# Public Key Cryptography: Merkle’s Puzzles

Up until the invention of public key cryptography in the 1970s, symmetric ciphers were the only encryption scheme available. A symmetric cipher refers to a cryptographic model where the communicating parties share a secret key to ensure the privacy of communications. Depending on the key, a symmetric encryption algorithm performs various permutations and substitutions to transform a given message into ciphertext. To recover the original message the decryption algorithm would perform the same operations in reverse, using the secret key. However, the key had to be sent through a secure channel, e.g. a trusted courier, registered mail, or best of all – agreed verbally in advance. There did not seem to be any other way for secure information exchange.

In 1974 Ralph Merkle, a computer science student at UC Berkeley attended a course in computer security. As part of the course he submitted a proposal addressing the issue of the distribution of a secret key [1]. The proposal was rejected several times, which led Merkle to drop the course and continue working on his idea independently [2]. Assuming that only an insecure channel is available for transmitting information, Merkle suggested that communications be based on a set of ‘puzzles’, solution to which would contain a unique puzzle ID and a unique puzzle key. If user $B$ wishes to set up a connection with user $A$, they require $A$ to generate a large number, say $N=1,000,000$, of puzzles and transmit them to $B$. After picking at random and solving one of the puzzles, $B$ transmits only the puzzle ID to $A$, and keeps the puzzle key private, which can then be used for subsequent message encryption with a symmetric cipher. Since the puzzle ID is sent through an insecure channel, it is available to an adversary, however, only  $A$ can match the puzzle ID to the puzzle key. The only way for the adversary to obtain the key is to solve every puzzle until a matching ID is found. This would on average require to solve $\frac{1}{2} N$ puzzles.

Figure: First $A$ transmits a large number of puzzles $\text{X}_1, \text{X}_2, \ldots, \text{X}_N$ to $B.$ Next, $B$ randomly selects and solves one puzzle $\text{X}_j$ in order to obtain an $(\text{ID}_j,\text{key}_j)$ pair. Finally, $B$ sends only the puzzle $ID=ID_j$ back to $A.$

A puzzle could be a symmetric encryption of the puzzle key and ID, solvable by brute-force, i.e. by trying all possible keys. If one puzzle takes 2 minutes to solve, with $1,000,000$ puzzles it would take an adversary an average of around a year to obtain the secret key. Merkle’s proposal was a new paradigm of which the puzzles were just an example. Indeed, soon after the proposal, cryptographic systems based on hard mathematical problems such as discrete logarithm [3] and integer factorization [4] were developed. The model is now known as public key cryptography, and it revolutionised the field by eliminating the need for a secure channel to transmit the key.

### References

[1] Merkle, R. C. Secure communications over insecure channels. Communications of the ACM 21.4 (1978): 294-299.

[2] Diffie, W. The first ten years of public-key cryptography. Proceedings of the IEEE 76.5 (1988): 560-577.

[3] Diffie, W. and Hellman, M. E. New directions in cryptography. Information Theory, IEEE Transactions on 22.6 (1976): 644-654.

[4] Rivest, R. L., Shamir, A. and Adleman, L. A method for obtaining digital signatures and public-key cryptosystems. Communications of the ACM 21.2 (1978): 120-126.

Author: Mante Zemaityte, mante.zemaityte@manchester.ac.uk