Bitcoin & Crypto-Assets

Bitcoin & Crypto-Assets, April-June 2025

Department of Mathematics, University of Milan

If you have an e-mail @studenti.unimi.it please join the 202504-math channel in the Slack BBT workspace using your first and last name (no nicknames) and a profile picture. Updates and conversations about the course will be posted in the Slack channel.

The course is for the students of University of Milan (La Statale) only; anyone else, please consider https://dgi.io/workshop.

Learning objectives

The course is an introduction to bitcoin and the associated blockchain technology.

Starting from a computationally focused approach to elliptic curves over finite fields and presenting the discrete logarithm problem as the cornerstone of public-key cryptography, bitcoin is introduced as ingenious breakthrough innovation.

Its game theory, computer science (distributed systems, distributed consensus), and monetary theory elements are examined in the attempt to properly convey the interdisciplinarity of the topic and appreciate its relevance.

Technical and programming elements about digital signatures, blockchain, Merkle tree, addresses, transactions, and timestamping are also provided to assess features and limits of the Bitcoin protocol.

Prerequisites

There are no strict prerequisites, even if a computer science mindset and some familiarity with algebra and finance might help to appreciate the course. While a rigorous formal approach is almost impossible in a course touching on so many and so different knowledge areas, intellectual curiosity is stimulated about the interplay between maths, cryptography, economic incentives, technology, monetary theory, regulatory issues, and politics.

Contents

• Cash, Electronic Money, Central Bank Money, eCash
• Internet Money
• Bitcoin Transactions
• About Money
• Private Money and the Centralization Dilemma
• The Double Spending Problem
• Bitcoin as Digital Gold
• Bitcoin as Investment Asset
• Bitcoin Financial Services
• Discrete Logarithm Problem on Finite Cyclic Groups
• Modular Arithmetic
• Finite Fields
• Elliptic Curves Over Real Numbers
• DLP on Elliptic Curves Over Finite Fields
• Hash Functions
• Partial Hash Inversion
• Hash Pointer Data Structures: Blockchain and Merkle Tree
• Design of A Simplified Digital Currency
• Distributed Consensus
• Mining
• P2P Network
• Protocol Governance
• Elliptic curve digital signature algorithm
• Elliptic curve Schnorr signature algorithm: Mu(lti)Sig(nature), threshold signature, batch verification
• Addresses and WIFs
• Hierarchical deterministic wallets: BIP32, BIP43, BIP44
• Mnemonic phrase: BIP39 and Electrum
• TxIns, TxOs, UTxO, nLockTime
• Bitcoin script language
• Transactions
• Blocks
• Wallets and Bitcoin Core
• Testnet, regtest, and wallet workshop
• Money and innovation: monetary and token engineering
• Hayek money and dual asset ledger money
• Blockchain beyond bitcoin
• Finance and blockchain
• Timestamping, notarization, and anchoring
• Smart contracts
• Distributed ledger technology

Teaching method

• Slide based lessons with associated bibliography
• Python 3 programming assignments and technology assignments (workbooks)
• Workshop labs on Bitcoin Core, Electrum, and OpenTimestamps

Teaching language

Italian or English, the latter if foreign students are present.

Assessment

To pass the exam each student must solve all the assignments for every lesson. Please send the assignments to the e-mail that will be indicated with subject bbt202504-math #1 for the first assignment, bbt202504-math #2 for the second assignment, etc.

Schedule

1. 2025-04-28 Monday 15:30-18:30, aula di dottorato
   Bitcoin as Digital Gold (Part 1)
   slides
2. 2025-05-05 Monday 15:30-18:30, aula di dottorato
   Bitcoin as Digital Gold (Part 2)
   slides
   Assignment #1: Testnet transaction
   Discrete Logarithm Problem on Finite Fields and Elliptic Curves
   slides
3. 2025-05-12 Monday 15:30-18:30, aula di dottorato
   Discrete Logarithm Problem on Finite Fields and Elliptic Curves
   slides
   Assignment #2: Finite Fields and Elliptic Curves
   Hash Functions
   slides
   Assignment #3: Partial Hash Inversion
   Blockchain, Mining, and Distributed Consensus (Part 1)
   slides
4. 2025-05-19 Monday 15:30-18:30, online
   Blockchain, Mining, and Distributed Consensus (Part 2)
   slides
5. 2025-05-27 Tuesday 15:30-18:30, online
   Elliptic Curve Digital Signature Algorithm (ECDSA)
   Schnorr Signature Algorithm (BIP340)
   Optimization of The Elliptic Curve Scalar Multiplication
   slides
   Addresses, WIFs, and Bitcoin Message Signing
   slides
   Assignment #4: Bitcoin Message Signature
6. 2025-06-09 Monday 15:30-18:30, aula di dottorato
   Transactions and Scripts
   slides
   Assignment #5: Halving Blocks
7. 2025-06-10 Tuesday 15:30-18:30, aula di dottorato
   Deterministic Key Chains
   slides
   Wallets and Custody
   slides
   Timestamping and the OpenTimestamps Protocol
   slides Assignment #6: Timestamped document
8. 2025-06-16 Monday 15:30-18:30, aula di dottorato
   Beyond Bitcoin: Between Hype and Reality
   slides
   Financial Products and Services for the Blockchain Economy
   slides
9. 2025-06-17 Tuesday 15:30-18:30, online
   The Cryptocurrency Frontier in Monetary Engineering
   slides and video
   Bitcoin: oro digitale per nuovi standard monetari
   Hayek Money: The Cryptocurrency Price Stability Solution

