Merkle’s Puzzle-box Key Exchange

Cryptography is fantastic, but much of it suffers from being unintuitive and math-heavy. This can make it challenging to teach to those without a math or computer science background, but makes it particularly difficult to develop a sense of why something is secure.

There are a handful of cryptographic systems however, that are delightfully easy to illustrate and provide a great introduction to security concepts. One such system is Merkle’s Puzzles.

The Problem

Similar to the Diffie-Hellman Key Exchange, the goal of Merkle’s Puzzle Boxes is for two parties (we’ll call them Alice and Bob) to agree on a password to encrypt their messages with. The problem is that Alice and Bob can only communicate in a public channel where anyone could be listening in on them. How can they exchange a password securely?

The Process

Alice creates several puzzle boxes (since she has a computer, we’ll say she makes several thousand of them). Each puzzle box has three pieces:

  1. A random number identifying the box
  2. A long, random, secure password
  3. A hash of parts 1 and 2

Each “box” is then encrypted with a weak password that can be brute-forced without taking too long. Let’s say it’s a five character password.

Alice then sends all her encrypted puzzle boxes to Bob:

Alice sending puzzle boxes

Bob selects one box at random, and brute-forces the five character password. He knows he has the right password because the hash inside the box will match the hash of parts 1 and 2 in the box. He then announces back to Alice the number of the box he broke open.

Bob sending back number of chosen box

Alice (who has all the unlocked puzzle boxes already) looks up which box Bob has chosen, and they both begin encrypting their messages with the associated password.

Why is it Secure?

If we have an eavesdropper, Eve, listening in to the exchange, then she can capture all the puzzle boxes. She can also hear the number of the puzzle box Bob has broken in to when he announces it back to Alice. Eve doesn’t know, however, which box has that number. The only way to find out is to break in to every puzzle box until she finds the right one.

Eve breaking every puzzle box

This means while it is an O(1) operation for Bob to choose a password (he only has to break one box), it is an O(n) operation for Eve to find the right box by smashing all of them.

This also means if we double the number of puzzle-boxes then the exchange has doubled in security, because Eve must break (on average) twice as many boxes to find what she’s looking for.

Why don’t we use Puzzle-boxes online?

Merkle’s puzzles are a great way of explaining a key exchange, but computationally they have a number of drawbacks. First, making the system more secure puts a heavy workload on Alice. But more importantly, it assumes the attackers and defenders have roughly the same computational power.

An O(n) attack complexity means Eve only needs n times more CPU time than Bob to crack the password - so if the key exchange is configured to take three seconds, and there are a million puzzle boxes, then it would take 35 days for that same computer to crack. But if the attacker has a computer 100 times faster than Bob (say they have a big GPU cracking cluster) then it will only take them 0.35 days to break the password. A more capable attacker like a nation state could crack such a system almost instantly.

If Eve is recording the encrypted conversation then she can decrypt everything after the fact once she breaks the puzzle box challenge. This means even the original 35-day attack is viable, let alone the GPU-cluster attack. As a result, we use much more secure algorithms like Diffie-Hellman instead.

Posted 7/17/17