Problem: discrete_logarithm

Your task is to implement a solver for the Discrete Logarithm Problem and verify its correctness in Dafny.  Preconditions: The inputs g (generator), h (target), and p (modulus) are non-negative integers (simulating u64). Assume p > 1. Requirements: Implement the method discrete_log_naive which returns Option<int>. Goal: Find an integer $x$ such that $g^x \equiv h \pmod p$, where $0 \le x < p$. Algorithm: Use a naive linear search. Iterate x from 0 to p-1. Maintain a running product for $g^x \pmod p$ to avoid recomputing the power from scratch. Minimality: If multiple solutions exist, you must return the smallest x. Unsolvable Case: If no such x exists in the range, return None. Verification Challenges: (1) Incremental Modular Exponentiation: You are not calling a power function inside the loop; you are updating current_val = (current_val * g) % p. You must prove the invariant that at iteration i, current_val == spec_pow_mod(g, i, p). This requires the modular multiplication lemma. (2) Negative Proof (None): The hardest part is the None case. If the loop finishes without returning, you must prove the universal quantifier: $\forall k \in [0, p), g^k \not\equiv h \pmod p$. (3) Minimality (Some): When you return Some(x), you must prove that no earlier index k < x was a solution. Since your loop searches strictly from 0 upwards, this follows from the control flow, but you must capture it in the loop invariant ("for all checked $k$, $g^k \neq h$").

// Following is the block for necessary definitions
// <preamble>
// Recursive definition of Modular Exponentiation: (b^e) % m
function spec_pow_mod(b: int, e: int, m: int): int
  decreases e
  requires m > 1
{
  if e <= 0 then
    1
  else
    (b * spec_pow_mod(b, e - 1, m)) % m
}

// The predicate that defines a valid Discrete Logarithm x
predicate is_discrete_log(g: int, h: int, p: int, x: int)
  requires p > 1
{
  spec_pow_mod(g, x, p) == h % p
}
// </preamble>

// Following is the block for potential helper specifications
// <helpers>

// </helpers>

// Following is the block for proofs of lemmas
// <proofs>

// </proofs>

// Following is the block for the main specification
// <spec>
// Dafny uses a datatype for Option
datatype Option<T> = Some(value: T) | None

method discrete_log_naive(g: int, h: int, p: int) returns (res: Option<int>)
  // Simulate u64: inputs are non-negative
  requires g >= 0 && h >= 0 && p >= 0
  requires p > 1
  
  ensures match res
    case Some(x) => 
      // 1. It is a valid solution
      is_discrete_log(g, h, p, x) &&
      // 2. It is within bounds
      0 <= x < p &&
      // 3. It is the *smallest* solution
      (forall k :: 0 <= k < x ==> !is_discrete_log(g, h, p, k))
    case None =>
      // We assert that NO solution exists in the range [0, p)
      forall k :: 0 <= k < p ==> !is_discrete_log(g, h, p, k)
// </spec>
// <code>
{
  // Implement and verify the naive algorithm to compute the discrete logarithm
}
// </code>

method Main() {
}