Mathematical analysis of digital signature algorithms

In this thesis, classical digital signature algorithms, blind signature schemes, and basic hash-based quantum-resistant signature algorithms are comprehensively investigated. First, the fundamental concepts forming the mathematical basis of digital signature mechanisms are introduced, and asymmetric cryptographic algorithms such as RSA, Rabin, and ElGamal, which are built upon these concepts, are explained. Subsequently, classical digital signature schemes including RSA, Rabin, ElGamal, Fiat–Shamir, Guillou–Quisquater, Schnorr, DSA, and Nyberg–Rueppel, along with their elliptic curve–based variants, are examined. In addition, the blind signature versions of the RSA, Rabin, Schnorr, DSA, and Nyberg–Rueppel algorithms are analyzed in detail. Within the framework of the threats posed by quantum computers to classical cryptographic systems, fundamental hash-based quantum-resistant signature algorithms such as the Lamport and Winternitz schemes are studied. Furthermore, the Chaum–van Antwerpen signature algorithm, which operates under the undeniable signature model, and the GMR fail-stop signature algorithm, which can halt the signing process upon attack detection, are also investigated. Throughout the thesis, the algebraic operational steps of each signature algorithm are presented in detail, and their correctness is rigorously proven through original derivations. Moreover, the probabilistic and deterministic properties of the algorithms, together with their underlying security assumptions, are analyzed, and illustrative examples are provided for each algorithm to enhance clarity.