# Notes

## Section 10: Cryptography and Cryptanalysis

Basic theory [of cryptography]

As was recognized in the 1920s the only way to make a completely secure cryptographic system is to use a so-called one-time pad and to have a key that is as long as the message, and is chosen completely at random separately for each message. As soon as there are a limited number of possible keys then in principle one can always try each of them in turn, looking in each case to see whether they imply an original message that is meaningful in the language in which the message is written. And as Claude Shannon argued in the 1940s, the length of message needed to be reasonably certain that only one key will satisfy this criterion is equal to the length of the key divided by the redundancy of the language in which the message is written—equal to about 0.5 for English (see below).

In a cryptographic system with keys of length n there will typically be a total of kn possible keys. If one guesses a key it will normally take a time polynomial in n to check whether the key is correct, and thus the problem of cryptanalysis is in the class known in theoretical computer science as NP or non-deterministic polynomial time (see page 1142). It is suspected but not established that there exist at least some problems in NP that cannot be solved in polynomial time, potentially indicating that for an appropriate system it might be impossible to do cryptanalysis in any time polynomial in n. (See page 1089.)

From Stephen Wolfram: A New Kind of Science [citation]