# An Intrusion Resistant SCADA Framework Based on Quantum and Post-Quantum Scheme
## 2 Related Work
### Post Quantum Digital Signature Scheme
#### 2.3.4. HORS and HORST  
In 2002, [PERSON] et al. [32] proposed a few-time signature scheme using hash functions. It uses two parameters \(t=2^{\tau}\) for \(\tau\in\mathbb{N}\) and \(k\in\mathbb{N}\). The hash value of message is \(m\) such that \(m=k\)\(log\)\(t=k\tau\). HORS uses two keys, a secret key consisting of \(t\) random values obtained from a pseudorandom number generator. The public key is obtained from the \(t\) hashes of the random values in the secret key, and the signature contains \(k\) secret key values [33]. In 2015, [PERSON] et al. [13] proposed HORS tree (HORST) for the SPHINCS protocol, which we have used in our proposed framework. HORST reduces the public key size and signature of the HORS scheme. However, it increases the runtime of the algorithm. The public key generated in HORST is the root of a binary hash tree of height \(log\)\(t\). The leaves of the tree are public-key elements of the root key. The signature generated has \(k\) secret values and authentication path for each value [13]. SPHINCS-256, exhibited in our proposed scheme, comprises Merkle tree, WOTS, and HORST signature scheme. Due to the rapid inception of quantum computing, quantum algorithms have the potential to crack long term security protocols, and improving the efficiency of quantum algorithms can crack post-quantum schemes. Therefore, we need to build up a security system that mitigates such threats. Table 2 provides a synthesis of all the research work related to the proposed algorithm. \begin{table} \begin{tabular}{p{42.7 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt}} \hline \hline
**Authors** & **Algorithm/ Organization** & **Overview** & **Advantages** & **Disadvantages** \\ \hline \multirow{4}{*}{American Gas Association} & \multirow{4}{*}{AGA-12} & \begin{tabular}{c} AGA-12 was \\ developed to provide \\ security as well as \\ to save time and \\ computation cost. \\ \end{tabular} & It provides confidentiality, integrity and authentication. & It fails to provide availability and is vulnerable against quantum attacks. \\ \hline \multirow{4}{*}{[PERSON] et al. 1984} & \multirow{4}{*}{BB84} & It is [PERSON]’s \\  & & uncertainty principle \\  & & based \\  & & QKD proposed & It is resistant against \\  & & by [PERSON] and \\  & & [PERSON] in 1984. & mainly, [PERSON]’s algorithm. \\  & & It uses two polarization \\  & & states for each basis. \\ \hline \multirow{4}{*}{[PERSON] 1992} & \multirow{4}{*}{B92} & In 1992, [PERSON] proposed & There are possible \\  & & a simplified version of the \\  & & BB84 protocol named B92. & It handles only one \\  & & of two polarization \\  & & states for each basis. \\ \hline \multirow{4}{*}{[PERSON] 1992} & \multirow{4}{*}{LOTSS} & It uses any secure \\  & & cryptographic hash functions \\  & & to generate a public key \\  & & to sign a message \\ \hline \multirow{4}{*}{[PERSON] 1979} & \multirow{4}{*}{WOTS} & It is an optimized \\  & & version of LOTSS. & \\  & & & \\ \hline \multirow{4}{*}{[PERSON] 1979} & \multirow{4}{*}{MSS} & It is a stateful hash-based \\  & & signature scheme \\  & & which creates a single \\  & & binary hash tree to generate \\  & & signature and key pair, \\  & & using OTS scheme. \\ \hline \multirow{4}{*}{[PERSON] et al. 2002} & \multirow{4}{*}{HORS and \begin{tabular}{c} HORS \\ and \\ HORST \\ with a binary hash tree. \\ \end{tabular} } & \
\begin{tabular}{c} HORS is a few-time \\ signature scheme \\ using hash functions. \\ \end{tabular} & Both of them efficiently \\  & & generates quantum safe \\  & & signatures. \\  & & HORST is HORS \\  & & with a binary hash tree. \\ \end{tabular}  & \begin{tabular}{c} HORST reduces the public \\ key size and signature \\ of the HORS scheme. \\ \end{tabular}  \\ \hline \multirow{4}{*}{[PERSON] et al. 2015} & \multirow{4}{*}{SPHINCS} & It is a stateless hash \\  & & signature scheme using \\  & & hyper-tree with OTS \\  & & and FTS. \\ \end{tabular} & It provides \(2^{128}\) security \\  & & & against quantum gates.