The word "bit" has two meanings in cryptography. What are they, and how do they relate?

A storage bit is a binary digit used to represent data; Shannon's information bit measures information content. Storage bits are always at least as many as information bits.

1. The everyday "storage" bit (Bit_normal): a binary digit, the unit you count when you store or transmit data. To represent the ten digits 0–9 you solve $2^x = 10$, so $x = \log_2 10 \approx 3.32$; in practice you round up to 4 bits per digit.

2. Shannon's "information" bit (Bit_information), the older meaning from 1948: a measure of how much information a symbol carries. A perfect random generator that emits one of 8 equally likely values (A…H), each independent of the last (a memoryless source), produces exactly 3 bits of information per symbol.

How they relate: $\text{Bit}_{normal} \geq \text{Bit}_{information}$, with strict inequality whenever the source is not perfectly random and uniform. The gap is redundancy, and it is exactly what compression removes. A million random decimal digits stored at 4 bits each take 4 million storage bits, but an ideal compressor would need only about 3.32 million, because that is all the information they contain.

Why it matters for crypto: a "128-bit key" counts storage bits. Its security depends on how many information bits it carries, i.e. how randomly it was generated. A key derived from a weak password may have 128 storage bits but only 30 bits of entropy.

Go deeper:

• Entropy (information theory) — Shannon's bit as a unit of information, distinct from the binary digit.