Understanding Large Numbers in Cryptography: How Key Sizes Compare to

Introduction
In cryptography, key size is a major factor in security, with modern encryption involving mind-bogglingly large numbers. To better understand just how big these numbers are, let’s compare the scale of various key sizes with other large quantities, like the number of stars in the universe. By the end, you’ll have a clearer perspective on why large key sizes are essential for strong cryptography.

1. Ordering Large Numbers: Key Sizes and Cosmic Scales

Here’s an ordering of cryptographic key sizes alongside some other vast numbers in the universe, listed from smallest to largest:

Number of Species of Birds on Earth: Around 10,000 species.
Number of Seconds Since the Beginning of the Universe: Roughly 101710^{17}1017 seconds.
Number of Stars in Our Galaxy: Estimated at 101110^{11}1011 stars.
Number of Web Pages Indexed by Google: Approximately 101210^{12}1012 pages.
Number of Possible 40-bit Keys: 2402^{40}240, or about 101210^{12}1012 possible keys.
Number of Possible 90-bit Keys: 2902^{90}290, roughly 102710^{27}1027 possible keys.
Number of Stars in the Universe: Around 102410^{24}1024 stars.
Number of Possible 128-bit Keys: 21282^{128}2128, approximately 103810^{38}1038 possible keys.

2. Why Large Key Sizes Are Essential in Cryptography

These massive numbers demonstrate why encryption relies on large key sizes for security. The higher the key length, the more possible combinations there are, making brute-force attacks practically impossible even with advanced computational power.

40-bit Keys: Now considered weak due to the limited number of possibilities, which modern computers can crack quickly.
128-bit Keys: Provides strong security for modern applications, as even supercomputers would take billions of years to try every combination.

3. Thinking in Exponents: Grasping Large Scales

To further put this into perspective, imagine some additional quantities:

Number of Grains of Sand on Earth: Estimated at 101810^{18}1018.
Number of Atoms in the Human Body: Roughly 102710^{27}1027.
Number of Planck Times Since the Big Bang: Around 106110^{61}1061.

Such comparisons illustrate that modern cryptographic keys operate on a scale beyond most natural quantities, adding essential security to digital communications.

Conclusion
Large numbers are a cornerstone of cryptography, with key sizes designed to be practically impossible to break. By comparing key sizes with cosmic and universal scales, we gain insight into why cryptography relies on such vast figures to keep data secure.