Deterministic generation of per-signature randomness has been a widely accepted solution to mitigate the catastrophic risk of randomness failure in Fiat--Shamir type signature schemes. However, recent studies have practically demonstrated that such de-randomized schemes, including EdDSA, are vulnerable to differential fault attacks, which enable adversaries to recover the entire secret signing key, by artificially provoking randomness reuse or corrupting computation in other ways. In order to balance concerns of both randomness failures and the threat of fault injection, some signature designs are advocating a ``hedged'' derivation of the per-signature randomness, by hashing the secret key, message, and a nonce. Despite the growing popularity of the hedged paradigm in practical signature schemes, to the best of our knowledge, there has been no attempt to formally analyze the fault resilience of hedged signatures. We perform a formal security analysis of the fault resilience of signature schemes constructed via the Fiat--Shamir transform. We propose a model to characterize bit-tampering fault attacks, and investigate their impact across different steps of the signing operation. We prove that, for some types of faults, attacks are mitigated by the hedged paradigm, while attacks remain possible for others. As concrete case studies, we then apply our results to XEdDSA, a hedged version of EdDSA used in the Signal messaging protocol, and to Picnic2, a hedged Fiat--Shamir signature scheme in Round 2 of the NIST Post-Quantum standardization process.

In this paper, we optimize Bleichenbacher’s statistical attack technique against (EC)DSA and other Schnorr-like signature schemes with biased or partially exposed nonces. Previous approaches to Bleichenbacher’s attack suffered from very large memory consumption during the so-called “range reduction” phase. Using a carefully analyzed and highly parallelizable approach to this range reduction based on the Schroeppel–Shamir algorithm for knapsacks, we manage to overcome the memory barrier of previous work while maintaining a practical level of efficiency in terms of time complexity.As a separate contribution, we present new fault attacks against the qDSA signature scheme of Renes and Smith (ASIACRYPT 2017) when instantiated over the Curve25519 Montgomery curve, and we validate some of them on the AVR microcontroller implementation of qDSA using actual fault experiments on the ChipWhisperer-Lite evaluation board. These fault attacks enable an adversary to generate signatures with 2 or 3 bits of the nonces known.Combining our two contributions, we are able to achieve a full secret key recovery on qDSA by applying our version of Bleichenbacher’s attack to these faulty signatures. Using a hybrid parallelization model relying on both shared and distributed memory, we achieve a very efficient implementation of our highly scalable range reduction algorithm. This allows us to complete Bleichenbacher’s attack in the 252-bit prime order subgroup of Curve25519 within a reasonable time frame and using relatively modest computational resources both for 3-bit nonce exposure and for the much harder case of 2-bit nonce exposure. Both of these computations, and particularly the latter, set new records in the implementation of Bleichenbacher’s attack.