**EC Cryptography Tutorials - Herong's Tutorial Examples** - v1.02, by Dr. Herong Yang

ECDSA (Elliptic Curve Digital Signature Algorithm)

This chapter provides tutorial notes on ECDSA (Elliptic Curve Digital Signature Algorithm). Topics includes ECDSA digital signature generation process and verification process; security issue of the private key with same random number k is used; find possible public keys from a digital signature; installing pycoin Python library; using pycoin to generate and manage EC keys; using pycoin to generate and verify ECDSA digital signatures.

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ECDSA (Elliptic Curve Digital Signature Algorithm)

ECDSA Digital Signature Generation

ECDSA Digital Signature Verification

ECDSA Problem If k Used Twice

Find ECDSA Public Key from Signature

Download and Install pycoin

pycoin.ecdsa.ellipticcurve Module

pycoin.ecdsa.generator_secp256k1 Object

Generate EC Key Pair with pycoin.ecdsa

pycoin.ecdsa.ecdsa.sign() - Signature Generation

pycoin.ecdsa.ecdsa.sign() - Signature Verification

'openssl dgst' - Signing and Verification

Takeaways:

- ECDSA is an algorithm that uses elliptic curve subgroup properties to generate digital signatures of any given messages.
- A ECDSA digital signature contains two parts (r,s), where r represents a random number, and s represents the signature proof of r.
- Do not use the same random number r to generate digital signatures of 2 messages. Eve can recover your private key from those two signatures!
- Eve can also recover two possible public key values from a single signature. But this is not a risk, because everyone knows the public key anyway.
- pycoin Python library for the named elliptic curve subgroup, secp256k1, and methods to generate and verify ECDSA digital signatures. It also offers methods to generate private and public key pairs.
- "openssl dgst -sign" command can be used to generate a digital signature for a given file.
- "openssl dgst -verify" command can be used to verify a digital signature for a given file.

Table of Contents

Geometric Introduction to Elliptic Curves

Algebraic Introduction to Elliptic Curves

Abelian Group and Elliptic Curves

Discrete Logarithm Problem (DLP)

Generators and Cyclic Subgroups

tinyec - Python Library for ECC

ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

►ECDSA (Elliptic Curve Digital Signature Algorithm)