Python library

btclib.org github.com/btclib-org/btclib

Python scripts, Excel spreadsheets, and regtest lab material

github.com/btclib-org/bbt

Introductory reading

• Ferdinando Ametrano,
  “Bitcoin: oro digitale, finanza e tulipani”,
  https://docs.google.com/document/d/1gecm0uT43tl8d4WFYNs9H_v3p70PPfPmQITR4GxSWkE

Technology references

• Satoshi Nakamoto,
  “Bitcoin: A Peer-to-Peer Electronic Cash System” (2008),
  https://bitcoin.org/bitcoin.pdf
• Andreas M. Antonopoulos and David A. Harding,
  “Mastering Bitcoin: Programming the Open Blockchain” (3rd edition, 2023),
  O’Reilly Media,
  https://github.com/bitcoinbook/bitcoinbook
• Jimmy Song,
  “Programming Bitcoin: Learn How to Program Bitcoin from Scratch” (2020)
  Oreilly & Associates Inc, 978-1492031499,
  https://github.com/jimmysong/programmingbitcoin
• A. Narayanan, et al.,
  “Bitcoin and Cryptocurrency Technologies: A Comprehensive Introduction” (2016),
  Princeton University Press, 978-0691171692,
  https://www.coursera.org/learn/cryptocurrency, https://bitcoinbook.cs.princeton.edu, https://www.lopp.net/pdf/princeton_bitcoin_book.pdf
• Pedro Franco,
  “Understanding Bitcoin: Cryptography, Engineering and Economics” (2014),
  Wiley, 978-1119019169
• Ferdinando Ametrano,
  “Bitcoin, Blockchain, and Distributed Ledgers: Between Hype and Reality” (2017),
  https://ssrn.com/abstract=2832249
• Roger Wattenhofer,
  “Blockchain Science: Distributed Ledger Technology” (3rd edition, 2020),
  Independently published, 978-1793471734

Cryptography references

• Christof Paar, Jan Pelzl,
  “Understanding Cryptography”,
  Springer, 978-3642041006
  https://www.cryptography-textbook.com/, https://youtube.com/channel/UC1usFRN4LCMcfIV7UjHNuQg/videos
• Dan Boneh, “A Graduate Course in Applied Cryptography”,
  Stanford University
  https://www.coursera.org/learn/crypto, https://toc.cryptobook.us/
• Lawrence C. Washington,
  “Elliptic Curves: Number Theory And Cryptography” (2008),
  Chapman and Hall, 978-1420071467
• Standards for Efficient Cryptography Group
  SEC 1: Elliptic Curve Cryptography, March 2009. Version 2.0.
  https://www.secg.org/sec1-v2.pdf
• Standards for Efficient Cryptography Group
  SEC 2: Recommended Elliptic Curve Domain Parameters, March 2009. Version 2.0.
  https://www.secg.org/sec2-v2.pdf
• Guidelines for Efficient Cryptography
  GEC 2: Test Vectors for SEC 1, September 1999. Version 0.3.
  https://www.secg.org/

Monetary theory references

• Friedrich A. Hayek,
  “Denationalisation of Money: The Argument Refined”,
  https://mises.org/library/denationalisation-money-argument-refined
• Saifedean Ammous,
  “The Bitcoin Standard: The Decentralized Alternative to Central Banking” (2018),
  Wiley, 978-1119473862
• Ferdinando Ametrano,
  “Hayek Money: The Cryptocurrency Price Stability Solution” (2014),
  https://ssrn.com/abstract=2425270
• Robert Sams,
  “A Note on Cryptocurrency Stabilisation: Seigniorage Shares” (2015),
  https://github.com/rmsams/stablecoins/blob/master/paper.pdf
• Ferdinando Ametrano,
  “Bitcoin: oro digitale per nuovi standard monetari (2020),
  published in “Dal sesterzio al bitcoin”, Rubbettino Editore (edited by Angelo Miglietta, and Alberto Mingardi)
  https://drive.google.com/file/d/1-1k3wlL6ElZzJMjSakTjTNetJI5ws6wL