W . T . Tutte William Thomas Tutte , known as Bill Tutte ( ; May 14 , 1917 – May 2 , 2002 ) , was a British , later Canadian , codebreaker and mathematician . During World War II he made a brilliant and fundamental advance in Cryptanalysis of the Lorenz cipher , a major German cipher system . The intelligence obtained from these decrypts had a significant impact on the Allied victory in Europe . He also had a number of significant mathematical accomplishments , including foundation work in the fields of graph theory and matroid theory . Tutte ’ s research in the field of graph theory proved to be of remarkable importance . At a time when graph theory was still a primitive subject , Tutte commenced the study of matroid s and developed them into a theory by expanding from the work that Hassler Whitney had first developed around the mid 1930s . Even though Tutte ’ s contributions to graph theory have been influential to modern graph theory and many of his theorems have been used to keep making advances in the field , most of his terminology was not in agreement with their conventional usage and thus his terminology is not used by graph theorists today . " Tutte advanced graph theory from a subject with one text ( D . König ’ s ) toward its present extremely active state " Early life and education . Tutte was born in Newmarket in Suffolk , the son of a gardener . At the age of 18 , he studied chemistry and mathematics at Trinity College , Cambridge University . As a student he ( along with three of his friends ) became the first to solve the problem of squaring the square . Together the four created the pseudonym Blanche Descartes , under which Tutte published occasionally for years . World War II . Soon after the outbreak of World War II , Tutte ' s tutor , Patrick Duff , suggested him for war work at the Government Code and Cypher School at Bletchley Park ( BP ) . He was interviewed and sent on a training course in London before going to Bletchley Park , where he joined the Research Section . At first he worked on the Hagelin cipher that was being used by the Italian Navy . This was a rotor cipher machine that was available commercially , so the mechanics of enciphering was known , and decrypting messages only required working out how the machine was set up . In the summer of 1941 , Tutte was transferred to work on a teleprinter cipher system that had been dubbed " Tunny " . Telegraphy used the 5 - bit International Telegraphy Alphabet No . 2 ( ITA2 ) . Other than that messages were preceded by a 12 - letter indicator , which implied a 12 - wheel rotor cipher machine , nothing was known about the mechanism of enciphering . The first step , therefore , had to be to diagnose the machine by establishing the logical structure and hence the functioning of the machine . Tutte played a pivotal role in achieving this , and it was not until shortly before the allied victory in Europe in 1945 , that Bletchley Park acquired a Tunny Lorenz cipher machine . Tutte ' s breakthroughs led eventually to bulk decrypting of Tunny - enciphered messages between German High Command ( OKW ) in Berlin and their army commands throughout occupied Europe , that played a crucial part in shortening the war . Diagnosing the cipher machine . On 31 August 1941 , two versions of the same message were sent using identical keys which constituted a " depth " . This allowed John Tiltman , Bletchley Park ' s veteran and remarkably gifted cryptanalyst , to deduce that it was a Vernam cipher which uses the Exclusive Or ( XOR ) function ( symbolised by " ⊕ " ) , and to extract the two messages and hence obtain the obscuring key . After a fruitless period of Research Section cryptanalysts trying to work out how the Tunny machine worked , this and some other keys were handed to Tutte who was asked to " see what you can make of these " . At his training course , Tutte had been taught the Kasiski examination technique of writing out a key on squared paper , starting a new row after a defined number of characters that was suspected of being the frequency of repetition of the key . If this number was correct , the columns of the matrix would show more repetitions of sequences of characters than chance alone . Tutte knew that the Tunny indicators used 25 letters ( excluding J ) for 11 of the positions , but only 23 letters for the other . He therefore tried Kasiski ' s technique on the first impulse of the key characters , using a repetition of 25 × 23 = 575 . He did not observe a large number of column repetitions with this period , but he did observe the phenomenon on a diagonal . He therefore tried again with 574 , which showed up repeats in the columns . Recognising that the prime factor s of this number are 2 , 7 and 41 , he tried again with a period of 41 and " got a rectangle of dots and crosses that was replete with repetitions " . It was clear , however , that the first impulse of the key was more complicated than that produced by a single wheel of 41 key impulses . Tutte called this component of the key formula _ 1 ( " chi " ) . He figured that there was another component , which was XOR - ed with this , that did not always change with each new character , and that this was the product of a wheel that he called formula _ 2 ( " psi " ) . The same applied for each of the five impulses ( formula _ 1formula _ 1formula _ 1formula _ 1formula _ 1 and formula _ 2formula _ 2formula _ 2formula _ 2formula _ 2 ) . So for a single character , the whole