Attacking (EC)DSA scheme with ephemeral keys sharing specific bits

In this paper, we present a deterministic attack on (EC)DSA signature scheme, providing that several signatures are known such that the corresponding ephemeral keys share a certain amount of bits without knowing their value. By eliminating the shared blocks of bits between the ephemeral keys, we get a lattice of dimension equal to the number of signatures having a vector containing the private key. We compute an upper bound for the distance of this vector from a target vector, and next, using Kannan's enumeration algorithm, we determine it and hence the secret key. The attack can be made highly efficient by appropriately selecting the number of shared bits and the number of signatures

Lattice Attacks against Elliptic-Curve Signatures with Blinded Scalar Multiplication

International audienceElliptic curve cryptography is today the prevailing approach to get efficient public-key cryptosystems and digital signatures. Most of elliptic curve signature schemes use a \emph{nonce} in the computation of each signature and the knowledge of this nonce is sufficient to fully recover the secret key of the scheme. Even a few bits of the nonce over several signatures allow a complete break of the scheme by lattice-based attacks. Several works have investigated how to efficiently apply such attacks when partial information on the nonce can be recovered through side-channel attacks. However, these attacks usually target unprotected implementation and/or make ideal assumptions on the recovered information, and it is not clear how they would perform in a scenario where common countermeasures are included and where only noisy information leaks via side channels. In this paper, we close this gap by applying such attack techniques against elliptic-curve signature implementations based on a blinded scalar multiplication. Specifically, we extend the famous Howgrave-Graham and Smart lattice attack when the nonces are blinded by the addition of a random multiple of the elliptic-curve group order or by a random Euclidean splitting. We then assume that noisy information on the blinded nonce can be obtained through a template attack targeting the underlying scalar multiplication and we show how to characterize the obtained likelihood scores under a realistic leakage assumption. To deal with this scenario, we introduce a filtering method which given a set of signatures and associated likelihood scores maximizes the success probability of the lattice attack. Our approach is backed up with attack simulation results for several signal-to-noise ratio of the exploited leakage

Secret Key Leakage from Public Key Perturbation of DLP-based Cryptosystems

Finding efficient countermeasures for cryptosystems against fault attacks is challenged by a constant discovery of flaws in designs. Even elements, such as public keys, that do not seem critical must be protected. From the attacks against RSA, we develop a new attack of DLP-based cryptosystems, built in addition on a lattice analysis to recover DSA public keys from partially known nonces. Based on a realistic fault model, our attack only requires 16 faulty signatures to recover a 160-bit DSA secret key within a few minutes on a standard PC. These results significantly improves the previous public element fault attack in the context of DLP-based cryptosystems