# Post-Quantum Digital Signature: Verkle-Based HORST
## 1 Introduction  
Classical cryptographic systems, especially those based on asymmetric algorithms like RSA and elliptic curve cryptography (ECC), face serious challenges due to the rapid advancement of quantum computing. These systems rely on hard mathematical problems like discrete logarithms and integer factorization, and they have a significant role in contemporary digital infrastructure. These issues can be effectively resolved by quantum algorithms, especially [PERSON]'s algorithm, making traditional cryptography systems vulnerable to quantum computing attacks [1, 2]. Post-quantum cryptography (PQC), which focuses on creating cryptographic methods that are secure against quantum computer attacks, has become a crucial field of study in response to this. Hash-based digital signatures are seen to be one of the most promising post-quantum cryptography approaches. These signatures only depend on the integrity of cryptographic hash functions, which, according to current theories, are resilient to quantum assaults. Hash-based signatures are still feasible with sufficiently large hash sizes, even if [PERSON]'s approach only cuts security levels in half, despite the fact that it can accelerate brute-forceattacks against hash functions [3]. Hash-based signatures are a top contender for post-quantum safe cryptography because of their reliability, ease of use, and well-understood security features. Beyond traditional post-quantum cryptography, quantum digital signatures (QDS) have evolved as an alternative solution that uses quantum key distribution (QKD) and quantum-resistant primitives. Recent developments show progress in gaining information-theoretic security, including effective lattice-based QDS constructs [4] and measurement-device-independent QDS protocols [5]. However, QDS's applicability for legacy systems is currently limited due to its requirement for quantum communication infrastructure. On the other hand, Verkle-based HORST offers a hash-based, classical approach that is both quantum-resistant and compatible with current technology. Introduced in 1979, the Lamport-Diffie one-time signature (OTS) system laid the groundwork for hash-based signatures [6]. Lamport-Diffie signatures offer a straightforward yet efficient framework for quantum-safe authentication by generating signatures using pairs of secret keys and matching hash values. However, a lot of applications cannot use Lamport-Diffie signatures because of their large key sizes. The Winternitz OTS (WOTS), which uses hash chains rather than individual key pairs to reduce key sizes, was proposed as a solution to this limitation [7]. Shorter signatures are possible with this method, but signing and verification need more processing power. Additional advancements resulted in the creation of WOTS+, a WOTS version that adds randomization to the hash chains to boost security [8]. Because of this randomization, WOTS+ is more resistant to some kinds of cryptographic attacks by reducing the impact of multi-target attacks. The fact that one-time signatures like WOTS and WOTS+ can only securely sign a single message limits their usefulness for many real-world applications, even with current improvements. By allowing the signature of several messages with a single public key, the Merkle Signature Scheme (MSS) overcomes this restriction [9]. In order to do this, MSS arranges public keys for one-time signatures into a Merkle tree, with the master public key acting as the tree's root. This method preserves quantum resistance while enabling the effective authentication of several signatures. However, MSS is stateful, which means the signer must track key usage to maintain security. Additionally, scalability in large-scale systems may be limited by the fact that the proof sizes in MSS increase exponentially with the number of leaves in the Merkle tree. To address these issues, stateless hash-based systems such as HORS (Hash to Obtain Random Subset) and its tree-based variant, HORST (HORS with Trees), were created [10; 11].