Discret Logarithm

The discrete logarithm is a central mathematical problem in cryptography. It involves finding an integer $ x $ such that:

$$ g^x \equiv y \pmod{n} $$

where:

— $ g $ is a generator of a finite cyclic group (often $ \mathbb{Z}_n^* $),

— $ y $ is an element of this group,

— $ n $ is a prime number (or an integer defining the order of the group).

This is the discrete equivalent of the classical logarithm, where is seek the exponent $ x $ that transforms $ g $ into $ y $ in a finite space.

Its primary applications are in cryptography. It forms the basis of algorithms like Diffie-Hellman (key exchange) and ElGamal (encryption/signature).

It is also used in Zero-Knowledge Proof for authentication protocols that do not require disclosure of secrets.

The Discrete Logarithm Problem (DLP) consists of finding $ x $ in the equation: $$ g^x \equiv y \pmod{n} $$ where $ g, y, n $ are known.

This problem is considered difficult for well-chosen groups, making it a cornerstone of asymmetric cryptography.

For small $ n $, use:

— Exhaustive brute-force search: Test $ x $ from 1 to $ n-1 $

— Baby-step Giant-step algorithm:

1. Calculate $ g^j $ for $ j = 0 $ to $ m-1 $ (baby-steps)

2. Calculate $ y \cdot g^{-km} $ for $ k = 0 $ to $ m-1 $ (giant-steps)

3. Look for a collision

Many researchers have attempted to find methods for quickly solving the discrete logarithm problem. Several algorithms can be efficient in certain cases:

— Baby-step Giant-step: Generic algorithm in O(√n)

— Index Calculus: Efficient in (Z/nZ)^*, but not in elliptic curves.

— Pohlig-Hellman algorithm: Exploits the factorization of n (group order)

— Quantum computing attacks: Shor's algorithm solves the DLP in polynomial time on a quantum computer.

For $ (\mathbb{Z}/n\mathbb{Z})^* $:

— Choose a large prime $ n $ (e.g., 2048 bits).

— Verify that $ n-1 $ has a large prime factor.

— Select a generator $ g $ of prime order.

For elliptic curves:

— Use standardized curves (e.g., NIST P-256, Curve25519)

— Avoid curves with special properties (e.g., supersingular curves)