What is the factorial function and how is it relevant to cryptography?

n! (n factorial) is the product of all positive integers from 1 to n, and it defines the size of permutation spaces — critical for understanding how large a block cipher's theoretical key space could be.

* The permutation count (2ⁿ)! dwarfs the 2ⁿ permutations a key can actually select. *

Definition:

• $n! = n \times (n-1) \times (n-2) \times \dots \times 2 \times 1$
• $0! = 1$ (by convention)
• Example: $5! = 120$, $10! = 3{,}628{,}800$

Cryptographic relevance:

• A block cipher with $n$-bit blocks operates on $2^n$ possible values
• The total number of possible permutations is $(2^n)!$
• For AES ($n = 128$): $(2^{128})!$ is unimaginably large

Stirling's approximation for large $n$:

$$n! \approx \sqrt{2\pi n}\,\left(\frac{n}{e}\right)^n$$

For large $n$, computing $n!$ directly is impossible, but Stirling's formula gives a good approximation for the logarithmic (bit-length) calculations that matter in cryptography.

Go deeper:

• Factorial (Wikipedia) — the function and Stirling's approximation for huge n.