Formally Verifying Cryptographic Primitives: Implementing Shannon Ciphers in Lean 4
This technical breakdown details the formal verification of the One-Time Pad protocol using the Lean 4 proof assistant. By leveraging dependent types and modular arithmetic libraries, we establish a machine-checked model of Shannon cipher correctness.
The Lean 4 Verification Environment
Lean 4, developed by Leonardo de Moura at Microsoft Research, functions as both a pure functional programming language and an interactive theorem prover. Because programs in Lean are free of side effects and treat functions as first-class values, they are highly suited for mathematical proofs. The compiler acts as the validator: if a proof compiles successfully, the underlying mathematical statements are guaranteed to be correct under the Lean trust model. Systems engineers can leverage this paradigm to break complex cryptographic specifications into modular, machine-checked proofs.
Type-Safe Bitstring Modeling via Dependent Types
To verify cryptographic primitives like the One-Time Pad (OTP), we must first establish a mathematically rigorous representation of bitstrings. Rather than relying on standard runtime arrays, we use dependent types to encode the length of the bitstring directly into its type signature. This ensures compile-time safety across cryptographic operations.
To model the key, message, and ciphertext spaces as bitstrings of equal length, we import Lean's finite field library, `Mathlib.Data.ZMod.Basic`, which implements arithmetic modulo $n$ via the `ZMod n` type. For bitstrings, we configure arithmetic modulo 2 to represent binary values.
The custom `BitString` type is defined as follows:
```lean def BitString (L : ℕ) : Type := Vector (ZMod 2) L ```
This definition utilizes Lean's `Vector α n` type, creating a vector of length `n` with elements of type `α`. Because `L` is parameterized as a natural number (`ℕ`), any mismatch in bitstring lengths during cryptographic operations will trigger a compilation error, eliminating a broad class of runtime bounds issues.
Defining the XOR Operator
The One-Time Pad relies on bitwise exclusive-OR (XOR) operations. In Lean, we define the `xor` function signature over two `BitString` instances of identical length `L`. To simplify usage, the parameter `L` is marked as implicit using curly braces:
```lean def xor {L : ℕ} (x y : BitString L) : (BitString L) ```
To implement the function body, we perform component-wise modular addition using `zipWith`. The predefined vector function `zipWith` has the following signature:
```lean def zipWith (f : α → β → φ) (as : Vector α n) (bs : Vector β n) : Vector φ n ```
This function signature showcases Lean's use of currying, where multi-argument functions are represented as sequences of single-argument functions (denoted by the arrow `→`). This allows for partial application and clean functional composition when mapping the addition modulo 2 operator over the two input vectors.
Shannon Cipher Verification Requirements
To prove that the One-Time Pad is mathematically secure, the implementation must be mapped to the formal definition of a Shannon cipher as described in Applied Cryptography literature. A valid Shannon cipher requires three components:
- An encryption function
- A decryption function
- A correctness proof demonstrating that decryption is the exact inverse of encryption
For the One-Time Pad, encryption and decryption are symmetric XOR operations. Proving correctness requires establishing several algebraic properties of XOR, specifically its commutativity, associativity, identity element, and self-inverse property. Once these lemmas are verified by the compiler, we can formally prove that decrypting an encrypted ciphertext yields the original message, establishing a fully verified Shannon cipher.