Quantum Security Analysis of CSIDH

International audienceCSIDH is a recent proposal for post-quantum non-interactive key-exchange, presented at ASIACRYPT 2018. Based on supersingular elliptic curve isogenies, it is similar in design to a previous scheme by Couveignes, Rostovtsev and Stolbunov, but aims at an improved balance between efficiency and security. In the proposal, the authors suggest concrete parameters in order to meet some desired levels of quantum security. These parameters are based on the hardness of recovering a hidden isogeny between two elliptic curves, using a quantum subexponential algorithm of Childs, Jao and Soukharev. This algorithm combines two building blocks: first, a quantum algorithm for recovering a hidden shift in a commutative group. Second, a computation in superposition of all isogenies originating from a given curve, which the algorithm calls as a black box.In this paper, we give a comprehensive security analysis of CSIDH. Our first step is to revisit three quantum algorithms for the abelian hidden shift problem from the perspective of non-asymptotic cost. There are many possible tradeoffs between the quantum and classical complexities of these algorithms and all of them should be taken into account by security levels. Second, we complete the non-asymptotic study of the black box in the hidden shift algorithm.This allows us to show that the parameters proposed by the authors of CSIDH do not meet their expected quantum security

He Gives C-Sieves on the CSIDH

Recently, Castryck, Lange, Martindale, Panny, and Renes proposed CSIDH (pronounced sea-side ) as a candidate post-quantum commutative group action. It has attracted much attention and interest, in part because it enables noninteractive Diffie--Hellman-like key exchange with quite small communication. Subsequently, CSIDH has also been used as a foundation for digital signatures. In 2003--04, Kuperberg and then Regev gave asymptotically subexponential quantum algorithms for hidden shift problems, which can be used to recover the CSIDH secret key from a public key. In late 2011, Kuperberg gave a follow-up quantum algorithm called the collimation sieve ( c-sieve for short), which improves the prior ones, in particular by using exponentially less quantum memory and offering more parameter tradeoffs. While recent works have analyzed the concrete cost of the original algorithms (and variants) against CSIDH, nothing of this nature was previously available for the c-sieve. This work fills that gap. Specifically, we generalize Kuperberg\u27s collimation sieve to work for arbitrary finite cyclic groups, provide some practical efficiency improvements, give a classical (i.e., non-quantum) simulator, run experiments for a wide range of parameters up to the actual CSIDH-512 group order, and concretely quantify the complexity of the c-sieve against CSIDH. Our main conclusion is that the proposed CSIDH parameters provide relatively little quantum security beyond what is given by the cost of quantumly evaluating the CSIDH group action itself (on a uniform superposition). For example, the cost of CSIDH-512 key recovery is only about $2^{16}$ quantum evaluations using $2^{40}$ bits of quantumly accessible classical memory (plus relatively small other resources). This improves upon a prior estimate of $2^{32.5}$ evaluations and $2^{31}$ qubits of quantum memory, for a variant of Kuperberg\u27s original sieve. Under the plausible assumption that quantum evaluation does not cost much more than what is given by a recent best case analysis, CSIDH-512 can therefore be broken using significantly less than $2^{64}$ quantum T-gates. This strongly invalidates its claimed NIST level 1 quantum security, especially when accounting for the MAXDEPTH restriction. Moreover, under analogous assumptions for CSIDH-1024 and -1792, which target higher NIST security levels, except near the high end of the MAXDEPTH range even these instantiations fall short of level 1

On the Practicality of Post-Quantum TLS Using Large-Parameter CSIDH

The isogeny-based scheme CSIDH is considered to be the only efficient post-quantum non-interactive key exchange (NIKE) and poses small bandwidth requirements, thus appearing to be an attractive alternative for classical Diffie--Hellman schemes. A crucial CSIDH design point, still under debate, is its quantum security when using prime fields of 512 to 1024 bits. Most work has focused on prime fields of that size and the practicality of CSIDH with large parameters, 2000 to 9000 bits, has so far not been thoroughly assessed, even though analysis of quantum security suggests these parameter sizes. We fill this gap by providing two CSIDH instantiations: A deterministic and dummy-free instantiation based on SQALE, aiming at high security against physical attacks, and a speed-optimized constant-time instantiation that adapts CTIDH to larger parameter sizes. We provide implementations of both variants, including efficient field arithmetic for fields of such size, and high-level optimizations. Our deterministic and dummy-free version, dCSIDH, is almost twice as fast as SQALE, and, dropping determinism, CTIDH at these parameters is thrice as fast as dCSIDH. We investigate their use in real-world scenarios through benchmarks of TLS using our software. Although our instantiations of CSIDH have smaller communication requirements than post-quantum KEM and signature schemes, both implementations still result in too-large handshake latency (tens of seconds), which hinder further consideration of using CSIDH in practice for conservative parameter set instantiations

SCALLOP:Scaling the CSI-FiSh

International audienceWe present SCALLOP: SCALable isogeny action based on Oriented supersingular curves with Prime conductor, a new group action based on isogenies of supersingular curves. Similarly to CSIDH and OSIDH, we use the group action of an imaginary quadratic order’s class group on the set of oriented supersingular curves. Compared to CSIDH, the main benefit of our construction is that it is easy to compute the class-group structure; this data is required to uniquely represent—and efficiently act by — arbitrary group elements, which is a requirement in, e.g., the CSI-FiSh signature scheme by Beullens, Kleinjung and Vercauteren. The index-calculus algorithm used in CSI-FiSh to compute the class-group structure has complexity L(1/2), ruling out class groups much larger than CSIDH-512, a limitation that is particularly problematic in light of the ongoing debate regarding the quantum security of cryptographic group actions.Hoping to solve this issue, we consider the class group of a quadratic order of large prime conductor inside an imaginary quadratic field of small discriminant. This family of quadratic orders lets us easily determine the size of the class group, and, by carefully choosing the conductor, even exercise significant control on it—in particular supporting highly smooth choices. Although evaluating the resulting group action still has subexponential asymptotic complexity, a careful choice of parameters leads to a practical speedup that we demonstrate in practice for a security level equivalent to CSIDH-1024, a parameter currently firmly out of reach of index-calculus-based methods. However, our implementation takes 35 seconds (resp. 12.5 minutes) for a single group-action evaluation at a CSIDH-512-equivalent (resp. CSIDH-1024-equivalent) security level, showing that, while feasible, the SCALLOP group action does not achieve realistically usable performance yet

Improved Low-qubit Hidden Shift Algorithms

Hidden shift problems are relevant to assess the quantum security of various cryptographic constructs. Multiple quantum subexponential time algorithms have been proposed. In this paper, we propose some improvements on a polynomial quantum memory algorithm proposed by Childs, Jao and Soukharev in 2010. We use subset-sum algorithms to significantly reduce its complexity. We also propose new tradeoffs between quantum queries, classical time and classical memory to solve this problem