key K consisted of two components : At Bletchley Park mark impulses were signified by x and space impulses by • . For example the letter " H " would be coded as • • x • x . Tutte ' s derivation of the " chi " and " psi " components was made possible by the fact that dots were more likely than not to be followed by dots , and crosses more likely than not to be followed by crosses . This was a product of a weakness in the German key setting , which they later eliminated . Once Tutte had made this breakthrough , the rest of the Research Section joined in to study the other impulses , and it was established that the five " chi " wheels all advanced with each new character and that the five " psi " wheels all moved together under the control of two " mu " or " motor " wheels . Over the following two months , Tutte and other members of the Research Section worked out the complete logical structure of the machine with its set of wheels bearing cams that could either be in a position ( raised ) that added x to the stream of key characters , or in the alternative position that added in • . Diagnosing the functioning of the Tunny machine in this way was a truly remarkable cryptanalytical achievement which , in the citation for Tutte ' s induction as an Officer of the Order of Canada , was described as : Tutte ' s statistical method . To decrypt a Tunny message required knowledge not only of the logical functioning of the machine , but the start positions of each rotor for the particular message . The search was on for a process that would manipulate the ciphertext or key to produce a frequency distribution of characters that departed from the uniformity that the enciphering process aimed to achieve . While on secondment to the Research Section in July 1942 , Alan Turing worked out that the XOR combination of the values of successive characters in a stream of ciphertext and key , emphasised any departures from a uniform distribution . The resultant stream ( symbolised by the Greek letter " delta " Δ ) was called the difference because XOR is the same as modulo 2 subtraction . The reason that this provided a way into Tunny , was that although the frequency distribution of characters in the ciphertext could not be distinguished from a random stream , the same was not true for a version of the ciphertext from which the " chi " element of the key had been removed . This is because , where the plaintext contained a repeated character and the " psi " wheels did not move on , the differenced " psi " character ( Δformula _ 2 ) would be the null character ( ' / ' at Bletchley Park ) . When XOR - ed with any character , this character has no effect . Repeated characters in the plaintext were more frequent both because of the characteristics of German ( EE , TT , LL and SS are relatively common ) , and because telegraphists frequently repeated the figures - shift and letters - shift characters as their loss in an ordinary telegraph message could lead to gibberish . To quote the General Report on Tunny : Turingery introduced the principle that the key differenced at one , now called ΔΚ , could yield information unobtainable from ordinary key . This Δ principle was to be the fundamental basis of nearly all statistical methods of wheel - breaking and setting . Tutte exploited this amplification of non - uniformity in the differenced values and by November 1942 had produced a way of discovering wheel starting points of the Tunny machine which became known as the " Statistical Method " . The essence of this method was to find the initial settings of the " chi " component of the key by exhaustively trying all positions of its combination with the ciphertext , and looking for evidence of the non - uniformity that reflected the characteristics of the original plaintext . Because any repeated characters in the plaintext would always generate • , and similarly ∆formula _ 2 ⊕ ∆formula _ 2 would generate • whenever the " psi " wheels did not move on , and about half of the time when they did - some 70 % overall . As well as applying differencing to the full 5 - bit characters of the ITA2 code , Tutte applied it to the individual impulses ( bits ) . The current " chi " wheel cam settings needed to have been established to allow the relevant sequence of characters of the " chi " wheels to be generated . It was totally impracticable to generate the 22 million characters from all five of the " chi " wheels , so it was initially limited to 41 × 31 = 1271 from the first two . After explaining his findings to Max Newman , Newman was given the job of developing an automated approach to comparing ciphertext and key to look for departures from randomness . The first machine was dubbed Heath Robinson but the much faster Colossus computer soon took over . Because of this work , Canada ' s Communications Security Establishment named an internal organisation aimed at promoting research into cryptology , the Tutte Institute for Mathematics and Computing ( TIMC ) in his honour in 2011 . Doctorate and career . Tutte completed a doctorate in mathematics from Cambridge in 1948 under the supervision of Shaun Wylie , who had also worked at Bletchley Park on TUNNY . The same year , invited by Harold Scott MacDonald Coxeter , he accepted a position at the University of Toronto . In 1962 , he moved to the University of Waterloo in Waterloo , Ontario where he stayed for the rest of his academic career . He officially retired in 1985 but remained active