International Association
for Cryptologic Research

<p> Side-channel attacks targeting cryptography may leak only partial or indirect information about the secret keys. There are a variety of techniques in the literature for recovering secret keys from partial information. In this work, we survey several of the main families of partial key recovery algorithms for RSA, (EC)DSA, and (elliptic curve) Diffie-Hellman, the classical public-key cryptosystems in common use today. We categorize the known techniques by the structure of the information that is learned by the attacker, and give simplified examples for each technique to illustrate the underlying ideas. </p>

In this paper, we study both the implications and potential impact of backdoored parameters for two RSA-based pseudorandom number generators: the ISO-standardized Micali-Schnorr generator and a closely related design, the RSA PRG. We observe, contrary to common understanding, that the security of the Micali-Schnorr PRG is not tightly bound to the difficulty of inverting RSA. We show that the Micali-Schnorr construction remains secure even if one replaces RSA with a publicly evaluatable PRG, or a function modeled as an efficiently invertible random permutation. This implies that any cryptographic backdoor must somehow exploit the algebraic structure of RSA, rather than an attacker’s ability to invert RSA or the presence of secret keys. We exhibit two such backdoors in related constructions: a family of exploitable parameters for the RSA PRG, and a second vulnerable construction for a finite-field variant of Micali-Schnorr. We also observe that the parameters allowed by the ISO standard are incompletely specified, and allow insecure choices of exponent. Several of our backdoor constructions make use of lattice techniques, in particular multivariate versions of Coppersmith’s method for finding small solutions to polynomials modulo integers.

We present an efficient quantum algorithm for solving the semidirect discrete logarithm problem ($\SDLP$) in \emph{any} finite group. The believed hardness of the semidirect discrete logarithm problem underlies more than a decade of works constructing candidate post-quantum cryptographic algorithms from non-abelian groups. We use a series of reduction results to show that it suffices to consider $\SDLP$ in finite simple groups. We then apply the celebrated Classification of Finite Simple Groups to consider each family. The infinite families of finite simple groups admit, in a fairly general setting, linear algebraic attacks providing a reduction to the classical discrete logarithm problem. For the sporadic simple groups, we show that their inherent properties render them unsuitable for cryptographically hard $\SDLP$ instances, which we illustrate via a Baby-Step Giant-Step style attack against $\SDLP$ in the Monster Group. Our quantum $\SDLP$ algorithm is fully constructive, up to the computation of maximal normal subgroups, for all but three remaining cases that appear to be gaps in the literature on constructive recognition of groups; for these cases $\SDLP$ is no harder than finding a linear representation. We conclude that $\SDLP$ is not a suitable post-quantum hardness assumption for any choice of finite group.

In early 2021, Apple announced the AirTag: a quarter-sized low-powered device that utilizes the privacy-preserving FindMy network to find physical objects. The release of Airtags has been highly controversial, in part because stalkers have misused them to track potential victims. In response to this threat, Apple came up with a strategy to detect stalkers at the cost of innocent AirTag users's privacy. Their methodology is currently in the process of being standardized by the IETF. In this talk, we will show that the hard trade-off presented by Apple is not necessary and that it is possible to efficiently achieve both privacy and stalker detection. We hope that by bringing this pressing issue to the attention of the community, we can spur more meaningful discussion on what privacy properties offline-finding networks should provide and incentivize the adoption of more privacy-preserving protocols.

In recent work, Backendal, Haller, and Paterson identified several exploitable vulnerabilities in the cloud storage provider MEGA. They demonstrated an RSA key recovery attack in which a malicious server could recover a client's private RSA key after 512 client login attempts. We show how to exploit additional information revealed by MEGA's protocol vulnerabilities to give an attack that requires only six client logins to recover the secret key. Our optimized attack combines several cryptanalytic techniques. In particular, we formulate and give a solution to a variant of the hidden number problem with small unknown multipliers, which may be of independent interest. We show that our lattice construction for this problem can be used to give improved results for the implicit factorization problem of May and Ritzenhofen.

We introduce a new lattice basis reduction algorithm with approximation guarantees analogous to the LLL algorithm and practical performance that far exceeds the current state of the art. We achieve these results by iteratively applying precision management techniques within a recursive algorithm structure and show the stability of this approach. We analyze the asymptotic behavior of our algorithm, and show that the heuristic running time is $O(n^{\omega}(C+n)^{1+\varepsilon})$ for lattices of dimension $n$, $\omega\in (2,3]$ bounding the cost of size reduction, matrix multiplication, and QR factorization, and $C$ bounding the log of the condition number of the input basis $B$. This yields a running time of $O\left(n^\omega (p + n)^{1 + \varepsilon}\right)$ for precision $p = O(\log \|B\|_{max})$ in common applications. Our algorithm is fully practical, and we have published our implementation. We experimentally validate our heuristic, give extensive benchmarks against numerous classes of cryptographic lattices, and show that our algorithm significantly outperforms existing implementations.

