Course

Unit 1

Hash Functions, SHA-3, and Keccak

Cryptographic hash functions; security properties of hash functions; preimage resistance; second preimage resistance; collision resistance; birthday attack and birthday bound; Boolean algebra and bitwise operations; XOR operations; permutations and nonlinear Boolean functions; entropy and randomness; Merkle-Damgård construction; sponge construction; absorbing and squeezing phases; capacity and rate; SHA-3 family; SHA3-224, SHA3-256, SHA3-384, SHA3-512; SHAKE extendable-output functions; Keccak-f permutation; Keccak state representation; lanes, planes, sheets, and rounds; theta, rho, pi, chi, and iota steps; domain separation; role of hash functions in digital signatures, key derivation, randomness generation, and post-quantum cryptographic protocols.

Unit 2

Quantum Computing and Cryptography

Introduction to quantum computing; complex numbers; vectors and matrices; vector spaces; inner products; tensor products; qubits; superposition; measurement; entanglement; quantum gates; matrix representation of quantum gates; basic quantum circuits; probability amplitudes; quantum algorithms relevant to cryptography; modular exponentiation; integer factorization; discrete logarithm problem; Shor’s algorithm and its impact on RSA, Diffie-Hellman, elliptic curve cryptography, and elliptic curve Diffie-Hellman; Grover’s algorithm and its impact on symmetric-key cryptography and hash functions; classical versus quantum attack complexity; harvest-now-decrypt-later attacks; need for post-quantum cryptography; quantum-secure asymmetric cryptography; post-quantum security levels; overview of post-quantum cryptographic families.

Unit 3

Lattice-Based Cryptography

Linear algebra review; vectors and matrices; matrix-vector multiplication; modular arithmetic modulo q; vector norms; Euclidean distance; introduction to lattices; lattice basis; lattice dimension; fundamental parallelepiped; shortest vector problem; closest vector problem; approximate lattice problems; probability distributions; Gaussian and discrete error distributions; Learning With Errors problem; search-LWE and decision-LWE; simple LWE-based encryption; correctness and noise growth; polynomial arithmetic; rings and quotient rings; polynomial multiplication; cyclic and negacyclic convolution; Ring Learning With Errors; Ring-LWE encryption; Module-LWE; number theoretic transform overview; hardness assumptions including LWE, Ring-LWE, Module-LWE, SIS, Ring-SIS, and Module-SIS; practical lattice-based cryptography; key encapsulation mechanisms; CRYSTALS-Kyber / ML-KEM; lattice-based digital signatures; CRYSTALS-Dilithium / ML-DSA; overview of Falcon; parameter selection; correctness, security, and efficiency trade-offs; implementation issues; side-channel considerations.

Unit 4

Code-Based Cryptography

Introduction to finite fields; binary vector spaces; linear algebra over finite fields; matrices over GF(2); error-correcting codes; linear codes; generator matrix; parity-check matrix; encoding and decoding; Hamming weight; Hamming distance; minimum distance; syndrome computation; syndrome decoding problem; combinatorial complexity of decoding; McEliece cryptosystem; Niederreiter cryptosystem; suitable choices of codes; Goppa codes overview; quasi-cyclic codes; QC-MDPC codes; HQC and code-based key encapsulation mechanisms; probability of decoding failure; advantages and limitations of code-based cryptography; large public key sizes; implementation and performance considerations.

Unit 5

Hash-Based Signatures and PQC Standardization

Hash-based cryptography; one-time signatures; Lamport one-time signatures; Winternitz one-time signatures; combinatorics of one-time and many-time signatures; few-time and many-time signatures; binary trees; Merkle trees; authentication paths; XMSS overview; LMS overview; SPHINCS+ / SLH-DSA; stateless hash-based signatures; probability of forgery; security reductions; collision resistance and preimage resistance in hash-based signatures; signature size and performance considerations; comparison between lattice-based, code-based, and hash-based schemes; NIST post-quantum cryptography standardization process; ML-KEM, ML-DSA, and SLH-DSA; HQC as a backup code-based KEM; hybrid cryptography; combining classical and post-quantum algorithms; PQC in TLS, VPNs, cloud systems, IoT, and embedded devices; crypto-agility; public key infrastructure in the post-quantum era; migration planning; implementation challenges; side-channel attacks; performance benchmarking; future directions in post-quantum cryptography.

Text Books / References

• Daniel J. Bernstein, Johannes Buchmann, and Erik Dahmen, editors, Post-Quantum Cryptography, Springer, 2009.
• Christof Paar and Jan Pelzl, Understanding Cryptography, Springer, 2024.
• Jonathan Katz and Yehuda Lindell, Introduction to Modern Cryptography, CRC Press.
• Douglas R. Stinson and Maura Paterson, Cryptography: Theory and Practice, CRC Press.
• Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone, Handbook of Applied Cryptography, CRC Press, 1996.
• Neal Koblitz, A Course in Number Theory and Cryptography, Springer-Verlag, 1994.
• NIST, FIPS 203: Module-Lattice-Based Key-Encapsulation Mechanism Standard.
• NIST, FIPS 204: Module-Lattice-Based Digital Signature Standard.
• NIST, FIPS 205: Stateless Hash-Based Digital Signature Standard.
• NIST Post-Quantum Cryptography Standardization Project documents and reports.
• Guido Bertoni, Joan Daemen, Michaël Peeters, and Gilles Van Assche, Keccak and SHA-3 specification documents.

Post-Quantum Cryptography is an advanced course that introduces cryptographic systems designed to remain secure against adversaries equipped with quantum computers. Classical public-key systems such as RSA, Diffie-Hellman, and elliptic curve cryptography are vulnerable to quantum algorithms, particularly Shors algorithm. This course focuses on the mathematical foundations, design principles, and practical constructions of quantum-secure cryptographic systems.The course begins with hash functions and SHA-3/Keccak, which are essential for post-quantum signatures, key derivation, randomness generation, and secure protocol design. It then introduces the basic mathematical model of quantum computing and explains its impact on classical cryptography. The course subsequently develops the main families of post-quantum cryptography, including lattice-based, code-based, and hash-based schemes. The course also covers post-quantum standardization, implementation challenges, and migration to quantum-safe systems.

Course Objectives

• Understand the mathematical and cryptographic foundations required for post-quantum cryptographic systems.
• Analyse the impact of quantum computing on classical public-key cryptography and motivate quantum-secure alternatives.
• Study the design and security of lattice-based, code-based, and hash-based post-quantum cryptographic schemes.
• Understand post-quantum cryptography standardization, implementation issues, and migration strategies for real-world systems.

Course Outcomes

CO-PO Mapping

Correlation Levels: 3 = High, 2 = Moderate, 1 = Low