# Performance Analysis of Post-Quantum Cryptography Algorithms for Digital Signature
## 3 Post-Quantum Cryptography
### Post-Quantum Ciphers  
Post-quantum cryptography is a proactive response to the threats brought on by the emergence of Shore and Grover algorithms. Due to the potential ability to break hitherto widely used algorithms like RSA or ECC, there have to be solutions supporting asymmetrical data encryption but resistant to attacks by quantum computers. This is precisely the purpose behind the development of post-quantum algorithms. They are divided into four main categories.  
* Lattice-Based Cryptography is one of the leading candidates of the post-quantum cryptography. They are based on the complexity of problems such as the Shortest Vector Problem (SVP), as well as Learning With Errors (LWE) [13]. The security of these solutions comes from the difficulty of solving some lattice problems in multidimensional spaces. The problem, for example, is finding the constants by which the basis vector should be multiplied, which is not an easy task for quantum computers. Some of the well-known algorithms belonging to this group are NTRU, CRYSTALS-Dilithium, and Falcon.
* Hash-Based Cryptography is a group that bases its security on appropriate hash functions and schemes such as the Merkle signature scheme [13]. Currently, no efficient quantum algorithm is known to be able to break a cryptographic hash function; however, the disadvantage of such a solution may be the limit of signatures generated using a particular key pair. An example of an algorithm that belongs to this group is SPHINCS+.
* Code-Based Cryptography is a group that bases its security on correction codes [14]. Examples of algorithms with such properties are McEliece and its variation Niederreiter. The difficulty of these solutions lies in decoding a general linear code.
* Isogeny-Based Cryptography is the last of the four main groups of the post-quantum algorithms. The solutions within it are based on the properties of elliptic curves and the Claw Finding problem. This involves the operation with elliptic curves that have the property of isogeny [15].  
Additionally, it is worth comparing well-known post-quantum algorithms, the effectiveness of which is analyzed by the authors in this paper. The CRYSTALS-Dilithium, Falcon, and SPHINCS+ algorithms are briefly discussed, each representing a different approach to post-quantum cryptography.  
CRYSTALS-Dilithium is a lattice-based algorithm. It was designed for digital signatures, and it was a major finalist in the NIST competition in this exact category, which is its main advantage also in the context of this study. The security of this algorithm lies in the difficulty level of lattice problems over modular lattices [16]. As a lattice-based algorithm, it benefits from the complexity of the Shortest Vector Problem and Learning With Errors problems, which are also considered unbreakable using quantum computers.
* Falcon is an acronym for 'Fast-Fourier Lattice-based Compact Signatures over NTRU'. It was also another candidate in the NIST competition. It uses the lattice structure of NTRU and the Fast Fourier Transform (FFT) to improve performance. Its main advantage is the small size of the generated signatures, which is especially important in the context of high-volume data transmission [17].
* SPHINCS+ represents a different approach to the previous examples, as it is an algorithm based on hash functions. Its advantage is the ability to be based on different hash functions, e.g., SHA2 or SHAKE, allowing it to be adapted according to specific security requirements. On the other hand, its disadvantage is the relatively large size of the generated signatures [18].  
Each of the mentioned algorithms represents a sufficient security level. These three algorithms were selected by NIST during the standardization process of algorithms to be intended for digital signatures. Considering the availability of all three algorithms presented, they have been implemented to use for digital signature purposes and tested.