Introduction to cryptography

I teach Introduction to cryptography [NDMI100] in the winter semester of 2022/2023. The lecture will cover the basics of both theoretical and practical cryptography, focusing on protocols currently used on the Internet.

Lectures will be on Wednesdays from 15:40 in S5.

If you want to consult anything, please write an e-mail to and we will discuss possibilities.

date topics recording
15. 2. Cryptographic primitives: symmetric and asymmetric ciphers, hash functions, random generators. Protocols and roles. Kerckhoffs principle. Simple protocols: multi-party communication, signatures (symmetric and asymmetric) hybrid ciphers, challenge-response authentication. Designing an auction protocol: padding, nonces, sequence numbers, signatures, session IDs.
Exercise: How to toss a coin over a phone call?
22. 2. No lecture today, but please watch recordings from 2020: part 1 (from 53:10 to 1:32:23), part 2 (from beginning to 45:42). Basic types of cryptographic attacks. Security level. Different kinds of "Birthday attacks". One-time pad and Vernam's cipher. Perfect security and its limits. Secret sharing and threshold schemes (construction with polynomials).
1. 3. Introduction to symmetric ciphers. Block ciphers: trivial examples, an attempt to define security, ideal block ciphers. General constructions: iterated ciphers, substitution-permutation networks, Feistel networks. DES: history, structure, critique, work-arounds (3-DES). AES a.k.a. Rijndael: history, structure.
Exercise: Why is the security level of 2-DES only 57 bits?
Further reading:: EFF DES cracker, details of AES.
8. 3. AES: critique. How to (mis)use a block cipher: padding, modes ECB, CBC, CTR. Information leaks in CBC and CTR modes. Padding oracle attacks on CBC. Stream cipher sketches: LFSR-based constructions, eSTREAM project, Trivium, RC4, ChaCha20.
Further reading: Trivium, ChaCha20, ciphertext stealing.
Illustrations: Block cipher modes (plaintext, ECB, CBC), visual secret sharing (layer 1, layer 2, both layers put together),
15. 3. Hash functions: requirements, Merkle-Damgård construction by iterating compression function, length extension attacks. Commitment using hash functions. Birthday attacks on hash functions: the tortoise, the hare, and the turtle. How to obtain a compression function: Davies-Meyer construction from a block cipher. Practical hash functions: MD-5 (broken), SHA-1 (broken), SHA-2, and SHA-3 (sketch). video
(sorry for noise)
22. 3. No lecture today, but please watch recordings from 2020: part 1 (from 1:20:00 to the end), part 2, part 3 (from start to 32:00). Sponge hashes: analysis for random permutations, Keccak, SHA-3, SHAKE-n. Parallel hashing: Merkle trees and Sakura. Symmetric signatures (MACs): construction from hash functions (KMAC and HMAC), different ways of combining a MAC with a cipher. CBC-MAC. Information-theoretic MAC from 2-independent linear functions or from polynomials. Practical polynomial MACs: GCM mode, Poly1305 (sketch).
29. 3. Generating random bits: pseudo-random generators from symmetric ciphers, physical randomness (LavaRand, circular oscillators, …), combining both – Fortuna, RDRAND, /dev/random. Putting all symmetric cryptography together: a secure channel. Basics of algorithmic number theory: Complexity of arithmetics. Euclid's algorithm and Bézout's coefficients. Multiplicative group modulo N. Little Fermat's and Euler's theorem. Lagrange's theorem on subgroups (without proof). video
5. 4. More number theory: Chinese remainder theorem and its constructive version. Computing Euler's function. Factoring vs. primality testing. Fermat and Rabin-Miller tests (R-M without proof), Carmichael numbers, note on Agarwal's deterministic test. Generating large primes. Multiplicative group is cyclic, discrete logarithms, how to find a generator. Discrete square roots, Euler's criterion. video
12. 4. RSA cipher: Definition, acceleration tricks. Properties of RSA: swapping roles of keys, commutativity, homomorphism with respect to multiplication (application: blind signatures). Attacks: short messages, small secret exponent, meet in the middle, similar primes, similar messages, multiple keys with the same modulus, effects of computation errors. Semantic security: RSA preserves Jacobi symbol, but determining the lowest-significant bit or the half of the interval is hard. RSA padding and a note on Bleichenbacher attack. RSA versus factoring. video
19. 4. Diffie-Hellman key exchange, signing the result/transcript, quadratic residues, problems with subgroups, DH in general groups. ElGamal cryptosystem and its derivation from DH. Elliptic curves over reals and over finite fields (sketch only). Asymmetric signatures: attack types, signatures using RSA, hashing is essential. ElGamal signatures, notes on EC(DSA), consequences of weak random numbers, and deterministic DSA.
Further reading on elliptic curves: blog post series by Andrea Corbellini, the SafeCurves project.
26. 4. Putting bits together: secure channel protocol, forward security. Implementation issues: how to not develop software, real-world attacks, various kinds of side channels, storing and destroying secrets, keeping state. Lessons from physical security. video
3. 5. Real-world cryptography. Handling passwords: hashing, rainbow tables, salting, key derivation functions (PBKDF2). Interactive authentication: challenge-response, the SCRAM protocol. DNSSEC (secure domain name system). Verifying identity: directory services, certification authorities, trust on first use, web of trust. Double ratchet protocols for protecting two-party messaging. video
10. 5. No lecture today: The rector's sports day.
17. 5. Real-world cryptography continued: TLS 1.3 (and its predecessors) – record protocol, handshake protocol, alert protocol, key exchange and authentication, pre-shared keys and session resume, interesting extensions. The Internet PKI: X.509, certification authorities and their behavior. video


Exam dates are listed in SIS, please sign up there. If no date suits you, let me know and we can arrange another one. If you want to consult anything before the exam, feel free to ask.

You are expected to know the theory presented at the lecture (constructions, theorems, and proofs) and use it to analyse (i.e., break) simple protocols.


This page is maintained by Martin Mareš