International Association
for Cryptologic Research

A popular way to build post-quantum signature schemes is by first constructing an identification scheme (IDS) and applying the Fiat-Shamir transform to it. In this work we tackle two open questions related to the general applicability of techniques around this approach that together allow for efficient post-quantum signatures with optimal security bounds in the QROM. First, we consider a recent work by Aguilar-Melchor, Hülsing, Joseph, Majenz, Ronen, and Yue (Asiacrypt'23) that showed that an optimal bound for three-round commit & open IDS by Don, Fehr, Majenz, and Schaffner (Crypto'22) can be applied to the five-round Syndrome-Decoding in the Head (SDitH) IDS. For this, they first applied a transform that replaced the first three rounds by one. They left it as an open problem if the same approach applies to other schemes beyond SDitH. We answer this question in the affirmative, generalizing their round-elimination technique and giving a generic security proof for it. Our result applies to any IDS with $2n+1$ rounds for $n>1$. However, a scheme has to be suitable for the resulting bound to not be trivial. We find that IDS are suitable when they have a certain form of special-soundness which many commit & open IDS have. Second, we consider the hypercube technique by Aguilar-Melchor, Gama, Howe, Hülsing, Joseph, and Yue (Eurocrypt'23). An optimization that was proposed in the context of SDitH and is now used by several of the contenders in the NIST signature on-ramp. It was conjectured that the technique applies generically for the MPC-in-the-Head (MPCitH) technique that is used in the design of many post-quantum IDS if they use an additive secret sharing scheme but this was never proven. In this work we show that the technique generalizes to MPCitH IDS that use an additively homomorphic MPC protocol, and we prove that security is preserved. We demonstrate the application of our results to the identification scheme of RYDE, a contender in the recent NIST signature on-ramp. While RYDE was already specified with the hypercube technique applied, this gives the first QROM proof for RYDE with an optimally tight bound.

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 paper presents a code-based signature scheme based on the well-known syndrome decoding (SD) problem. The scheme builds upon a recent line of research which uses the Multi-Party-Computation-in-the-Head (MPCitH) approach to construct efficient zero-knowledge proofs, such as Syndrome Decoding in the Head (SDitH), and builds signature schemes from them using the Fiat-Shamir transform. At the heart of our proposal is a new approach, Hypercube-MPCitH, to amplify the soundness of any MPC protocol that uses additive secret sharing. An MPCitH protocol with N parties can be repeated D times using parallel composition to reach the same soundness as a protocol run with N^D parties. However, the former comes with D times higher communication costs, often mainly contributed by the usage of D `auxiliary' states (which in general have a significantly bigger impact on size than random states). Instead of that, we begin by generating N^D shares, arranged into a D-dimensional hypercube of side N containing only one `auxiliary' state. We derive from this hypercube D sharings of size N which are used to run D instances of an N party MPC protocol. Hypercube-MPCitH leads to a protocol with 1/N^D soundness error, requiring N^D offline computation, but with only N*D online computation, and only one `auxiliary'. As the (potentially offline) share generation phase is generally inexpensive, this leads to trade-offs that are superior to just using parallel composition. Our novel method of share generation and aggregation not only improves certain MPCitH protocols in general but also shows in concrete improvements of signature schemes. Specifically, we apply it to the work of Feneuil, Joux, and Rivain (CRYPTO'22) on code-based signatures, and obtain a new signature scheme that achieves a 8.1x improvement in global runtime and a 30x improvement in online runtime for their shortest signatures size (8,481 Bytes). It is also possible to leverage the fact that most computations are offline to define parameter sets leading to smaller signatures: 6,784 Bytes for 26 ms offline and 5,689 Bytes for 320 ms offline. For NIST security level 1, online signature cost is around 3 million cycles (<1 ms on commodity processors), regardless of signature size.

We extend and consolidate the security justification for the Dilithium signature scheme. In particular, we identify a subtle but crucial gap that appears in several ROM and QROM security proofs for signature schemes that are based on the Fiat-Shamir with aborts paradigm, including Dilithium. The gap lies in the CMA-to-NMA reduction and was uncovered when trying to formalize a variant of the QROM security proof by Kiltz, Lyubashevsky, and Schaffner (Eurocrypt 2018). The gap was confirmed by the authors, and there seems to be no simple patch for it. We provide new, fixed proofs for the affected CMA-to-NMA reduction, both for the ROM and the QROM, and we perform a concrete security analysis for the case of Dilithium to show that the claimed security level is still valid after addressing the gap. Furthermore, we offer a fully mechanized ROM proof for the CMA-security of Dilithium in the EasyCrypt proof assistant. Our formalization includes several new tools and techniques of independent interest for future formal verification results.

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+.

