Post Quantum Cryptography from Mutant Prime Knots

📝 Abstract

By resorting to basic features of topological knot theory we propose a (classical) cryptographic protocol based on the `difficulty’ of decomposing complex knots generated as connected sums of prime knots and their mutants. The scheme combines an asymmetric public key protocol with symmetric private ones and is intrinsecally secure against quantum eavesdropper attacks.

💡 Analysis

By resorting to basic features of topological knot theory we propose a (classical) cryptographic protocol based on the `difficulty’ of decomposing complex knots generated as connected sums of prime knots and their mutants. The scheme combines an asymmetric public key protocol with symmetric private ones and is intrinsecally secure against quantum eavesdropper attacks.

📄 Content

arXiv:1010.2055v1 [math-ph] 11 Oct 2010 Post Quantum Cryptography from Mutant Prime Knots Annalisa Marzuoli (1) and Giandomenico Palumbo (2) Dipartimento di Fisica Nucleare e Teorica, Universit`a degli Studi di Pavia and Istituto Nazionale di Fisica Nucleare, Sezione di Pavia via A. Bassi 6, 27100 Pavia (Italy) (1) E-mail: annalisa.marzuoli@pv.infn.it (2) E-mail: giandomenico.palumbo@pv.infn.it Abstract By resorting to basic features of topological knot theory we pro- pose a (classical) cryptographic protocol based on the ‘difficulty’ of decomposing complex knots generated as connected sums of prime knots and their mutants. The scheme combines an asymmetric public key protocol with symmetric private ones and is intrinsecally secure against quantum eavesdropper attacks. PACS2008: 89.70.-a (Information and communication theory) 02.10.kn (Knot theory) 03.67Dd (Quantum Cryptography and communication security) MSC2010: 68QXX (Theory of computing) 57M27 (Invariants of knots and 3-manifolds) 68Q17 (Computational difficulty of problems) 1 1 Introduction Knots and links (collections of knotted circles), beside being fascinating mathematical objects, are encoded in the modeling of a number of physi- cal, chemical and biological systems. In particular it was in the late 1980 that knot theory was recognized to have a deep, unexpected interaction with quantum field theory [1]. In earlier periods of the history of science, geometry and physics interacted very strongly at the ‘classical’ level (as in Einstein’s General Relativity theory), but the main feature of this new, ‘quantum’ connection is the fact that geometry is involved in a global and not purely local way, i.e. only ‘topological’ features do matter. Over the years math- ematicians have proposed a number of ‘knot invariants’ aimed to classify systematically all possible knots. Most of these invariants are polynomial expressions (in one or two variables) with coefficients in the relative integers. It was Vaughan Jones in [2] who discovered the most famous polynomial in- variant, the Jones invariant, and solved the Tait’s conjectures for alternating knots. In the seminal paper by Edward Witten [1], the Jones polynomial was actually recognized to be associated with the vacuum expectation value of a ‘Wilson loop operator’ in a quantum Chern–Simons theory (see the rewiews [3], [4] for comprehensive accounts on these topics). Seemly far from the above remarks, the search for new algorithmic prob- lems and techniques which should improve ‘quantum’ with respect to clas- sical computation is getting more and more challenging in the last decade. Most quantum algorithms are based on the standard quantum circuit model [5], and are designed to solve problems which are essentially number the- oretic such as the Shor’s algorithm [6] (see [7] for a general review on the basics of quantum algorithms). However, other types of problems, typically classified in the field of enumerative combinatorics and ubiquitous in many areas of mathematics and physics, share the feature to be ‘intractable’ in the framework of classical information theory. In particular the evaluation of the Jones polynomial has been shown to be #P–hard, namely computationally intractable in a very strong sense [8]. In this perspective, efficient quantum algorithms for computing approximately knot invariants (of the Jones’ type or extensions of it) have been successfully addressed in the last few years [9], [10, 11, 12], [13] and indeed such problem has been recognized to be ‘universal’ in the quantum complexity class BQP (Bounded error Quantum Polynomial), namely the hardest problem that a quantum computer can efficiently handle [14]. 2 Notwithstanding the improvements outlined above both in field–theoretic settings and in quantum complexity theory, the basic unsolved problem in topological knot theory still remains the ‘recognition problem’. Namely, given two knots, how can we check if they are ‘equivalent’ (in the sense to be for- malized in the next section). Invariants of (oriented) knots might be useful to this task, but there exist particular classes of knots –the ‘mutants’ of a given knot– that cannot be distinguished in principle since by definition all of them possess the same Jones’ type invariants, a result derived by resorting to standard tools in combinatorial topology (see e.g. [15]) but recognizable also in the field–theoretic framework as a property of expectation values of Wilson loop operators [16]. As is well known, group–based cryptography has became in the last few years a very fruitful branch of cryptoanalysis [17], [18]. In particular, the key–agreement protocol proposed in [19] can be implemented using the braid group Bn (a non–Abelian group on (n-1) generators that can be associated to geometric configurations of n interlaced strands whose endpoints are fixed on two parallel straight lines in the plane). Knots and braids are indeed closely interconnected since we can get a (multi–component) knot by ‘clo

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