International Association
for Cryptologic Research

Provable security is a cornerstone of modern cryptography, aiming to provide formal and precise security guarantees. However, for various reasons, security proofs are not always properly verified, possibly leading to unwarranted security claims and, in the worst case, deployment of insecure constructions. To further enhance trust and assurance, machine-checked cryptography makes these proofs more formal and rigorous. Unfortunately, the complexity of writing machine-verifiable proofs remains prohibitively high in many interesting use-cases. In this paper, we investigate the sources of this complexity, specifically examining how the style of security definitions influences the difficulty of constructing machine-verifiable proofs in the context of game-playing security. Concretely, we present a new security proof for the generic construction of a PKAE scheme from a NIKE and AE scheme, written in a code-based, game-playing style `a la Bellare and Rogaway, and compare it to the same proof written in the style of state-separating proofs, a methodology for developing modular game-playing security proofs. Additionally, we explore a third “blended” style designed to avoid anticipated difficulties with the formalization. Our findings suggest that the choice of definition style impacts proof complexity—including, we argue, in detailed pen- and-paper proofs—with trade-offs depending on the proof writer’s goals.

SPHINCS+ is a post-quantum signature scheme that, at the time of writing, is being standardized as SLH-DSA. It is the most conser- vative option for post-quantum signatures, but the original tight proofs of security were flawed — as reported by Kudinov, Kiktenko and Fe- dorov in 2020. In this work, we formally prove a tight security bound for SPHINCS+ using the EasyCrypt proof assistant, establishing greater confidence in the general security of the scheme and that of the param- eter sets considered for standardization. To this end, we reconstruct the tight security proof presented by Hülsing and Kudinov (in 2022) in a modular way. A small but important part of this effort involves a com- plex argument relating four different games at once, of a form not yet formalized in EasyCrypt (to the best of our knowledge). We describe our approach to overcoming this major challenge, and develop a general formal verification technique aimed at this type of reasoning. Enhancing the set of reusable EasyCrypt artifacts previously produced in the formal verification of stateful hash-based cryptographic construc- tions, we (1) improve and extend the existing libraries for hash func- tions and (2) develop new libraries for fundamental concepts related to hash-based cryptographic constructions, including Merkle trees. These enhancements, along with the formal verification technique we develop, further ease future formal verification endeavors in EasyCrypt, especially those concerning hash-based cryptographic constructions.

This work presents a novel machine-checked tight security proof for XMSS --- a stateful hash-based signature scheme that is (1) standardized in RFC 8391 and NIST SP 800-208, and (2) employed as a primary building block of SPHINCS+, one of the signature schemes recently selected for standardization as a result of NIST's post-quantum competition. In 2020, Kudinov, Kiktenko, and Fedoro pointed out a flaw affecting the tight security proofs of SPHINCS+ and XMSS. For the case of SPHINCS+, this flaw was fixed in a subsequent tight security proof by Hülsing and Kudinov. Unfortunately, employing the fix from this proof to construct an analogous tight security proof for XMSS would merely demonstrate security with respect to an insufficient notion. At the cost of modeling the message-hashing function as a random oracle, we complete the tight security proof for XMSS and formally verify it using the EasyCrypt proof assistant. (Note that this merely extends the use of the random oracle model, as this model is already required in other parts of the security analysis to justify the currently standardized parameter values). As part of this endeavor, we formally verify the crucial step common to the security proofs of SPHINCS+ and XMSS that was found to be flawed before, thereby confirming that the core of the aforementioned security proof by Hülsing and Kudinov is correct. As this is the first work to formally verify proofs for hash-based signature schemes in EasyCrypt, we develop several novel libraries for the fundamental cryptographic concepts underlying such schemes --- e.g., hash functions and digital signature schemes --- establishing a common starting point for future formal verification efforts. These libraries will be particularly helpful in formally verifying proofs of other hash-based signature schemes such as LMS or SPHINCS+.