International Association for Cryptologic Research

International Association
for Cryptologic Research


Steven D. Galbraith


A Simpler and More Efficient Reduction of DLOG to CDH for Abelian Group Actions
Steven Galbraith Yi-Fu Lai Hart Montgomery
Abelian group actions appear in several areas of cryptography, especially isogeny-based post-quantum cryptography. A natural problem is to relate the analogues of the computational Diffie-Hellman (CDH) and discrete logarithm (DLOG) problems for abelian group actions. Galbraith, Panny, Smith and Vercauteren (Mathematical Cryptology '21) gave a quantum reduction of DLOG to CDH, assuming a CDH oracle with perfect correctness. Montgomery and Zhandry (Asiacrypt '22, best paper award) showed how to convert an unreliable CDH circuit into one that is correct with overwhelming probability. However, while a theoretical breakthrough, their reduction is quite inefficient: if the CDH oracle is correct with probability $q$ then their algorithm to amplify the success requires on the order of $1/q^{21}$ calls to the CDH oracle. We revisit this line of work and give a much simpler and tighter algorithm. Our method only takes on the order of $1/q^{4}$ CDH oracle calls and is much conceptually simpler than the Montgonery-Zhandry reduction. Our algorithm is also fully black-box, whereas the Montgomery-Zhandry algorithm is slightly non-black-box. Our main tool is a thresholding technique that replaces the comparison of distributions in Montgomery-Zhandry with testing equality of thresholded sets. We also give evidence that $1/q^{2}$ calls to the CDH oracle (or perhaps even more) is necessary, showing that it will be potentially difficult to substantially improve our method further.
SIDH Proof of Knowledge 📺
We show that the soundness proof for the De Feo--Jao--Plût identification scheme (the basis for supersingular isogeny Diffie--Hellman (SIDH) signatures) contains an invalid assumption, and we provide a counterexample for this assumption---thus showing the proof of soundness is invalid. As this proof was repeated in a number of works by various authors, multiple pieces of literature are affected by this result. Due to the importance of being able to prove knowledge of an SIDH key (for example, to prevent adaptive attacks), soundness is a vital property. Surprisingly, the problem of proving knowledge of a specific isogeny turns out to be considerably more difficult than was perhaps anticipated. The main results of this paper are a sigma protocol to prove knowledge of a walk of specified length in a supersingular isogeny graph, and a second one to additionally prove that the isogeny maps some torsion points to some other torsion points (as seen in SIDH public keys). Our scheme also avoids the SIDH identification scheme soundness issue raised by Ghantous, Pintore and Veroni. In particular, our protocol provides a non-interactive way of verifying correctness of SIDH public keys, and related statements, as protection against adaptive attacks. Post-scriptum: Some months after this work was completed and made public, the SIDH assumption was broken in a series of papers by several authors. Hence, in the standard SIDH setting, some of the statements studied here now have trivial polynomial time non-interactive proofs. Nevertheless our first sigma protocol is unaffected by the attacks, and our second protocol may still be useful in present and future variants of SIDH that escape the attacks.
Compact, Efficient and UC-Secure Isogeny-Based Oblivious Transfer 📺
Oblivious transfer (OT) is an essential cryptographic tool that can serve as a building block for almost all secure multiparty functionalities. The strongest security notion against malicious adversaries is universal composability (UC-secure). An important goal is to have post-quantum OT protocols. One area of interest for post-quantum cryptography is isogeny-based crypto. Isogeny-based cryptography has some similarities to Diffie-Hellman, but lacks some algebraic properties that are needed for discrete-log-based OT protocols. Hence it is not always possible to directly adapt existing protocols to the isogeny setting. We propose the first practical isogeny-based UC-secure oblivious transfer protocol in the presence of malicious adversaries. Our scheme uses the CSIDH framework and does not have an analogue in the Diffie-Hellman setting. The scheme consists of a constant number of isogeny computations. The underlying computational assumption is a problem that we call the computational reciprocal CSIDH problem, and that we prove polynomial-time equivalent to the computational CSIDH problem.
Identification Protocols and Signature Schemes Based on Supersingular Isogeny Problems
Steven D. Galbraith Christophe Petit Javier Silva
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.
Integral Matrix Gram Root and Lattice Gaussian Sampling without Floats 📺
Léo Ducas Steven D. Galbraith Thomas Prest Yang Yu
Many advanced lattice based cryptosystems require to sample lattice points from Gaussian distributions. One challenge for this task is that all current algorithms resort to floating-point arithmetic (FPA) at some point, which has numerous drawbacks in practice: it requires numerical stability analysis, extra storage for high-precision, lazy/backtracking techniques for efficiency, and may suffer from weak determinism which can completely break certain schemes. In this paper, we give techniques to implement Gaussian sampling over general lattices without using FPA. To this end, we revisit the approach of Peikert, using perturbation sampling. Peikert's approach uses continuous Gaussian sampling and some decomposition $\BSigma = \matA \matA^t$ of the target covariance matrix $\BSigma$. The suggested decomposition, e.g. the Cholesky decomposition, gives rise to a square matrix $\matA$ with real (not integer) entries. Our idea, in a nutshell, is to replace this decomposition by an integral one. While there is in general no integer solution if we restrict $\matA$ to being a square matrix, we show that such a decomposition can be efficiently found by allowing $\matA$ to be wider (say $n \times 9n$). This can be viewed as an extension of Lagrange's four-square theorem to matrices. In addition, we adapt our integral decomposition algorithm to the ring setting: for power-of-2 cyclotomics, we can exploit the tower of rings structure for improved complexity and compactness.
