Sure, we can break down the math behind protocols like Diffie‑Hellman, RSA and elliptic‑curve schemes, then pinpoint where side‑channels, weak primes or quantum algorithms could slip through. We’ll keep the analysis tight and focus on the practical attack vectors that matter.

Let’s begin with the group choice. In a classic Diffie‑Hellman scheme you pick a prime p and a generator g of a large prime‑order subgroup. If p−1 has small factors, an attacker can use the Pohlig‑Hellman algorithm or even a small‑subgroup attack, so we need p−1 with only large prime factors. Also check that g is truly a generator, not a low‑order element; otherwise a malicious server can force the shared key into a tiny set. Once the parameters pass that test, we move to the exchange itself, watching for timing leaks, or if the private exponents are poorly chosen—small or non‑random, it’s a recipe for a log‑jam. So that’s the first layer of scrutiny.

Next, run through the actual handshake: generate private exponents, compute g^a and g^b, exchange them, compute shared secret. Verify that each side checks the peer’s value for membership in the subgroup to prevent small‑subgroup or invalid‑curve attacks. Then watch the multiplication and exponentiation routines for side‑channel leakage—timing, power, cache. If those stay flat, the exchange should be secure.