## 2 Literature Review
### Falcon  
Falcon, with its foundation based on lattice-based cryptography, addresses an important need for cryptographic algorithms that maintains effectiveness and practicality while dealing with the power of quantum computers [24; 25]. One of the outstanding features of this scheme is its remarkable efficiency in both signature generation and verification processes [44]. This efficiency sets Falcon apart by providing a solution that is not only secure against quantum adversaries but also practical for real-world deployment. The ability to achieve robust security without sacrificing performance positions Falcon as a vital candidate in the post-quantum cryptographic landscape. Its selection as another finalist in the NIST Post-Quantum Cryptography Standardization process underscores its credibility and the community's confidence in its accuracy in cryptography [34]. Being an NIST finalist demonstrates its potential to become a recommended standard for secure communication in the future. Beyond its theoretical prowess, however, some researchers have reported problems with configuration and key generation timing, which can be a block to its widespread adoption [45]. Despite these challenges, its small public and private key sizes make it an attractive option for many applications, and ongoing research and development efforts are aimed at addressing these configuration issues. Although Falcon is undoubtedly a great choice in post-quantum cryptography, it is essential to note that NIST has selected CRYSTALS-Dilithium as its first choice for PQC digital signature schemes [34]. While Falcon has earned its place as an NIST finalist in standardization process and has received significant attention and interest within the cryptographic community, NIST's preference for CRYSTALS-Dilithium shows the thorough assessment and selection process used to identify the best options for ensuring our digital future in a post-quantum era. Falcon features parameter sets tailored to different security levels, providing flexibility in selecting an appropriate configuration based on specific requirements. The parameters, summarized in Table 5, play a crucial role in shaping Falcon's security and efficiency. The parameter sets include values for \(n\) (dimension), \(\phi\) (modulus polynomial), \(q\) (modulus), \(\beta^{2}\) (security parameter), signature size, and public key size. These parameters allow users to customize Falcon's configuration to achieve the desired balance between security and performance. During its setup process, Falcon employs a unique mathematical polynomial and a numerical input, employing these elements in a manner outlined in Algorithms 7-9. These algorithms work together to produce both the confidential secret key and the publicly accessible key, establishing the cryptographic foundation upon which the Falcon system operates securely and effectively. ```
A monic polynomial \(\phi\in\mathbb{Z}[x]\), a modulus \(q\) Ensure: A secret key (sk), a public key (pk) \(\mathbf{f},\mathbf{g},\mathbf{F},\mathbf{G},\mathbf{f}\leftarrow\mathrm{NTRUGen} (\phi,q)\) \(\mathbf{B}\leftarrow\left[\frac{\phi-f}{\mathcal{G}|-\mathrm{F}}\right]\) \(\mathbf{\hat{B}}\leftarrow\mathrm{FFT}(\mathbf{B})\) \(\mathbf{G}\leftarrow\mathbf{\tilde{B}}\times\mathbf{\tilde{B}}^{*}\) \(\mathrm{T}\leftarrow\mathrm{f}\mathrm{f}\mathrm{LDL}(\mathbf{G})\) for each leaf of T do  leaf.value \(\leftarrow\frac{\sigma}{\sqrt{\mathrm{leaf.value}}}\) endfor \(\mathrm{sk}\leftarrow(\mathbf{\tilde{B}},\mathrm{T})\) \(h\gets gf^{-1}\bmod q\) \(\mathrm{pk}\gets h\) returnsk, pk
```  
**Algorithm 7** Falcon key generation KeyGen(\(\phi\), q) [46]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Falcon 256** & **Falcon 512** & **Falcon 1024** \\ \hline \(n\) & 256 & 512 & 1024 \\ \(\phi\) & \(x^{n+1}\) & \(x^{n+1}\) & \(x^{n+1}\) \\ \(q\) & 12,289 & 12,289 & 12,289 \\ \(\beta\) & 16,468,416 & 43,533,782 & 87,067,565 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameter sets for Falcon.