In this work, we consider the formal verification of the public-key encryption scheme of Saber, one of the selected few post-quantum cipher suites currently considered for potential standardization. We formally verify this public-key encryption scheme's IND-CPA security and delta-correctness properties, i.e., the properties required to transform the public-key encryption scheme into an IND-CCA2 secure and delta-correct key encapsulation mechanism, in EasyCrypt. To this end, we initially devise hand-written proofs for these properties that are significantly more detailed and meticulous than the presently existing proofs. Subsequently, these hand-written proofs serve as a guideline for the formal verification. The results of this endeavor comprise hand-written and computer-verified proofs which demonstrate that Saber's public-key encryption scheme indeed satisfies the desired security and correctness properties.

In 2020, Kudinov, Kiktenko, and Fedorov pointed out a flaw in the tight security proof of the SPHINCS+ construction. This work gives a new tight security proof for SPHINCS+. The flaw can be traced back to the security proof for the Winternitz one-time signature scheme (WOTS) used within SPHINCS+. In this work, we give a stand-alone description of the WOTS variant used in SPHINCS+ that we call WOTS-TW. We provide a security proof for WOTS-TW and multi-instance WOTS-TW against non-adaptive chosen message attacks where the adversary only learns the public key after it made its signature query. Afterwards, we show that this is sufficient to give a tight security proof for SPHINCS+. We recover almost the same bound for the security of SPHINCS+, with only a factor w loss compared to the previously claimed bound, where w is the Winternitz parameter that is commonly set to 16. On a more technical level, we introduce new lower bounds on the quantum query complexity for generic attacks against properties of cryptographic hash functions and analyse the constructions of tweakable hash functions used in SPHINCS+ with regard to further security properties.

In known security reductions for the Fujisaki-Okamoto transformation, decryption failures are handled via a reduction solving the rather unnatural task of finding failing plaintexts given the private key, resulting in a Grover search bound. Moreover, they require an implicit rejection mechanism for invalid ciphertexts to achieve a reasonable security bound in the QROM. We present a reduction that has neither of these deficiencies: We introduce two security games related to finding decryption failures, one capturing the computationally hard task of using the public key to find a decryption failure, and one capturing the statistically hard task of searching the random oracle for key-independent failures like, e.g., large randomness. As a result, our security bounds in the QROM are tighter than previous ones with respect to the generic random oracle search attacks: The attacker can only partially compute the search predicate, namely for said key-independent failures. In addition, our entire reduction works for the explicit-reject variant of the transformation and improves significantly over all of its known reductions. Besides being the more natural variant of the transformation, security of the explicit reject mechanism is also relevant for side channel attack resilience of the implicit-rejection variant. Along the way, we prove several technical results characterizing preimage extraction and certain search tasks in the QROM that might be of independent interest.

The random oracle model (ROM) enjoys widespread popularity, mostly because it tends to allow for tight and conceptually simple proofs where provable security in the standard model is elusive or costly. While being the adequate replacement of the ROM in the post-quantum security setting, the quantum-accessible random oracle model (QROM) has thus far failed to provide these advantages in many settings. In this work, we focus on adaptive reprogrammability, a feature of the ROM enabling tight and simple proofs in many settings. We show that the straightforward quantum-accessible generalization of adaptive reprogramming is feasible by proving a bound on the adversarial advantage in distinguishing whether a random oracle has been reprogrammed or not. We show that our bound is tight by providing a matching attack. We go on to demonstrate that our technique recovers the mentioned advantages of the ROM in three QROM applications: 1) We give a tighter proof of security of the message compression routine as used by XMSS. 2) We show that the standard ROM proof of chosen-message security for Fiat-Shamir signatures can be lifted to the QROM, straightforwardly, achieving a tighter reduction than previously known. 3) We give the first QROM proof of security against fault injection and nonce attacks for the hedged Fiat-Shamir transform.

This work presents new speed records for XMSS (RFC 8391) signature verification on embedded devices. For this we make use of a probabilistic method recently proposed by Perin, Zambonin, Martins, Custódio, and Martina (PZMCM) at ISCC 2018, that changes the XMSS signing algorithm to search for rapidly verifiable signatures. We improve the method, ensuring that the added signing cost for the search is independent of the message length. We provide a statistical analysis of the resulting verification speed and support it by experiments. We present a record setting RFC compatible implementation of XMSS verification on the ARM Cortex-M4. At a signing time of about one minute on a general purpose CPU, we create signatures that are verified about 1.44 times faster than traditionally generated signatures. Adding further well-known implementation optimizations to the verification algorithm we reduce verification time by over a factor two from 13.85 million to 6.56 million cycles. In contrast to previous works, we provide a detailed security analysis of the resulting signature scheme under classical and quantum attacks that justifies our selection of parameters. On the way, we fill a gap in the security analysis of XMSS as described in RFC 8391 proving that the modified message hashing in the RFC does indeed mitigate multi-target attacks. This was not shown before and might be of independent interest.

