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Birkhäuser Basel eBooks

Canonical Non-commutative Cryptography

Published Nov 4, 2008• 0 citations
Publisher

Abstract

In this chapter, we discuss various cryptographic primitives that use non-commutative (semi)groups as platforms, but at the same time do not depart from the canonical paradigm of a public-key protocol based on a one-way function. We include here the ground-breaking Anshel-Anshel-Goldfeld protocol [1] as well as protocols that are closer in spirit to classical protocols based on commutative (semi)groups.

DOI
10.1007/978-3-7643-8827-0_4

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In this chapter, we discuss various cryptographic primitives that use non-commutative (semi)groups as platforms, but at the same time do not depart from the canonical paradigm of a public-key protocol based on a one-way function. We include here the ground-breaking Anshel-Anshel-Goldfeld protocol [1] as well as protocols that are closer in spirit to classical protocols based on commutative (semi)groups.
Quantum cryptographic primitives beyond key distribution remain a less well understood area of research. In classical cryptography, one-way functions underpin nearly all standard cryptographic protocols, motivating the search for meaningful quantum analogues and for a clear understanding of the physical and computational mechanisms that could enforce one-wayness. In this article, we review quantum one-way functions and a range of closely related quantum-state primitives, including one-way state generators, pseudorandom quantum states, and efficiently indistinguishable pairs of states. We discuss both computational and information-theoretic notions of quantum one-wayness, emphasizing the different adversarial models and security assumptions that underlie these constructions. We compare and contrast the various proposed primitives, and clarify their conceptual relationships. Particular emphasis is placed on questions of physical realizability, experimental feasibility, and robustness to noise. Finally, we outline open problems and future directions toward the development of practical quantum cryptographic primitives beyond key distribution, and the emergence of a broader quantum-cryptographic ecosystem.
Following its etymology, the term cryptography denotes the techniques used to ensure the confidentiality of information in its storage or transmission. Besides confidentiality, cryptography also encompasses other important functions such as authentication, signature, non-repudiation or secret sharing, to name just a few. The purpose of this section is to give a short introduction to classical cryptography, but only for areas that are relevant to quantum cryptography. For our purposes, we will only deal with confidentiality and authentication. The former is the most important function that quantum key distribution helps to achieve, while the latter is a requirement for quantum key distribution to work. Also, we will cover some topics on the security of classical cryptographic schemes so as to give some insight when comparing classical and quantum cryptography. The study of cryptography ranges from the analysis of cryptographic algorithms (or primitives ) to the design of a solution to a security problem. A cryptographic algorithm is a mathematical tool that provides a solution to a very specific problem and may be based on premises or on other primitives. A solution can be designed by combining primitives, and the requirements and functions of the primitives have to be combined properly. We will first review the confidentiality provided by ciphers that use the same key for both encryption and decryption, hence called secret-key ciphers . Then, we will discuss authentication techniques. Finally, we will show some aspects of public-key cryptography .
This chapter discusses cryptographic protocols, which provide a means for cryptographic primitives to be combined in ways that allow complex sets of security goals to be tailored to particular application environments. It describes a simple, but artificial, application in order to demonstrate that there are many different ways in which a cryptographic protocol can be designed, and also how sensitive the security of cryptographic protocols can be. It then considers the important family of AKE protocols (authentication and key establishment).
We show that some problems in cryptography can be solved without using one-way functions. The latter are usually regarded as a central concept of cryptography, but the very existence of one-way functions depends on difficult conjectures in complexity theory, most notably on the notorious “P ̸ = NP ” conjecture. This is why cryptographic primitives that do not employ one-way functions are often called “unconditionally secure”. In this paper, we suggest protocols for secure computation of the sum, product, and some other functions of two or more elements of an arbitrary constructible ring, without using any one-way functions. A new input that we offer here is that, in contrast with other proposals, we conceal “intermediate results ” of a computation. For example, when we compute the sum of k numbers, only the final result is known to (one of) the parties; partial sums are not known to anybody. Other applications of our method include voting/rating over insecure channels and a rather elegant and efficient solution of the “two millionaires problem”. While it is fairly obvious that a secure (bit) commitment between two parties is