Perfect Secrecy

Cryptography · Axiom Academy

Shannon's mathematical definition of unbreakable encryption and the theory of information-theoretic security 1. Shannon's Definition of Perfect Secrecy Perfect secrecy means that observing a ciphertext provides zero information about the plaintext. Mathematically, the probability of a message given the ciphertext equals the probability of the message without knowing the ciphertext. In other words: learning the ciphertext tells you nothing new about what the plaintext might be. Every possible message remains equally likely before and after seeing the ciphertext. Shannon showed that perfect secrecy has several equivalent mathematical formulations. Each perspective reveals different insights about what makes encryption unbreakable. First formulation: Ciphertext reveals nothing about plaintext (independence). Second formulation: All messages are equally likely given any ciphertext. Third formulation: Every plaintext-ciphertext pair has equal probability. 3. Shannon's Impossibility Theorem Shannon proved a fundamental limitation: perfect secrecy requires the key space to be at least as large as the message space. This has profound implications for practical cryptography. The number of possible keys must be at least as large as the number of possible messages. Why this matters: If you want to send a 1 GB message with perfect secrecy, you need a 1 GB key. For a 128-bit message, you need at least a 128-bit key. There's no way around this fundamental limit.

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