Dawn of Mathematical Cryptology: Probabilists Vs Algebraists Or Algebraists & Probabilists?
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Dawn of Mathematical Cryptology: Probabilists vs Algebraists or Algebraists & Probabilists? Marek Grajek freelance consultant and historian, Poland [email protected] 1 Introduction Khalil1 (also known as Ahmed al-Farahidi), living in Basra in eight century. Their works Traditional cryptology, before the advent of linked early cryptography with the equally early the ciphering machines, relied mostly on the methods of mathematical statistics. It should be linguistical methods, and the role of mathematics noted that Al-Kindi’s interest in statistics was not in the codebreaking was limited to counting the limited to the secret writings. He proposed also a frequency of letters, their pairs and triplets. statistical approach to the medical treatment Machine cryptology changed everything; only evaluation. mathematicians were able to interpret even the During the mediaeval and early modern bare numbers of combinations resulting from the periods attacks based on the letter frequency use of the ciphering machine. The first successful were still popular due to the popularity of the application of advanced mathematics in nomenclators; monoalphabetic substitution cryptology, Marian Rejewski’s success with representing a part of the nomenclator made it Enigma, marked a change of paradigm; his attack vulnerable to statistical attack. Later on, when the was based on the algebra and the group theory. codes and nomenclators started to lose their However, soon after the outbreak of WW2 the popularity in favour of simpler and more Germans had changed the Enigma operational practical ciphers, statistical attacks have gained procedures, rendering most Polish methods of in importance; codebreakers started analysing attack ineffective. British mathematicians, who not only the frequency of the single letters, but took over from the Poles, had to revert to the old also their pairs, triplets and entire, popular words. and proven methods based on probability and Invention and fast adoption of the telegraph statistics, which dominated their work during, has changed this picture for a moment. Early and well after WW2. It was only 30 years after telegraph required not only hiding the message the end of this conflict that the role of the content, but also its compacting. Codes provided algebraic methods was restored. an easy and practical answer; second half of the This paper presents the early period of the nineteenth century was heavily dominated by the development of the mathematical cryptology, use of codes, which, from the codebreaker’s focusing on the clash of two approaches to the perspective, required the application of the codebreaking; that based on statistics and linguistical rather than statistical methods of probability on the one side, and algebraic attack. Use of the radio during the Great War has methods on the other. brought another game changer. Both sides used radio on a mass scale. Ease of interception of the radio messages forced the application of 2 Historical context cryptography at the equally mass scale, and the Although traditional, historical codebreaking traditional codes were getting more and more has always been based on linguistics rather than impractical; during World War One ciphers mathematics, at least since eight century a replaced the codes as the mainstream of component of simple application of math was cryptography, and, consequently, statistics present therein. Al-Kindi, an Arab polymath replaced the linguistics as the mainstream of living in Baghdad in ninth century, described in cryptanalysis. his “Manuscript on Deciphering Cryptographic However, statistical methods used in Messages” an attack on the monoalphabetic cryptanalysis represented rather elementary substitution ciphers based on the natural applications of mathematics, which could be frequency of letters in the language of the clear dealt with by amateurs. Immediately after the end text. As far as we know his work was based on of World War One agencies of major countries the earlier (and presently lost) writings of Al- dealing with cryptology did not realize the need Proceedings of the 3rd International Conference on Historical Cryptology, HistoCrypt 2020 70 to employ or train mathematicians. If some and in the last days of 1932 he managed to solve mathematicians happened to be employed in the his equations, reengineering thus Enigma crypto world, it was only due to their general machine in a purely mathematical way. intellectual discipline, and not to the particular Terms used in the description above do not skills resulting from their scientific discipline. leave a shadow of doubt that Rejewski was using Werner Kunze was employed at the German an absolutely pioneering approach. System of foreign ministry cipher office in 1919, but it was equations represents a term functioning in the only in 1936 that he became the head of its newly purely algebraic context. Permutations are used created mathematical section. William Friedman in the context of the theory of groups. Neither published in 1930 an offer to employ three reminds ideas or notions used in the probability “government mathematicians” at some obscure or statistics, dominating codebreaking up to that agency of the US Army. From the memories of moment. It is somewhat surprising that Rejewski Solomon Kullback, Frank Rowlett and Abraham had not started his attack from the statistical Sinkov, whom he selected from among the approach, considering his earlier professional candidates, we learn that for the first several plans. Just after having completed his studies in years nature of their occupations was rather mathematics at the Poznań University, he distant from mathematics. decided to continue education in the actuarial statistics at Göttingen. One of his relatives was It seems that the first agency dealing with among the founders of the first life insurance cryptology that consciously and purposefully company in Wielkopolska, and Marian Rejewski decided to employ and train mathematicians was obviously planned to start a career in the the Polish Cipher Bureau in 1928. Effects of that insurance business. decision are well known among the historians of cryptology; after the half year training in Algebraic and group theoretic approach, used cryptology in Poznań, in 1929, and three years by Rejewski in his breakthrough attack at the long period of apprenticeship in the Enigma cipher, had numerous advantages over codebreaking, Marian Rejewski was asked to the statistical attacks used against the earlier hand take a look at the real objective of this effort – the ciphers. Its crucial advantage was that it worked. Enigma cipher. It took him less than three months Codebreaking agencies of the major countries to break the cipher and, simultaneously, to initially declared helplessness when confronted change the nature of cryptology forever. with the Enigma cipher. William Clarke, one of the veterans of British Room 40, remarked in a 3 Probabilists vs Algebraists memorandum written in 1937 that “only one cloud obscures the horizon – possibility of When in October 1932 Maksymilian Ciężki general application of the ciphering machines. had asked Marian Rejewski to take a look at the One can argue that it would mean the complete materials that Polish Cipher Bureau was able to end of the codebreaking”. Frank Birch noted an gather about Enigma (Rejewski, 1967), opinion of one of G.C.&C.S. heads of section Rejewski, in spite of his over three years long stating that “(a)ll the German ciphers are training in cryptology, was still a mathematician unbreakable. (…) putting pundits on them rather than the codebreaker. One might say, represents a waste of time”. luckily for the civilized world; had he been the Rejewski’s approach was unique among the cryptologist, he would have probably tried to traditional codebreaking methods, as its success apply the traditional codebreaking methods, was deterministic rather than probabilistic. Most completely ineffective versus Enigma cipher. codebreaking methods used up to his Rejewski started his work identifying some breakthrough were offering only a promise of purely mathematical features of the cipher and success, without granting it. Success depended continued transforming his entire knowledge on many factors beyond the codebreaker’s about the machine and its cipher into a system of control: errors committed by the cipher clerk, equations. He was unable to solve these external evidence permitting to guess the content equations outright, as the variables they of the message, inspired guess of the probable contained represented unknown permutations cleartext. Algebraic approach invented by rather than the numbers. Theory permitting to Rejewski virtually granted the success, provided solve such type of equations was missing and that the codebreaker was able to accumulate Rejewski had to provide it himself, which he did, some 80-100 messages during a single day. Proceedings of the 3rd International Conference on Historical Cryptology, HistoCrypt 2020 71 Finally, with the proper tooling Rejewski’s He developed the new idea within a month and it method was extremely efficient and fast. In 1935 took the AVA company working for Polish Polish team decided to construct a simple intelligence service another month to deliver six electromechanical device named cyclometer. prototypes, but nobody was proud of this Cyclometer was used to simplify the preparation achievement. First – because in December the of the catalogue of so called cyclic Germans increased the number of rotors to five, characteristics.