Nadia Heninger

Joshua Fried, Pierrick Gaudry, Nadia Heninger et al.

We perform a special number field sieve discrete logarithm computation in a 1024-bit prime field. To our knowledge, this is the first kilobit-sized discrete logarithm computation ever reported for prime fields. This computation took a little over two months of calendar time on an academic cluster using the open-source CADO-NFS software. Our chosen prime $p$ looks random, and $p--1$ has a 160-bit prime factor, in line with recommended parameters for the Digital Signature Algorithm. However, our p has been trapdoored in such a way that the special number field sieve can be used to compute discrete logarithms in $\mathbb{F}\_p^*$ , yet detecting that p has this trapdoor seems out of reach. Twenty-five years ago, there was considerable controversy around the possibility of back-doored parameters for DSA. Our computations show that trapdoored primes are entirely feasible with current computing technology. We also describe special number field sieve discrete log computations carried out for multiple weak primes found in use in the wild. As can be expected from a trapdoor mechanism which we say is hard to detect, our research did not reveal any trapdoored prime in wide use. The only way for a user to defend against a hypothetical trapdoor of this kind is to require verifiably random primes.

Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic et al.

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.

Daniel Moghimi, Jo Van Bulck, Nadia Heninger et al.

The adversarial model presented by trusted execution environments (TEEs) has prompted researchers to investigate unusual attack vectors. One particularly powerful class of controlled-channel attacks abuses page-table modifications to reliably track enclave memory accesses at a page-level granularity. In contrast to noisy microarchitectural timing leakage, this line of deterministic controlled-channel attacks abuses indispensable architectural interfaces and hence cannot be mitigated by tweaking microarchitectural resources. We propose an innovative controlled-channel attack, named CopyCat, that deterministically counts the number of instructions executed within a single enclave code page. We show that combining the instruction counts harvested by CopyCat with traditional, coarse-grained page-level leakage allows the accurate reconstruction of enclave control flow at a maximal instruction-level granularity. CopyCat can identify intra-page and intra-cache line branch decisions that ultimately may only differ in a single instruction, underscoring that even extremely subtle control flow deviations can be deterministically leaked from secure enclaves. We demonstrate the improved resolution and practicality of CopyCat on Intel SGX in an extensive study of single-trace and deterministic attacks against cryptographic implementations, and give novel algorithmic attacks to perform single-trace key extraction that exploit subtle vulnerabilities in the latest versions of widely-used cryptographic libraries. Our findings highlight the importance of stricter verification of cryptographic implementations, especially in the context of TEEs.

Daniel Moghimi, Berk Sunar, Thomas Eisenbarth et al.

Trusted Platform Module (TPM) serves as a hardware-based root of trust that protects cryptographic keys from privileged system and physical adversaries. In this work, we perform a black-box timing analysis of TPM 2.0 devices deployed on commodity computers. Our analysis reveals that some of these devices feature secret-dependent execution times during signature generation based on elliptic curves. In particular, we discovered timing leakage on an Intel firmware-based TPM as well as a hardware TPM. We show how this information allows an attacker to apply lattice techniques to recover 256-bit private keys for ECDSA and ECSchnorr signatures. On Intel fTPM, our key recovery succeeds after about 1,300 observations and in less than two minutes. Similarly, we extract the private ECDSA key from a hardware TPM manufactured by STMicroelectronics, which is certified at Common Criteria (CC) EAL 4+, after fewer than 40,000 observations. We further highlight the impact of these vulnerabilities by demonstrating a remote attack against a StrongSwan IPsec VPN that uses a TPM to generate the digital signatures for authentication. In this attack, the remote client recovers the server's private authentication key by timing only 45,000 authentication handshakes via a network connection. The vulnerabilities we have uncovered emphasize the difficulty of correctly implementing known constant-time techniques, and show the importance of evolutionary testing and transparent evaluation of cryptographic implementations. Even certified devices that claim resistance against attacks require additional scrutiny by the community and industry, as we learn more about these attacks.

Ted Chinburg, Brett Hemenway, Nadia Heninger et al.

We draw a new connection between Coppersmith's method for finding small solutions to polynomial congruences modulo integers and the capacity theory of adelic subsets of algebraic curves. Coppersmith's method uses lattice basis reduction to construct an auxiliary polynomial that vanishes at the desired solutions. Capacity theory provides a toolkit for proving when polynomials with certain boundedness properties do or do not exist. Using capacity theory, we prove that Coppersmith's bound for univariate polynomials is optimal in the sense that there are \emph{no} auxiliary polynomials of the type he used that would allow finding roots of size $N^{1/d+ε}$ for monic degree-$d$ polynomials modulo $N$. Our results rule out the existence of polynomials of any degree and do not rely on lattice algorithms, thus eliminating the possibility of even superpolynomial-time improvements to Coppersmith's bound. We extend this result to constructions of auxiliary polynomials using binomial polynomials, and rule out the existence of any auxiliary polynomial of this form that would find solutions of size $N^{1/d+ε}$ unless $N$ has a very small prime factor.