## CryptoDB

### Christophe Petit

#### Publications

Year
Venue
Title
2021
EUROCRYPT
Supersingular isogeny Diffie-Hellman key exchange (SIDH) is a post-quantum protocol based on the presumed hardness of computing an isogeny between two supersingular elliptic curves given some additional torsion point information. Unlike other isogeny-based protocols, SIDH has been widely believed to be immune to subexponential quantum attacks because of the non-commutative structure of the endomorphism rings of supersingular curves. We contradict this commonly believed misconception in this paper. More precisely, we highlight the existence of an abelian group action on the SIDH key space, and we show that for sufficiently \emph{unbalanced} and \emph{overstretched} SIDH parameters, this action can be efficiently computed (heuristically) using the torsion point information revealed in the protocol. This reduces the underlying hardness assumption to a hidden shift problem instance which can be solved in quantum subexponential time. We formulate our attack in a new framework allowing the inversion of one-way functions in quantum subexponential time provided a malleability oracle with respect to some commutative group action. This framework unifies our new attack with earlier subexponential quantum attacks on isogeny-based protocols, and it may be of further interest for cryptanalysis.
2021
CRYPTO
SIDH is a post-quantum key exchange algorithm based on the presumed difficulty of finding isogenies between supersingular elliptic curves. However, SIDH and related cryptosystems also reveal additional information: the restriction of a secret isogeny to a subgroup of the curve (torsion-point information). Petit [31] was the first to demonstrate that torsion-point information could noticeably lower the difficulty of finding secret isogenies. In particular, Petit showed that "overstretched'' parameterizations of SIDH could be broken in polynomial time. However, this did not impact the security of any cryptosystems proposed in the literature. The contribution of this paper is twofold: First, we strengthen the techniques of [31] by exploiting additional information coming from a dual and a Frobenius isogeny. This extends the impact of torsion-point attacks considerably. In particular, our techniques yield a classical attack that completely breaks the $n$-party group key exchange of [2], first introduced as GSIDH in [17], for 6 parties or more, and a quantum attack for 3 parties or more that improves on the best known asymptotic complexity. We also provide a Magma implementation of our attack for 6 parties. We give the full range of parameters for which our attacks apply. Second, we construct SIDH variants designed to be weak against our attacks; this includes backdoor choices of starting curve, as well as backdoor choices of base-field prime. We stress that our results do not degrade the security of, or reveal any weakness in, the NIST submission SIKE [20].
2021
ASIACRYPT
We cryptanalyse the SIDH-based oblivious pseudorandom function from supersingular isogenies proposed at Asiacrypt'20 by Boneh, Kogan and Woo. To this end, we give an attack on an assumption, the auxiliary one-more assumption, that was introduced by Boneh et al. and we show that this leads to an attack on the oblivious PRF itself. The attack breaks the pseudorandomness as it allows adversaries to evaluate the OPRF without further interactions with the server after some initial OPRF evaluations and some offline computations. More specifically, we first propose a polynomial-time attack. Then, we argue it is easy to change the OPRF protocol to include some countermeasures, and present a second subexponential attack that succeeds in the presence of said countermeasures. Both attacks break the security parameters suggested by Boneh et al. Furthermore, we provide a proof of concept implementation as well as some timings of our attack. Finally, we examine the generation of one of the OPRF parameters and argue that a trusted third party is needed to guarantee provable security.
2021
ASIACRYPT
We present Séta, a new family of public-key encryption schemes with post-quantum security based on isogenies of supersingular elliptic curves. It is constructed from a new family of trapdoor one-way functions, where the inversion algorithm uses Petit's so called \emph{torsion attacks} on SIDH to compute an isogeny between supersingular elliptic curves given an endomorphism of the starting curve and images of torsion points. We prove the OW-CPA security of S\'eta and present an IND-CCA variant using the post-quantum OAEP transformation. Several variants for key generation are explored together with their impact on the selection of parameters, such as the base prime of the scheme. We furthermore formalise an uber'' isogeny assumption framework which aims to generalize computational isogeny problems encountered in schemes including SIDH, CSDIH, OSIDH and ours. Finally, we carefully select parameters to achieve a balance between security and run-times and present experimental results from our implementation.
2021
ASIACRYPT
