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

Still Wrong Use of Pairings in Cryptography

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.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 ine cient 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/e ciency issues. Finally, we give a compact and an up-to-date recipe of the correct use of pairings

Plumo: An Ultralight Blockchain Client

Syncing the latest state of a blockchain can be a resource-intensive task, driving (especially mobile) end users towards centralized services offering instant access. To expand full decentralized access to anyone with a mobile phone, we introduce a consensus-agnostic compiler for constructing ultralight clients, providing secure and highly efficient blockchain syncing via a sequence of SNARK-based state transition proofs, and prove its security formally. Instantiating this, we present Plumo, an ultralight client for the Celo blockchain capable of syncing the latest network state summary in just a few seconds even on a low-end mobile phone. In Plumo, each transition proof covers four months of blockchain history and can be produced for just $25 USD of compute. Plumo achieves this level of efficiency thanks to two new SNARK-friendly constructions, which may also be of independent interest: a new BLS-based offline aggregate multisignature scheme in which signers do not have to know the members of their multisignature group in advance, and a new composite algebraic-symmetric cryptographic hash function

Plumo: An Ultralight Blockchain Client

Syncing the latest state of a blockchain can be a resource-intensive task, driving (especially mobile) end users towards centralized services offering instant access. To expand full decentralized access to anyone with a mobile phone, we introduce a consensus-agnostic compiler for constructing {\em ultralight clients}, providing secure and highly efficient blockchain syncing via a sequence of SNARK-based state transition proofs, and prove its security formally. Instantiating this, we present Plumo, an ultralight client for the Celo blockchain capable of syncing the latest network state summary in just a few seconds even on a low-end mobile phone. In Plumo, each transition proof covers four months of blockchain history and can be produced for just $25 USD of compute. Plumo achieves this level of efficiency thanks to two new SNARK-friendly constructions, which may also be of independent interest: a new BLS-based offline aggregate multisignature scheme in which signers do not have to know the members of their multisignature group in advance, and a new composite algebraic-symmetric cryptographic hash function

A short-list of pairing-friendly curves resistant to Special TNFS at the 128-bit security level

https://www.iacr.org/docs/pub_2013-16.htmlThis paper is the IACR version. It can be made freely available on the homepages of authors, on their employer's institutional page, and in non-commercial archival repositories such as the Cryptology ePrint Archive, ArXiv/CoRR, HAL, etc.International audienceThere have been notable improvements in discrete logarithm computations in finite fields since 2015 and the introduction of the Tower Number Field Sieve algorithm (TNFS) for extension fields. The Special TNFS is very efficient in finite fields that are target groups of pairings on elliptic curves, where the characteristic is special (e.g.~sparse). The key sizes for pairings should be increased, and alternative pairing-friendly curves can be considered.We revisit the Special variant of TNFS for pairing-friendly curves. In this case the characteristic is given by a polynomial of moderate degree (between 4 and 38) and tiny coefficients, evaluated at an integer (a seed). We present a polynomial selection with a new practical trade-off between degree and coefficient size. As a consequence, the security of curves computed by Barbulescu, El~Mrabet and Ghammam in 2019 should be revised: we obtain a smaller estimated cost of STNFS for all curves except BLS12 and BN.To obtain TNFS-secure curves, we reconsider the Brezing--Weng generic construction of families of pairing-friendly curves and estimate the cost of our new Special TNFS algorithm for these curves. This improves on the work of Fotiadis and Konstantinou, Fotiadis and Martindale, and Barbulescu, El~Mrabet and Ghammam. We obtain a short-list of interesting families of curves that are resistant to the Special TNFS algorithm, of embedding degrees 10 to 16 for the 128-bit security level. We conclude that at the 128-bit security level, BLS-12 and Fotiadis--Konstantinou--Martindale curves with $k=12$ over a 440 to 448-bit prime field seem to be the best choice for pairing efficiency. We also give hints at the 192-bit security level

M-SIDH and MD-SIDH: countering SIDH attacks by masking information

The SIDH protocol is an isogeny-based key exchange protocol using supersingular isogenies, designed by Jao and De Feo in 2011. The protocol underlies the SIKE algorithm which advanced to the fourth round of NIST\u27s post-quantum standardization project in May 2022. The algorithm was considered very promising: indeed the most significant attacks against SIDH were meet-in-the-middle variants with exponential complexity, and torsion point attacks which only applied to unbalanced parameters (and in particular, not to SIKE). This security picture dramatically changed in August 2022 with new attacks by Castryck-Decru, Maino-Martindale and Robert. Like prior attacks on unbalanced versions, these new attacks exploit torsion point information provided in the SIDH protocol. Crucially however, the new attacks embed the isogeny problem into a similar isogeny problem in a higher dimension to also affect the balanced parameters. As a result of these works, the SIKE algorithm is now fully broken both in theory and in practice. Given the considerable interest attracted by SIKE and related protocols in recent years, it is natural to seek countermeasures to the new attacks. In this paper, we introduce two such countermeasures based on partially hiding the isogeny degrees and torsion point information in the SIDH protocol. We present a preliminary analysis of the resulting schemes including non-trivial generalizations of prior attacks. Based on this analysis we suggest parameters for our M-SIDH variant with public key sizes of 4434, 7037 and 9750 bytes respectively for NIST security levels 1, 3, 5

