Fundamentals and Frontiers of Post Quantum Cryptography

Namita Tiwari, Mayur Rahul, Vikash Yadav

The field of cryptography is facing a significant challenge due to fast updates in computer history. This chapter explores the key concepts, techniques, and challenges associated with post-quantum cryptography (PQC). The chapter begins by introducing the fundamental principles of quantum computing and the threats it poses to traditional cryptography systems. Each family is examined in detail, highlighting its underlying mathematical foundations, security assumptions, and implementation considerations. The chapter also discusses the ongoing standardization efforts in postquantum cryptography led by organizations such as the “National Institute of Standards and Technology” (NIST). Furthermore, the chapter explores the challenges and open research questions in the field, such as the efficient implementation of post-quantum algorithms, the impact on existing protocols and systems, and the integration of postquantum cryptography into real-world applications.

Daniel J. Bernstein

Shweta Agrawal

PAULO SÉRGIO DE ANDRADE

The rise of quantum computing poses a significant threat to contemporary public-key cryptography, which relies on problems like integer factorization and the discrete logarithm, both of which are solvable in polynomial time by Shor's algorithm. This has prompted a global effort to develop and standardize post-quantum cryptographic primitives that are resistant to attacks by both classical and quantum computers. Among the various families of post-quantum candidates, isogeny-based cryptography has emerged as a particularly compelling, albeit challenging, alternative. This paper explores the mathematical foundations of elliptic curve isogenies and their application as a basis for quantum-resistant cryptographic protocols. We delve into the structure of supersingular elliptic curves and the associated isogeny graphs, which form the basis for the conjectured hardness of the underlying computational problems. The Supersingular Isogeny Diffie-Hellman (SIDH) protocol and its derivative, Supersingular Isogeny Key Encapsulation (SIKE), are examined in detail as principal examples of this cryptographic paradigm. We analyze their key advantages, most notably their exceptionally small key sizes, which offer significant benefits for constrained environments. However, the security landscape of isogeny-based cryptography has been dramatically reshaped by recent cryptanalytic breakthroughs that completely broke the SIKE protocol. This paper analyzes the nature of these attacks, discussing the vulnerabilities they exploited and the profound implications for the field. We conclude by surveying the current state of isogeny-based cryptography in the aftermath of these developments, outlining the proposed countermeasures, alternative designs, and the open research questions that define the new frontier for this promising area of post-quantum cryptography.

Syed Muntasir Mamun

This seventh volume of the series establishes the first formal post-post-quantum taxonomy—a three-tier ontological and epistemological hierarchy that reclassifies all post-quantum cryptographic schemes according to the fundamental nature and justification of their security. The hierarchy progresses from computational (algorithmic hardness assumptions) through physical (thermodynamic, noise-engineered, and hardware-constrained irreversibility) to device-independent (information-theoretic guarantees independent of device implementation). Building directly upon the anatomical critiques, hybrid primitives (Vol II), QRAM-hardness (Vol III), parameterized adversary models (Vol IV), GKP-resistant countermeasures (Vol V), and NISQ empirical validation (Vol VI), we provide a complete reclassification of NIST PQC standards, hybrid protocols, and the novel primitives introduced across the series. Dedicated Mamun, SM (2026) ……………………………….1 Theoretical and Physical Boundaries of Lattice-Based Post-Quantum Cryptography: An Anatomical Critique of Post-Post-Quantum Taxonomy (Vol-VII) sections dissect the ontology (what security is) and epistemology (how security claims are known and justified) of each tier. K S C The taxonomy reveals that current lattice-based schemes remain anchored in the provisional computational tier, while the countermeasures developed in Vols III–VI elevate them into the physical tier. True device-independent security remains an aspirational frontier. Also available on Academia.edu: https://www.academia.edu/attachments/132771868/download_file?s=portfolio

Syed Ismail Abdul Lathif, Priya Saxena, Priyanshi Singh