as an emeritus professor . Tutte was instrumental in helping to found the Department of Combinatorics and Optimization at the University of Waterloo . His mathematical career concentrated on combinatorics , especially graph theory , which he is credited as having helped create in its modern form , and matroid theory , to which he made profound contributions ; one colleague described him as " the leading mathematician in combinatorics for three decades " . He was editor in chief of " The Journal of Combinatorial Theory " when it was started , and served on the editorial boards of several other mathematical research journals . His work in graph theory includes the structure of cycle and cut spaces , size of maximum matching s and existence of " k " - factor s in graphs , and Hamiltonian and non - Hamiltonian graphs . He disproved Tait ' s conjecture using the construction known as Tutte ' s fragment . The eventual proof of the four color theorem made use of his earlier work . The graph polynomial he called the " dichromate " has become famous and influential under the name " Tutte polynomial " and serves as the prototype of combinatorial invariants that are universal for all invariants that satisfy a specified reduction law . The first major advances in matroid theory were made by Tutte in his 1948 Cambridge Ph . D . thesis which formed the basis of an important sequence of papers published over the next two decades . Tutte ' s work in graph theory and matroid theory has been profoundly influential on the development of both the content and direction of these two fields . In matroid theory he discovered the highly sophisticated homotopy theorem as well as founding the studies of chain group s and regular matroid s , about which he proved deep results . In addition , Tutte developed an algorithm for determining whether a given binary matroid is graphic . The algorithm makes use of the fact that a planar graph is simply a graph whose circuit - matroid , the dual of its bond - matroid , is graphic . Tutte wrote a paper entitled " How to Draw a Graph " in which he proves that any face in a 3 - connected graph is enclosed by a peripheral cycle . Using this fact , Tutte developed an alternative proof to show that every Kuratowski graph is non - planar by showing that " K " and " K " each have three distinct peripheral cycles with a common edge . In addition to using peripheral cycles to prove that the Kuratowski graphs are non - planar , Tutte proved that there exists a convex embedding of any simple 3 - connected graph and devised an algorithm which constructs the plane drawing by solving a linear system . This algorithm makes use of the barycentric mappings of the peripheral circuits of a simple 3 - connected graph . The findings published in this paper have proved to be of much significance because the algorithms that Tutte developed have become popular planar graph drawing methods . In 1997 , Michael S . Floater published a paper entitled " Parameterization and smooth approximation of surface triangulations " which extends Tutte ’ s original theorem on the existence of a plane drawing of a 3 - connected graph bounded by a convex polygon . Floater shows that a plane drawing of a 3 - connected graph can be drawn without the boundary necessarily being a convex polygon . One of the reasons for which Tutte ’ s embedding is popular is that the necessary computations that are carried out by his algorithms are simple and guarantee a one - to - one correspondence of a graph and its embedding onto the Euclidean plane which is of importance when paramterizing a three dimensional mesh to the plane in geometric modeling . “ Tutte ' s theorem is the basis for solutions to other computer graphics problems , such as morphing Tutte was mainly responsible for developing the theory of enumeration of planar graphs , which has close links with chromatic and dichromatic polynomials . This work involved some innovated techniques of his own invention , requiring considerable manipulative dexterity in handling power series ( whose coefficients count appropriate kinds of graphs ) and the functions arising as their sums , as well as geometrical dexterity in extracting these power series from the graph - theoretic situation . Positions , honours and awards . Tutte ' s work in WW2 and subsequently in combinatorics brought him various positions , honours and awards : Asteroid 14989 Tutte ( 1997 UB7 ) was named in his honour . Personal life and death . The opportunity to work at the University of Waterloo appealed to Tutte because it offered the possibility of advancement . It also happened that both William and Dorothea enjoyed natural settings and the overall rural environment that was offered by Waterloo was of interest to Tutte and his wife . Tutte accepted the position and he and Dorothea bought a house in the small nearby town of West Montrose , Ontario . Both Bill and Dorothea enjoyed spending time in their garden and allowing others to enjoy the beautiful scenery that was contained within their property . They also had an extensive knowledge of all the birds in their garden , they could name every bird they encountered . Dorothea was a keen hiker and Bill organized hiking trips . Even near the end of his life Bill still was an avid walker , he could out - walk colleagues 20 years younger . After his wife died in 1994 , he returned to live in Newmarket , but then returned to Waterloo in 2000 , where he died two years later .