Still Wrong Use of Pairings in Cryptography

Several pairing-based cryptographic protocols are recently proposed with a wide variety of new novel applications including the ones in emerging technologies like cloud computing, internet of things (IoT), e-health systems and wearable technologies. There have been however a wide range of incorrect use of these primitives. The paper of Galbraith, Paterson, and Smart (2006) pointed out most of the issues related to the incorrect use of pairing-based cryptography. However, we noticed that some recently proposed applications still do not use these primitives correctly. This leads to unrealizable, insecure or too inefficient designs of pairing-based protocols. We observed that one reason is not being aware of the recent advancements on solving the discrete logarithm problems in some groups. The main purpose of this article is to give an understandable, informative, and the most up-to-date criteria for the correct use of pairing-based cryptography. We thereby deliberately avoid most of the technical details and rather give special emphasis on the importance of the correct use of bilinear maps by realizing secure cryptographic protocols. We list a collection of some recent papers having wrong security assumptions or realizability/efficiency issues. Finally, we give a compact and an up-to-date recipe of the correct use of pairings.Comment: 25 page

Discrete logarithms in curves over finite fields

A survey on algorithms for computing discrete logarithms in Jacobians of curves over finite fields

Finding Significant Fourier Coefficients: Clarifications, Simplifications, Applications and Limitations

Ideas from Fourier analysis have been used in cryptography for the last three decades. Akavia, Goldwasser and Safra unified some of these ideas to give a complete algorithm that finds significant Fourier coefficients of functions on any finite abelian group. Their algorithm stimulated a lot of interest in the cryptography community, especially in the context of `bit security'. This manuscript attempts to be a friendly and comprehensive guide to the tools and results in this field. The intended readership is cryptographers who have heard about these tools and seek an understanding of their mechanics and their usefulness and limitations. A compact overview of the algorithm is presented with emphasis on the ideas behind it. We show how these ideas can be extended to a `modulus-switching' variant of the algorithm. We survey some applications of this algorithm, and explain that several results should be taken in the right context. In particular, we point out that some of the most important bit security problems are still open. Our original contributions include: a discussion of the limitations on the usefulness of these tools; an answer to an open question about the modular inversion hidden number problem

Pairing-based identification schemes

We propose four different identification schemes that make use of bilinear pairings, and prove their security under certain computational assumptions. Each of the schemes is more efficient and/or more secure than any known pairing-based identification scheme

Identity based cryptography from bilinear pairings

This report contains an overview of two related areas of research in cryptography which have been prolific in significant advances in recent years. The first of these areas is pairing based cryptography. Bilinear pairings over elliptic curves were initially used as formal mathematical tools and later as cryptanalysis tools that rendered supersingular curves insecure. In recent years, bilinear pairings have been used to construct many cryptographic schemes. The second area covered by this report is identity based cryptography. Digital certificates are a fundamental part of public key cryptography, as one needs a secure way of associating an agent’s identity with a random (meaningless) public key. In identity based cryptography, public keys can be arbitrary bit strings, including readable representations of one’s identity.Fundação para a Ci~Encia e Tecnologia - SFRH/BPD/20528/2004

Revisiting the security model for aggregate signature schemes

Aggregate signature schemes combine the digital signatures of multiple users on different messages into one single signature. The Boneh-Gentry-Lynn-Shacham (BGLS) aggregate signature scheme is one such scheme, based on pairings, where anyone can aggregate the signatures in any order. We suggest improvements to its current chosen-key security model. In particular, we argue that the scheme should be resistant to attackers that can adaptively choose their target users, and either replace other users' public keys or expose other users' private keys. We compare these new types of forgers to the original targeted-user forger, building up to the stronger replacement-and-exposure forger. Finally, we present a security reduction for a variant of the BGLS aggregate signature scheme with respect to this new notion of forgery. Recent attacks by Joux and others on the discrete logarithm problem in small-characteristic finite fields dramatically reduced the security of many type I pairings. Therefore, we explore security reductions for BGLS with type III rather than type I pairings. Although our reductions are specific to BGLS, we believe that other aggregate signature schemes could benefit from similar changes to their security models

Bit Security of the CDH Problems over Finite Field

It is a long-standing open problem to prove the existence of (deterministic) hard-core predicates for the Computational Diffie-Hellman (CDH) problem over finite fields, without resorting to the generic approaches for any one-way functions (e.g., the Goldreich-Levin hard-core predicates). Fazio et al. (FGPS, Crypto \u2713) make important progress on this problem by defining a weaker Computational Diffie-Hellman problem over $\mathbb{F}_{p^2}$, i.e., Partial-CDH problem, and proving, when allowing changing field representations, the unpredictability of every single bit of one of the coordinates of the secret Diffie-Hellman value

Isogeny-based post-quantum key exchange protocols

The goal of this project is to understand and analyze the supersingular isogeny Diffie Hellman (SIDH), a post-quantum key exchange protocol which security lies on the isogeny-finding problem between supersingular elliptic curves. In order to do so, we first introduce the reader to cryptography focusing on key agreement protocols and motivate the rise of post-quantum cryptography as a necessity with the existence of the model of quantum computation. We review some of the known attacks on the SIDH and finally study some algorithmic aspects to understand how the protocol can be implemented