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

Cryptography from tensor problems

We describe a new proposal for a trap-door one-way function. The new proposal belongs to the "multivariate quadratic" family but the trap-door is different from existing methods, and is simpler

An Elliptic Curve-based Signcryption Scheme with Forward Secrecy

An elliptic curve-based signcryption scheme is introduced in this paper that effectively combines the functionalities of digital signature and encryption, and decreases the computational costs and communication overheads in comparison with the traditional signature-then-encryption schemes. It simultaneously provides the attributes of message confidentiality, authentication, integrity, unforgeability, non-repudiation, public verifiability, and forward secrecy of message confidentiality. Since it is based on elliptic curves and can use any fast and secure symmetric algorithm for encrypting messages, it has great advantages to be used for security establishments in store-and-forward applications and when dealing with resource-constrained devices.Comment: 13 Pages, 5 Figures, 2 Table

Quantum e-commerce: A comparative study of possible protocols for online shopping and other tasks related to e-commerce

A set of quantum protocols for online shopping is proposed and analyzed to establish that it is possible to perform secure online shopping using different types of quantum resources. Specifically, a single photon based, a Bell state based and two 3-qubit entangled state based quantum online shopping schemes are proposed. The Bell state based scheme, being a completely orthogonal state based protocol, is fundamentally different from the earlier proposed schemes which were based on conjugate coding. One of the 3-qubit entangled state based scheme is build on the principle of entanglement swapping which enables us to accomplish the task without transmission of the message encoded qubits through the channel. Possible ways of generalizing the entangled state based schemes proposed here to the schemes which use multiqubit entangled states is also discussed. Further, all the proposed protocols are shown to be free from the limitations of the recently proposed protocol of Huang et al. (Quantum Inf. Process. 14, 2211-2225, 2015) which allows the buyer (Alice) to change her order at a later time (after initially placing the order and getting it authenticated by the controller). The proposed schemes are also compared with the existing schemes using qubit efficiency.Comment: It's shown that quantum e-commerce is not a difficult task, and it can be done in various way

Year 2010 Issues on Cryptographic Algorithms

In the financial sector, cryptographic algorithms are used as fundamental techniques for assuring confidentiality and integrity of data used in financial transactions and for authenticating entities involved in the transactions. Currently, the most widely used algorithms appear to be two-key triple DES and RC4 for symmetric ciphers, RSA with a 1024-bit key for an asymmetric cipher and a digital signature, and SHA-1 for a hash function according to international standards and guidelines related to the financial transactions. However, according to academic papers and reports regarding the security evaluation for such algorithms, it is difficult to ensure enough security by using the algorithms for a long time period, such as 10 or 15 years, due to advances in cryptanalysis techniques, improvement of computing power, and so on. To enhance the transition to more secure ones, National Institute of Standards and Technology (NIST) of the United States describes in various guidelines that NIST will no longer approve two-key triple DES, RSA with a 1024-bit key, and SHA-1 as the algorithms suitable for IT systems of the U.S. Federal Government after 2010. It is an important issue how to advance the transition of the algorithms in the financial sector. This paper refers to issues regarding the transition as Year 2010 issues in cryptographic algorithms. To successfully complete the transition by 2010, the deadline set by NIST, it is necessary for financial institutions to begin discussing the issues at the earliest possible date. This paper summarizes security evaluation results of the current algorithms, and describes Year 2010 issues, their impact on the financial industry, and the transition plan announced by NIST. This paper also shows several points to be discussed when dealing with Year 2010 issues.Cryptographic algorithm; Symmetric cipher; Asymmetric cipher; Security; Year 2010 issues; Hash function

An efficient and secure RSA--like cryptosystem exploiting R\'edei rational functions over conics

We define an isomorphism between the group of points of a conic and the set of integers modulo a prime equipped with a non-standard product. This product can be efficiently evaluated through the use of R\'edei rational functions. We then exploit the isomorphism to construct a novel RSA-like scheme. We compare our scheme with classic RSA and with RSA-like schemes based on the cubic or conic equation. The decryption operation of the proposed scheme turns to be two times faster than RSA, and involves the lowest number of modular inversions with respect to other RSA-like schemes based on curves. Our solution offers the same security as RSA in a one-to-one communication and more security in broadcast applications.Comment: 18 pages, 1 figur