# Post-Quantum Digital Signature: Verkle-Based HORST
## 11 Efficiency Comparison
### Performance Benchmarks  
There are multiple major improvements in signature size, verification performance, and memory economy when comparing Verkle-based HORST to traditional implementations. These benefits result from fundamental distinctions in the way that different structures manage cryptographic proofs.  
Compared to its Merkle-based equivalent, Verkle-based HORST provides an extensive theoretical increase in terms of signature size. Because Merkle-based HORST must include Merkle paths, it usually generates signatures that are roughly 12.8 kilobytes in size. However, by using KZG polynomial commitments, Verkle-based HORST may be able to decrease this to about 3.2 kilobytes. This 75% decrease in signature size is in line with performance improvements seen in comparable systems, such as Ethereum's implementation of the Verkle tree, where Verkle proofs outperform Merkle proofs in terms of compression.  
In Verkle-based HORST, the verification procedure provides additional positive effects. While Verkle-based HORST's KZG proofs can be verified in constant time, Merkle-based HORST needs verifiers to perform several hash operations along a Merkle path, which usually takes around 5.6 milliseconds per signature. This approach is especially appealing for high-throughput applications where verification speed is crucial since it may be able to cut the verification time to about 1.4 milliseconds. Benefits observed in previous KZG-based systems are mirrored in this fourfold increase in verification speed.  
There is a trade-off when it comes to key generation. Verkle-based HORST probably requires more computing power than Merkle-based HORST. The key generation time might be increased from 110 milliseconds to around 320 milliseconds due to the polynomial operations required for KZG commitments. However, as is the case with other cryptographic systems that use polynomial commitments, this one-time expense is balanced by the continuous advantages during the verification phase.  
Considerable improvements in storage efficiency may be achieved when comparing Verkle-based HORST to traditional Merkle-based structures. For conventional security settings (t = 1024 keys, k = 32 subset size), Merkle proofs need \(O(\text{klog}_{2}\,t)\) storage, or around 10 KB when 256-bit hashes are used. Verkle trees, on the other hand, use KZG polynomial commitments to reach \(O(1)\) proof sizes, typically lowering storage to 1-3 KB. This fourfold decrease is consistent with Verkle tree research [23], which displays substantially smaller proofs than Merkle trees at scale. Hybrid signature schemes like SPHINCS+ (\(\approx\)16 KB) and XMSS (\(\approx\)8 KB for t = 1024) provide intermediate storage efficiency due to their logarithmic proof growth. Verkle-based HORST's constant-size proofs make it especially well-suited for storage-constrained applications, like blockchain light clients and IoT devices, where 128-bit security and memory usage reduction are crucial.  
Another area in which Verkle-based HORST demonstrates theoretical advantage is memory efficiency. In contrast to an analogous solution that requires 48 megabytes of storage, a Verkle tree that supports one million signatures may only require 12 megabytes. Verkle-based HORST may be especially useful in resource-constrained settings where storage space is limited, such IoT devices or blockchain light clients, because of its fourfold memory consumption decrease.  
Verkle-based HORST is positioned as a potential development of hash-based signature schemes due to its smaller signature sizes, quicker verification, and lower memory needs. The consistent gains shown in other Verkle tree applications imply that comparable benefits might be gained in HORST implementations.