Statistical Analysis in Cryptography

In today’s world the need to protect vocal and written communication between individuals,

institutions, entities and commercial agencies is ever present and growing. Digital communication has, in part, been integrated into our social life. For many, the day begins with the perusal of e-mail and the tedious task of eliminating spam and other messages we do not consider worthy of our attention. We turn to the internet to read newspaper articles, to see what’s on at the cinema, to check flight arrivals, the telephone book, the state of our checking account and stock holdings, to send and receive money transfers, to shop on line, for students’ research and for many other reasons. But the digital society must adequately protect communication from intruders, whether persons or institutions which attack our privacy. Cryptography (from o&, hidden), the study and creation of secret writing systems in numbers or codes, is essential to the development of digital communication which is absolutely private insofar as being impossible to be read by anyone to whom it is not addressed. Cryptography seeks to study and create systems for ciphering and to verify and authenticate the integrity of data. One must make the distinction between cryptoanalysis, the research of methods an “enemy” might use to read the messages of others and cryptography. Cryptography and cryptoanalysis are what make up cryptology.

Until the 1950s cryptography was essentially used only for military and diplomatic communication. The decryption of German messages by the English and of Japanese messages by the Americans played a very important role in the outcome of the Second World War. The great mathematician Alan Turing made an essential contribution to the war effort with his decryption of the famous Enigma machine which was considered absolutely secure by the Germans. It was the Poles, however,

who had laid the basis for finding its weak link. Cryptography also played a vital role in the Pacific at the battle of Midway Regarding Italy, the naval battles of Punta Stilo and of Capo Matapan were strongly influenced by the interception and decryption of messages.

There are four disciplines which have important roles in cryptography:

1. Linguistics, in particular Statistical Linguistics

2. Statistics, in particular the Theory of the Tests for the Analysis of Randomness

and of Primality and Data Mining

3. Mathematics, in particular Discrete Mathematics

4. The Theory of Information

The technique of Data Mining seems to be of more use in the analysis of a great number of data which are exchanged on a daily basis such as satellite data. Technical developments are largely inter-disciplinary. This suggests that new applications will be found which will, in turn, lead to new queries and problems for the scholars of Number Theory,Modular Arithmetic, Polynomial Algebra, Information Theory and Statistics to apply to cryptography.

Until the 1950s the decryption of messages was based exclusively on statistical methods and specific techniques of cryptography. In substance, the working instruments of cryptography, both for the planning of coding systems and for reading messages which the sender intended remain secret, were statistical methods applied to linguistics. The key to decoding systems using poly-alphabetic substitution and simple and double transposition has always been the analysis of the statistical distribution of graphemes (letters, figures, punctuation marks, etc.). Mathematics was not fundamental to the work of the cryptoanalyst.

Today,with the advent of data processing technology, coding of messages is done by coding machines. The structure of reference is the algebra of Galois (GF(q)). The search for prime numbers, in particular tests of primality, are of notable interest to modern cryptology.