Fractal: Post-Quantum and Transparent Recursive Proofs from Holography

We present a new methodology to efficiently realize recursive composition of succinct non-interactive arguments of knowledge (SNARKs). Prior to this work, the only known methodology relied on pairing-based SNARKs instantiated on cycles of pairing-friendly elliptic curves, an expensive algebraic object. Our methodology does not rely on any special algebraic objects and, moreover, achieves new desirable properties: it is *post-quantum* and it is *transparent* (the setup is public coin). We exploit the fact that recursive composition is simpler for SNARKs with *preprocessing*, and the core of our work is obtaining a preprocessing zkSNARK for rank-1 constraint satisfiability (R1CS) that is post-quantum and transparent. We obtain this latter by establishing a connection between holography and preprocessing in the random oracle model, and then constructing a holographic proof for R1CS. We experimentally validate our methodology, demonstrating feasibility in practice

Security Analysis of Isogeny-Based Cryptosystems

Let $E$ be a supersingular elliptic curve over a finite field. In this document we study public-key encryption schemes which use non-constant rational maps from $E$. The purpose of this study is to determine if such cryptosystems are secure. Supersingular Isogeny Diffie-Hellman (SIDH) and other supersingular isogeny-based cryptosystems are considered. The content is naturally divided by cryptosystem, and in the case of SIDH, further divided by type of cryptanalysis: SIDH when the endomorphism ring of the base elliptic curve is given (as is done in practice), repeated use of keys in SIDH, and endomorphism ring constructing algorithms. In each case the relevent background material is presented to develop the theory. In studying the security of SIDH when the endomorphism ring of the base curve $E$ is known, one of the main results is the following. This theorem is then used to reduce the security of such an SIDH instantiation to the problem of finding particular endomorphisms in \End(E). \begin{thm} Given \begin{enumerate} \item a supersingular elliptic curve E/\FQ such that $p = N_1 N_2 - 1$ for coprime $N_1\approx N_2$, where $N_2$ is $\log p$-smooth, \item an elliptic curve $E'$ that is the codomain of an $N_1$-isogeny $\phi:E\rightarrow E'$, \item the action of $\phi$ on $E[N_2]$, and \item a $k$-endomorphism $\psi$ of $E$, where $\gcd(k, N_1) = 1$, and if \g is the greatest integer such that $g\mid N_2^2$ and $g\mid k$, then \h := \frac{k}{g} < N_1, \end{enumerate} there exists a classical algorithm with worst case runtime \tilde{O}(\h^3) which decides whether $\psi(\ker\phi) = \ker\phi$ or not, but may give false positives with probability $\approx \frac{1}{\sqrt{p}}$. Further, if \h is $\log{p}$-smooth, then the runtime is \tilde{O} (\sqrt{\h}). \end{thm} In studying the security of repeated use of SIDH public keys, the main result presented is the following theorem, which proves that performing multiple pairwise instances of SIDH prevents certain active attacks when keys are reused. \begin{thm} Assuming that the CSSI problem is intractable, it is computationally infeasible for a malicious adversary, with non-negligible probability, to modify a public key $(E_B,\phi_B(P_A),\phi_B(Q_A))$ to some $(E_B,R,S)$ which is malicious for SIDH. \end{thm} It is well known that the problem of computing hidden supersingular isogenies can be reduced to computing the endomorphism rings of the domain and codomain elliptic curves. A novel algorithm for computing an order in the endomorphism ring of a supersingular elliptic curve is presented and analyzed to have runtime $O(p^{1/2}(\log p)^2)$. In studying non-SIDH cryptosystems, four other isogeny-based cryptosystems are examined. The first three were all proposed by the same authors and use secret endomorphisms. These are each shown to be either totally insecure (private keys can be recovered directly from public keys) or impractical to implement efficiently. The fourth scheme is a novel proposal which attempts to combine isogenies with the learning with errors problem. This proposal is also shown to be totally insecure

On Small Degree Extension Fields in Cryptology

This thesis studies the implications of using public key cryptographic primitives that are based in, or map to, the multiplicative group of finite fields with small extension degree. A central observation is that the multiplicative group of extension fields essentially decomposes as a product of algebraic tori, whose properties allow for improved communication efficiency. Part I of this thesis is concerned with the constructive implications of this idea. Firstly, algorithms are developed for the efficient implementation of torus-based cryptosystems and their performance compared with previous work. It is then shown how to apply these methods to operations required in low characteristic pairing-based cryptography. Finally, practical schemes for high-dimensional tori are discussed. Highly optimised implementations and benchmark timings are provided for each of these systems. Part II addresses the security of the schemes presented in Part I, i.e., the hardness of the discrete logarithm problem. Firstly, an heuristic analysis of the effectiveness of the Function Field Sieve in small characteristic is given. Next presented is an implementation of this algorithm for characteristic three fields used in pairing-based cryptography. Finally, a new index calculus algorithm for solving the discrete logarithm problem on algebraic tori is described and analysed