What is the Significance of One-Way Functions for Cryptography?
Public-key cryptosystems are based on (presumed) trap-door one-way functions. The public key gives information about the particular instance of the function; the private key gives information about the trap door. Whoever knows the trap door can perform the function easily in both directions, but anyone lacking the trap door can perform the function only in the forward direction. The forward direction is used for encryption and signature verification; the inverse direction is used for decryption and signature generation.
In almost all public-key systems, the size of the key corresponds to the size of the inputs to the one-way function; the larger the key, the greater the difference between the efforts necessary to compute the function in the forward and inverse directions (for someone lacking the trap door). For a digital signature to be secure for years, for example, it is necessary to use a trap-door one-way function with inputs large enough that someone without the trap door would need many years to compute the inverse function.
All practical public-key cryptosystems are based on functions that are believed to be one-way, but no function has been proven to be so. This means that it is theoretically possible that an algorithm will be discovered that can compute the inverse function easily without a trap door; this development would render any cryptosystem based on that one-way function insecure and useless. On the other hand, further research in theoretical computer science may result in concrete lower bounds on the difficulty of inverting certain functions, and this would be a landmark event with significant positive ramifications for cryptography.