# AMT Probability and Cryptography

Alan M. Turing The Applications of Probability to Cryptography GCCS 1941/2, declassified and released 2012.

## CHAPTER 1

### Introduction

#### 1.1. Preamble

The theory of probability may be used in cryptography with most eﬀect when the type of cipher used is already fully understood, and it only remains to ﬁnd the actual keys. It is of rather less value when one is trying to diagnose the type of cipher, but if deﬁnite rival theories about the type of cipher are suggested it may be used to decide between them.

#### 1.4. A priori probabilities

The evidence concerning the possibility of an event occurring usually divides into a part about which statistics are available, or some mathematical method can be applied, and a less deﬁnite part about which one can only use one’s judgement. …

… It should be noticed that the whole argument is to some extent fallacious, as it is assumed that there are only two possibilities, viz. [that either the observed behaviour has come about because of the hypothesized rule, or it has come about at random]. The [actual] possibilities are of course endless, and it is therefore always necessary to bear in mind the possibility of there being other theories not yet suggested.

Applications of Probability theory have long been complicated by some significant differences of view. Some widely-held views were that Turing et al at Bletchley Park were Bayesians in the modern sense, and that anyway a very strong version of Bayesian theory, that Bayesian probability has universal application to all kinds of uncertainty, has since been ‘proved’. It was well-known that there were relevant documents held by GCHQ, and these were progressively released. This appears to be part of the last batch, but not for the first time.

It is clear from the preamble that AMT does not regard Bayesian probability theory as being of the greatest value for all kinds of problem, being of ‘less value’ when you do not fully understand the context. Under ‘a priori probabilities’ AMT notes that probabilistic reasoning only compares theories that are explicitly considered; it cannot take account of anything not considered. Although AMT does not present his theory in detail, it would seem impossible to deny this.

For example, consider a statement such as ‘X is true with probability 1’. This may be intended to suggest that X is probably true. Yet suppose that it is uttered by an academic with little experience of the real-world situation to which X pertains. Then all it can mean is that the academic has not been able to conceive of a situation in which X is not probably true. But we, having regard to the academic’s lack of experience, might not regard X as probably true.

Dave Marsay