Certainty and Uncertainty in Quantum Information Processing

This survey, aimed at information processing researchers, highlights intriguing but lesser known results, corrects misconceptions, and suggests research areas. Themes include: certainty in quantum algorithms; the "fewer worlds" theory of quantum mechanics; quantum learning; probability theory versus quantum mechanics.Comment: Invited paper accompanying invited talk to AAAI Spring Symposium 2007. Comments, corrections, and suggestions would be most welcom

Group theory in cryptography

This paper is a guide for the pure mathematician who would like to know more about cryptography based on group theory. The paper gives a brief overview of the subject, and provides pointers to good textbooks, key research papers and recent survey papers in the area.Comment: 25 pages References updated, and a few extra references added. Minor typographical changes. To appear in Proceedings of Groups St Andrews 2009 in Bath, U

Ramanujan graphs in cryptography

In this paper we study the security of a proposal for Post-Quantum Cryptography from both a number theoretic and cryptographic perspective. Charles-Goren-Lauter in 2006 [CGL06] proposed two hash functions based on the hardness of finding paths in Ramanujan graphs. One is based on Lubotzky-Phillips-Sarnak (LPS) graphs and the other one is based on Supersingular Isogeny Graphs. A 2008 paper by Petit-Lauter-Quisquater breaks the hash function based on LPS graphs. On the Supersingular Isogeny Graphs proposal, recent work has continued to build cryptographic applications on the hardness of finding isogenies between supersingular elliptic curves. A 2011 paper by De Feo-Jao-Pl\^{u}t proposed a cryptographic system based on Supersingular Isogeny Diffie-Hellman as well as a set of five hard problems. In this paper we show that the security of the SIDH proposal relies on the hardness of the SIG path-finding problem introduced in [CGL06]. In addition, similarities between the number theoretic ingredients in the LPS and Pizer constructions suggest that the hardness of the path-finding problem in the two graphs may be linked. By viewing both graphs from a number theoretic perspective, we identify the similarities and differences between the Pizer and LPS graphs.Comment: 33 page

Quantum Information Science

Quantum computing is implicated as a next-generation solution to supplement traditional von Neumann architectures in an era of post-Moores law computing. As classical computational infrastructure becomes more limited, quantum platforms offer expandability in terms of scale, energy-consumption, and native three-dimensional problem modeling. Quantum information science is a multidisciplinary field drawing from physics, mathematics, computer science, and photonics. Quantum systems are expressed with the properties of superposition and entanglement, evolved indirectly with operators (ladder operators, master equations, neural operators, and quantum walks), and transmitted (via quantum teleportation) with entanglement generation, operator size manipulation, and error correction protocols. This paper discusses emerging applications in quantum cryptography, quantum machine learning, quantum finance, quantum neuroscience, quantum networks, and quantum error correction