryptography

This work formally introduces and starts investigating the structure of matrix polynomial algebra extensions 
of a coefficient algebra by (elementary) matrix-variables over 
 a ground polynomial ring in not necessary commuting variables. 
 These matrix subalgebras of full matrix rings over polynomial rings show up 
 in noncommutative algebraic geometry. We carefully study their (one-sided or bilateral) noetherianity, obtaining a precise lift of the Hilbert Basis Theorem when the 
ground ring is either a commutative polynomial ring, a free noncommutative polynomial ring or a skew polynomial ring extension by a free commutative term-ordered monoid. 
We equally address the natural but rather delicate question of recognising which matrix polynomial algebras are Cayley-Hamilton algebras, 
which are interesting noncommutative algebras arising from the study of $\mathrm{Gl}_{n}$-varieties.

/pdf/the-structure-of-matrix-polynomial-algebras-14ybigg0.pdf

The structure of matrix polynomial algebras

Effective Buchberger-Zacharias-Weispfenning theory of skew polynomial extensions of subbilateral coherent rings

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Effective Buchberger-Zacharias-Weispfenning theory of skew polynomial extensions of subbilateral coherent rings