Safety in Numbers: On the Need for Robust Diffie-Hellman Parameter Validation
Steven D. Galbraith Jake Massimo Kenneth G. Paterson
We consider the problem of constructing Diffie-Hellman (DH) parameters which pass standard approaches to parameter validation but for which the Discrete Logarithm Problem (DLP) is relatively easy to solve. We consider both the finite field setting and the elliptic curve setting.For finite fields, we show how to construct DH parameters (p, q, g) for the safe prime setting in which $$p=2q+1$$ is prime, q is relatively smooth but fools random-base Miller-Rabin primality testing with some reasonable probability, and g is of order q mod p. The construction involves modifying and combining known methods for obtaining Carmichael numbers. Concretely, we provide an example with 1024-bit p which passes OpenSSL’s Diffie-Hellman validation procedure with probability $$2^{-24}$$ (for versions of OpenSSL prior to 1.1.0i). Here, the largest factor of q has 121 bits, meaning that the DLP can be solved with about $$2^{64}$$ effort using the Pohlig-Hellman algorithm. We go on to explain how this parameter set can be used to mount offline dictionary attacks against PAKE protocols. In the elliptic curve case, we use an algorithm of Bröker and Stevenhagen to construct an elliptic curve E over a finite field $${\mathbb {F}}_p$$ having a specified number of points n. We are able to select n of the form $$h\cdot q$$ such that h is a small co-factor, q is relatively smooth but fools random-base Miller-Rabin primality testing with some reasonable probability, and E has a point of order q. Concretely, we provide example curves at the 128-bit security level with $$h=1$$ , where q passes a single random-base Miller-Rabin primality test with probability 1/4 and where the elliptic curve DLP can be solved with about $$2^{44}$$ effort. Alternatively, we can pass the test with probability 1/8 and solve the elliptic curve DLP with about $$2^{35.5}$$ effort. These ECDH parameter sets lead to similar attacks on PAKE protocols relying on elliptic curves.Our work shows the importance of performing proper (EC)DH parameter validation in cryptographic implementations and/or the wisdom of relying on standardised parameter sets of known provenance.
SeaSign: Compact Isogeny Signatures from Class Group Actions 📺
Luca De Feo Steven D. Galbraith
We give a new signature scheme for isogenies that combines the class group actions of CSIDH with the notion of Fiat-Shamir with aborts. Our techniques allow to have signatures of size less than one kilobyte at the 128-bit security level, even with tight security reduction (to a non-standard problem) in the quantum random oracle model. Hence our signatures are potentially shorter than lattice signatures, but signing and verification are currently very expensive.
Obfuscated Fuzzy Hamming Distance and Conjunctions from Subset Product Problems
Steven D. Galbraith Lukas Zobernig
We consider the problem of obfuscating programs for fuzzy matching (in other words, testing whether the Hamming distance between an n-bit input and a fixed n-bit target vector is smaller than some predetermined threshold). This problem arises in biometric matching and other contexts. We present a virtual-black-box (VBB) secure and input-hiding obfuscator for fuzzy matching for Hamming distance, based on certain natural number-theoretic computational assumptions. In contrast to schemes based on coding theory, our obfuscator is based on computational hardness rather than information-theoretic hardness, and can be implemented for a much wider range of parameters. The Hamming distance obfuscator can also be applied to obfuscation of matching under the $$\ell _1$$ norm on $$\mathbb {Z}^n$$.We also consider obfuscating conjunctions. Conjunctions are equivalent to pattern matching with wildcards, which can be reduced in some cases to fuzzy matching. Our approach does not cover as general a range of parameters as other solutions, but it is much more compact. We study the relation between our obfuscation schemes and other obfuscators and give some advantages of our solution.
Improved Combinatorial Algorithms for the Inhomogeneous Short Integer Solution Problem
Shi Bai Steven D. Galbraith Liangze Li Daniel Sheffield
The paper is about algorithms for the inhomogeneous short integer solution problem: given $$(\mathbf A , \mathbf s )$$ ( A , s ) to find a short vector $$\mathbf{x }$$ x such that $$\mathbf A \mathbf{x }\equiv \mathbf s \pmod {q}$$ A x ≡ s ( mod q ) . We consider algorithms for this problem due to Camion and Patarin; Wagner; Schroeppel and Shamir; Minder and Sinclair; Howgrave–Graham and Joux (HGJ); Becker, Coron and Joux (BCJ). Our main results include: applying the Hermite normal form (HNF) to get faster algorithms; a heuristic analysis of the HGJ and BCJ algorithms in the case of density greater than one; an improved cryptanalysis of the SWIFFT hash function; a new method that exploits symmetries to speed up algorithms for Ring-SIS in some cases.
Elliptic Curve Paillier Schemes
Steven D. Galbraith

Program Committees

Eurocrypt 2023
Asiacrypt 2022
Asiacrypt 2021
PKC 2020
Asiacrypt 2019 (Program chair)
Asiacrypt 2018 (Program chair)
Crypto 2018
Asiacrypt 2017
Asiacrypt 2016
Crypto 2016
PKC 2015
Crypto 2015
Asiacrypt 2014
Eurocrypt 2013
PKC 2013
Eurocrypt 2012
Crypto 2012
Crypto 2011
Eurocrypt 2010
Crypto 2009
PKC 2008
Asiacrypt 2007
PKC 2007
Crypto 2007
Eurocrypt 2005