A Zero-Knowledge Proof Based on a Multivariate Polynomial Reduction of the Graph Isomorphism Problem

Zero-Knowledge Proofs ZKP provide a reliable option to verify that a claim is true without giving detailed information other than the answer. A classical example is provided by the ZKP based in the Graph Isomorphism problem (GI), where a prover must convince the verifier that he knows an isomorphism between two isomorphic graphs without publishing the bijection. We design a novel ZKP exploiting the NP-hard problem of finding the algebraic ideal of a multivariate polynomial set, and consequently resistant to quantum computer attacks. Since this polynomial set is obtained considering instances of GI, we guarantee that the protocol is at least as secure as the GI based protocol.

Two Mathematical Comments on the Thevenin Theorem: An “Algebraic Ideal” and the “Affine Nonlinearity”

We discuss the most important and simple concept of basic circuit theory—the concept of the unideal source—or the Thevenin circuit. It is explained firstly how the Thevenin circuit can be interpreted in the algebraic sense. Then, we critically consider the common opinion that it is a linear circuit, showing that linearity (or nonlinearity) depends on the use of the port. The difference between the cases of a source being an input or an internal element (as it is in Thevenin’s circuit) is important here. The distinction in the definition of linear operator in algebra (here in system theory) and in geometry is also important for the subject, and we suggest the wide use of the concept of “affine nonlinearity.” This kind of nonlinearity should be relevant for the development of complicated circuitry (perhaps in a biological modeling context) with nonprescribed definition of subsystems, when the interpretation of a port as input or output can become dependent on the local intensity of a process.

Properly algebraic and spectrum-finite ideals in Jordan systems

The two main results of this paper are:(i) The set of properly algebraic elements of a Jordan system (algebra, triple system or pair) over an uncountable field is an ideal.(ii) For a semiprimitive Banach Jordan system, the socle, the largest properly algebraic ideal, the largest properly spectrum-finite ideal and the largest von Neumann regular ideal all coincide.

Algebraic ideals of semiprime Banach algebras

If A is a semiprime Banach algebra, soc A, rad A the socle and radical of A, then Soc A ∩ rad A = (0). This elementary result enables us to prove some results concerning algebraic ideal and algebraic elements modulo the socle of A. We also deduce several conditions for A equivalent to the condition dim A <+∞.

Discrete structure spaces of ƒ-rings

Birkhoff and Pierce [2] introduced the concept of an ƒ-ring and showed that an l-ring is an f-ring if and only if it is a subdirect product of totallyordered rings. An l-ideal of an f-ring R is an algebraic ideal which is at the same time a lattice ideal of R. Structure spaces (i.e. sets of prime ideals endowed with the so-called hull-kernel or Stone topology) for ordinary rings have been studied by many authors. In this paper we consider certain analogues for ƒ-rings, and give characterisations of ƒ-rings for which these structure spaces are discrete.