In this work we show that the sponge construction can be used to construct quantum-secure pseudorandom functions. As our main result we prove that random sponges are quantum indistinguishable from random functions. In this setting the adversary is given superposition access to the input-output behavior of the construction but not to the internal function. Our proofs hold under the assumption that the internal function is a random function or permutation. We then use this result to obtain a quantum-security version of a result by Andreeva, Daemen, Mennink, and Van Assche (FSE’15) which shows that a sponge that uses a secure PRP or PRF as internal function is a secure PRF. This result also proves that the recent attacks against CBC-MAC in the quantum-access model by Kaplan, Leurent, Leverrier, and Naya-Plasencia (Crypto’16) and Santoli, and Schaffner (QIC’16) can be prevented by introducing a state with a non-trivial inner part.The proof of our main result is derived by analyzing the joint distribution of any q input-output pairs. Our method analyzes the statistical behavior of the considered construction in great detail. The used techniques might prove useful in future analysis of different cryptographic primitives considering quantum adversaries. Using Zhandry’s PRF/PRP switching lemma we then obtain that quantum indistinguishability also holds if the internal block function is a random permutation.

We revisit the construction of IND-CCA secure key encapsulation mechanisms (KEM) from public-key encryption schemes (PKE). We give new, tighter security reductions for several constructions. Our main result is an improved reduction for the security of the $$U^{\not \bot }$$ -transform of Hofheinz, Hövelmanns, and Kiltz (TCC’17) which turns OW-CPA secure deterministic PKEs into IND-CCA secure KEMs. This result is enabled by a new one-way to hiding (O2H) lemma which gives a tighter bound than previous O2H lemmas in certain settings and might be of independent interest. We extend this result also to the case of PKEs with non-zero decryption failure probability and non-deterministic PKEs. However, we assume that the derandomized PKE is injective with overwhelming probability.In addition, we analyze the impact of different variations of the $$U^{\not \bot }$$ -transform discussed in the literature on the security of the final scheme. We consider the difference between explicit ( $$U^{\bot }$$ ) and implicit ( $$U^{\not \bot }$$ ) rejection, proving that security of the former implies security of the latter. We show that the opposite direction holds if the scheme with explicit rejection also uses key confirmation. Finally, we prove that (at least from a theoretic point of view) security is independent of whether the session keys are derived from message and ciphertext ( $$U^{\not \bot }$$ ) or just from the message ( $$U^{\not \bot }_m$$ ).

There is a well-known gap between second-preimage resistance and preimage resistance for length-preserving hash functions. This paper introduces a simple concept that fills this gap. One consequence of this concept is that tight reductions can remove interactivity for multi-target length-preserving preimage problems, such as the problems that appear in analyzing hash-based signature systems. Previous reduction techniques applied to only a negligible fraction of all length-preserving hash functions, presumably excluding all off-the-shelf hash functions.

We propose SOFIA, the first $$\mathcal {MQ}$$MQ-based signature scheme provably secure in the quantum-accessible random oracle model (QROM). Our construction relies on an extended version of Unruh’s transform for 5-pass identification schemes that we describe and prove secure both in the ROM and QROM.Based on a detailed security analysis, we provide concrete parameters for SOFIA that achieve 128-bit post-quantum security. The result is SOFIA-4-128 with parameters carefully optimized to minimize signature size and maximize performance. SOFIA-4-128 comes with an implementation targeting recent Intel processors with the AVX2 vector-instruction set; the implementation is fully protected against timing attacks.

This paper suggests to use rounded Gaussians in place of discrete Gaussians in rejection-sampling-based lattice signature schemes like BLISS or Lyubashevsky’s signature scheme. We show that this distribution can efficiently be sampled from while additionally making it easy to sample in constant time, systematically avoiding recent timing-based side-channel attacks on lattice-based signatures.We show the effectiveness of the new sampler by applying it to BLISS, prove analogues of the security proofs for BLISS, and present an implementation that runs in constant time. Our implementation needs no precomputed tables and is twice as fast as the variable-time CDT sampler posted by the BLISS authors with precomputed tables.

This paper presents software demonstrating that the 20-year-old NTRU cryptosystem is competitive with more recent lattice-based cryptosystems in terms of speed, key size, and ciphertext size. We present a slightly simplified version of textbook NTRU, select parameters for this encryption scheme that target the 128-bit post-quantum security level, construct a KEM that is CCA2-secure in the quantum random oracle model, and present highly optimized software targeting Intel CPUs with the AVX2 vector instruction set. This software takes only 307 914 cycles for the generation of a keypair, 48 646 for encapsulation, and 67 338 for decapsulation. It is, to the best of our knowledge, the first NTRU software with full protection against timing attacks.