Is EC Private Key Secure

This section discusses the question of: how secure is the EC (Elliptic Curve) key pair? Or how hard is it someone to figure out the private key from a given EC public key?

Is the EC (Elliptic Curve) private key secure? Or is it hard for Eve figure out the private key from the public key?

The answer is yes, if the private number i and the subgroup order n are large enough. This is because the scalar multiplication i*G is a good trapdoor function.

For example, if Alice selects a 64-bit long key, i:

i = 264 = 9,223,372,036,854,775,808

Alice can calculate i*G with maximum of 128 operations. But Eve needs to try 9,223,372,036,854,775,808 point addition operations figure out i.

Let's assume that Eve uses a laptop with a CPU of 3 GHz speed. Eve has a very efficient program that can perform 1 point addition operation per CPU click. So Eve can perform 3,000,000,000 operations per second. Eve still needs about 3,000,000,000 seconds = 50,000,000 minutes = 833,333 hours = 34,722 days = 95 years to find i.

If Alice selects a 256-bit long key i, 0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff Eve needs about 3.8597363079105397e+67 seconds = 6.432893846517566e+65 minutes = 1.0721489744195944e+64 hours = 4.467287393414976e+62 days = 1.2239143543602674e+60 years to find i.

Last update: 2019.

Table of Contents

 About This Book

 Geometric Introduction to Elliptic Curves

 Algebraic Introduction to Elliptic Curves

 Abelian Group and Elliptic Curves

 Discrete Logarithm Problem (DLP)

 Finite Fields

 Generators and Cyclic Subgroups

 Reduced Elliptic Curve Groups

 Elliptic Curve Subgroups

 tinyec - Python Library for ECC

EC (Elliptic Curve) Key Pair

 EC Private and Public Key Pair

Is EC Private Key Secure

 EC Private Key Example - secp256k1

 Generate secp256k1 Keys with OpenSSL

 EC Key in PEM File Format

 EC Key File with Curve Name

 Create EC Public Key File

 ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

 ECDSA (Elliptic Curve Digital Signature Algorithm)

 ECES (Elliptic Curve Encryption Scheme)

 Terminology

 References

 Full Version in PDF/EPUB