Dual EC DRBG is widely believed to have been backdoored by the U.S. National Security Agency. But there was another number theoretic PRG proposed alongside Dual EC that has seen surprisingly little attention: the Micali-Schnorr generator, standardized as MS DRBG, which is based on the hardness of RSA. It appears in early drafts of the ANSI X9.82 standard (but was eventually removed in favor of Dual EC) and the final version of ISO 18031 (alongside Dual EC). The MS DRBG standard follows a pattern eerily reminiscent of Dual EC: it incorporates a series of recommended public parameters that are intended to be used in production as the RSA modulus N. Given the known vulnerabilities in Dual EC and the identical provenance, it is reasonable to ask whether MS DRBG is vulnerable to an analogous attack: Does knowledge of the factors of (or malicious construction of) the recommended moduli imply a practical attack on the MS DRBG generator? Surprisingly, this question is not easy to answer. The security proofs of course do not go through if the factors are known, but all obvious attack strategies fail. In this talk, we give historical background on MS DRBG and describe progress toward finding the backdoor (or proving it doesn't exist). We show that any backdoor must somehow exploit the algebraic structure of RSA, rather than just the attacker's ability to invert the RSA operation. We exhibit two such backdoors in related constructions. Ultimately we were unsuccessful in fully finding a plausible backdoor in MS DRBG (or proving one doesn't exist), but we hope this talk will bring more attention to this interesting open problem with potential real-world impact.

Lattice-based algorithms in cryptanalysis often search for a target vector satisfying integer linear constraints as a shortest or closest vector in some lattice. In this work, we observe that these formulations may discard non-linear information from the underlying application that can be used to distinguish the target vector even when it is far from being uniquely close or short. We formalize lattice problems augmented with a predicate distinguishing a target vector and give algorithms for solving instances of these prob- lems. We apply our techniques to lattice-based approaches for solving the Hidden Number Problem, a popular technique for recovering secret DSA or ECDSA keys in side-channel attacks, and demonstrate that our algorithms succeed in recovering the signing key for instances that were previously believed to be unsolvable using lattice approaches. We carried out extensive experiments using our estimation and solving framework, which we also make available with this work.

We report on two new records: the factorization of RSA-240, a 795-bit number, and a discrete logarithm computation over a 795-bit prime field. Previous records were the factorization of RSA-768 in 2009 and a 768-bit discrete logarithm computation in 2016. Our two computations at the 795-bit level were done using the same hardware and software, and show that computing a discrete logarithm is not much harder than a factorization of the same size. Moreover, thanks to algorithmic variants and well-chosen parameters, our computations were significantly less expensive than anticipated based on previous records. The last page of this paper also reports on the factorization of RSA-250.

Intel Software Guard Extensions (SGX) allows users to perform secure computation on platforms that run untrusted software. To validate that the computation is correctly initialized and that it executes on trusted hardware, SGX supports attestation providers that can vouch for the user’s computation. Communication with these attestation providers is based on the Extended Privacy ID (EPID) protocol, which not only validates the computation but is also designed to maintain the user’s privacy. In particular, EPID is designed to ensure that the attestation provider is unable to identify the host on which the computation executes. In this work we investigate the security of the Intel implementation of the EPID protocol. We identify an implementation weakness that leaks information via a cache side channel. We show that a malicious attestation provider can use the leaked information to break the unlinkability guarantees of EPID. We analyze the leaked information using a lattice-based approach for solving the hidden number problem, which we adapt to the zero-knowledge proof in the EPID scheme, extending prior attacks on signature schemes.

It is well known that constant-time implementations of modular exponentiation cannot use sliding windows. However, software libraries such as Libgcrypt, used by GnuPG, continue to use sliding windows. It is widely believed that, even if the complete pattern of squarings and multiplications is observed through a side-channel attack, the number of exponent bits leaked is not sufficient to carry out a full key-recovery attack against RSA. Specifically, 4-bit sliding windows leak only 40% of the bits, and 5-bit sliding windows leak only 33% of the bits.In this paper we demonstrate a complete break of RSA-1024 as implemented in Libgcrypt. Our attack makes essential use of the fact that Libgcrypt uses the left-to-right method for computing the sliding-window expansion. We show for the first time that the direction of the encoding matters: the pattern of squarings and multiplications in left-to-right sliding windows leaks significantly more information about the exponent than right-to-left. We show how to extend the Heninger-Shacham algorithm for partial key reconstruction to make use of this information and obtain a very efficient full key recovery for RSA-1024. For RSA-2048 our attack is efficient for 13% of keys.