BIP-39 Seed Phrase Combinations vs. Natural Quantities

Introduction

The BIP-39 standard uses a 2,048-word list to create seed phrases for cryptocurrency wallets, typically 12 or 24 words long. These generate vast numbers of combinations, which can be compared to natural quantities like the number of sand particles on Earth or stars in the observable universe. This page visualizes those comparisons.

Number of Combinations

• 12-Word Seed: \( 2,048^{12} \approx 5.363 \times 10^{39} \) combinations
• 24-Word Seed: \( 2,048^{24} \approx 2.860 \times 10^{79} \) combinations

Comparisons to Natural Quantities

• Sand Particles on Earth: \( 7.5 \times 10^{18} \) to \( 1 \times 10^{19} \)
• Stars in the Observable Universe: \( 1 \times 10^{22} \) to \( 1 \times 10^{24} \)
• Atoms in the Observable Universe: \( 1 \times 10^{80} \) to \( 1 \times 10^{82} \)

Visual Representation

This chart shows the relative magnitude of these numbers on a logarithmic scale (base 10). The x-axis represents the logarithmic values of the quantities, and the y-axis shows a logarithmic scale from 0 to 90, corresponding to the range of data.

Analysis

• \( 2024^{12} \approx 1.2589257 \times 10^{41} \) is larger than sand particles and stars but smaller than atoms.
• 12-word seed combinations (\( 5.363 \times 10^{39} \)) exceed sand and stars by factors of \( 10^{20} \) and \( 10^{15} \)-\( 10^{17} \), respectively.
• 24-word seed combinations (\( 2.860 \times 10^{79} \)) are on the order of atoms, differing by factors of 0.286 to 2.86.

These vast numbers highlight the cryptographic strength of BIP-39 seed phrases, far surpassing natural counts except at the atomic scale.