In 2016, Galbraith et al. presented an adaptive attack on the SIDH key exchange protocol. In SIKE, one applies a variant of the Fujisaki-Okamoto transform to force Bob to reveal his encryption key to Alice, which Alice then uses to re-encrypt Bob's ciphertext and verify its validity. Therefore, Bob can not reuse his encryption keys. There have been two other proposed countermeasures enabling static-static private keys: k-SIDH and its variant by Jao and Urbanik. These countermeasures are relatively expensive since they consist in running multiple parallel instances of SIDH. In this paper, firstly, we propose a new countermeasure to the GPST adaptive attack on SIDH. Our countermeasure does not require key disclosure as in SIKE, nor multiple parallel instances as in k-SIDH. We translate our countermeasure into a key validation method for SIDH-type schmes. Secondly, we use our key validation to design HealSIDH, an efficient SIDH-type static-static key interactive exchange protocol. Thirdly, we derive a PKE scheme SHealS using HealSIDH. SHealS uses larger primes compared to SIKE, has larger keys and ciphertexts, but only $4$ isogenies are computed in a full execution of the scheme, as opposed to $5$ isogenies in SIKE. We prove that SHealS is IND-CPA secure relying on a new assumption we introduce and we conjecture its IND-CCA security. We suggest HealS, a variant of SHealS using a smaller prime, providing smaller keys and ciphertexts. As a result, HealSIDH is a practically efficient SIDH based (interactive) key exchange incorporating a "direct" countermeasure to the GPST adaptive attack.
2020
JOFC
We present signature schemes whose security relies on computational assumptions relating to isogeny graphs of supersingular elliptic curves. We give two schemes, both of them based on interactive identification protocols. The first identification protocol is due to De Feo, Jao and Plût. The second one, and the main contribution of the paper, makes novel use of an algorithm of Kohel, Lauter, Petit and Tignol for the quaternion version of the $\ell$ ℓ -isogeny problem, for which we provide a more complete description and analysis, and is based on a more standard and potentially stronger computational problem. Both identification protocols lead to signatures that are existentially unforgeable under chosen message attacks in the random oracle model using the well-known Fiat-Shamir transform, and in the quantum random oracle model using another transform due to Unruh. A version of the first signature scheme was independently published by Yoo, Azarderakhsh, Jalali, Jao and Soukharev. This is the full version of a paper published at ASIACRYPT 2017.
2020
ASIACRYPT
We introduce a new signature scheme, \emph{SQISign}, (for \emph{Short Quaternion and Isogeny Signature}) from isogeny graphs of supersingular elliptic curves. The signature scheme is derived from a new one-round, high soundness, interactive identification protocol. Targeting the post-quantum NIST-1 level of security, our implementation results in signatures of $204$ bytes, secret keys of $16$ bytes and public keys of $64$ bytes. In particular, the signature and public key sizes combined are an order of magnitude smaller than all other post-quantum signature schemes. On a modern workstation, our implementation in C takes 0.6s for key generation, 2.5s for signing, and 50ms for verification. While the soundness of the identification protocol follows from classical assumptions, the zero-knowledge property relies on the second main contribution of this paper. We introduce a new algorithm to find an isogeny path connecting two given supersingular elliptic curves of known endomorphism rings. A previous algorithm to solve this problem, due to Kohel, Lauter, Petit and Tignol, systematically reveals paths from the input curves to a `special' curve. This leakage would break the zero-knowledge property of the protocol. Our algorithm does not directly reveal such a path, and subject to a new computational assumption, we prove that the resulting identification protocol is zero-knowledge.
2019
PKC
Braid groups are infinite non-abelian groups naturally arising from geometric braids. For two decades they have been proposed for cryptographic use. In braid group cryptography public braids often contain secret braids as factors and it is hoped that rewriting the product of braid words hides individual factors. We provide experimental evidence that this is in general not the case and argue that under certain conditions parts of the Garside normal form of factors can be found in the Garside normal form of their product. This observation can be exploited to decompose products of braids of the form ABC when only B is known.Our decomposition algorithm yields a universal forgery attack on WalnutDSATM, which is one of the 20 proposed signature schemes that are being considered by NIST for standardization of quantum-resistant public-key cryptography. Our attack on WalnutDSATM can universally forge signatures within seconds for both the 128-bit and 256-bit security level, given one random message-signature pair. The attack worked on 99.8% and 100% of signatures for the 128-bit and 256-bit security levels in our experiments.Furthermore, we show that the decomposition algorithm can be used to solve instances of the conjugacy search problem and decomposition search problem in braid groups. These problems are at the heart of other cryptographic schemes based on braid groups.
2019
ASIACRYPT
We present two new Verifiable Delay Functions (VDF) based on assumptions from elliptic curve cryptography. We discuss both the advantages and drawbacks of our constructions, we study their security and we demonstrate their practicality with a proof-of-concept implementation.
2018
EUROCRYPT
2018
PKC
We present a practical cryptanalysis of WalnutDSA, a digital signature algorithm trademarked by SecureRF. WalnutDSA uses techniques from permutation groups, matrix groups and braid groups, and is designed to provide post-quantum security in lightweight IoT device contexts. The attack given in this paper bypasses the E-Multiplication$^{\text {TM}}$TM and cloaked conjugacy search problems at the heart of the algorithm and forges signatures for arbitrary messages in approximately two minutes. We also discuss potential countermeasures to the attack.
2017
ASIACRYPT
2017
ASIACRYPT
2016
EUROCRYPT
2016
PKC
2016
ASIACRYPT
2012
EUROCRYPT
2012
ASIACRYPT

PKC 2022
PKC 2021
PKC 2020
Crypto 2019
PKC 2017
PKC 2016