The Enigma Machine. Chapter The Enigma Cipher Machine. i crypttext 2017/4/4 9:57 page 45 #53

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1 crypttext 2017/4/4 9:57 page 45 #53 Chapter 4 The Engma Machne Ths book s a survey of cryptology, not a hstory book. However, any survey of cryptology should nclude or lead readers to dscover for themselves at least a lttle hstory of the subject, as ths book does, not just because of how the subject has been formed by ts hstory, but also because the dea of wrtng secret messages has led to hstorcal matters, both fctonal (e.g., a cpher n a Sherlock Holmes mystery) and actual (e.g., encrypted messages sent by the Zodac kller), that are (at least to us) nherently nterestng. We have placed many of our hstorcal references n exercses to prompt readers to search for ths nformaton themselves, because, ths beng a survey book rather than a hstory book, what we could nclude here would not do the nformaton justce relatve to what s already readly avalable n more specalzed books and artcles n prnt and, usually, onlne. A szable part of the hstory of cryptology s related to government communcatons, especally mltary durng tmes of conflct. Man usually functons optmally out of necessty, not convenence, and whle the Zodac kller dd not need to send encrypted messages, governments and mltares do, especally durng tmes of conflct. So although ths s not a hstory book, we feel t s approprate to go nto some detal about the Engma cpher machne, the cryptanalyss of whch by the Alles durng World War II s one of the greatest achevements of the human ntellect. Ths feat, born out of necessty, drectly contrbuted to a swfter end to the greatest war n hstory, and saved many lves on both sdes. 4.1 The Engma Cpher Machne In 1918, German electrcal engneer Arthur Scherbus appled for a patent for a mechancal cpher machne. Ths machne, later marketed commer- 45

2 crypttext 2017/4/4 9:57 page 46 #54 46 CHAPTER 4. THE ENIGMA MACHINE cally under the name Engma, was desgned wth electrc current runnng through revolvng wred wheels, called rotors. Scherbus o ered hs machne to the German mltary, and whle they dd not fnd any defcences n t, they dd not choose at that tme to purchase any. Only years later, after learnng that ther World War I cphers had routnely been broken, dd the Germans adopt varous models of the Engma, whch they used as ther prmary resource for encrypted communcatons throughout World War II. In ths secton, we wll present some techncal detals of two of these models, the Wehrmacht Engma, used by the German army, and the Kregsmarne M4 Engma, used by the German navy. Before presentng any techncal detals of an Engma, we should note that descrptons of these detals to the extent as they are ncluded n ths book are rarely found n lterature amed at nontechncal audences. On the other hand, mages of the varous components of an Engma abound, and are readly avalable through a smple Internet search. As such, n ths book we wll not nclude mages of the varous components of an Engma. However, we do very strongly encourage our readers to do each part of Exercse 1 as the component of an Engma gven n the exercse s descrbed n ths secton. The components of an Engma are lsted n Exercse 1 n bascally the same order n whch they are descrbed n ths secton. An Engma conssted of four components: a 26-letter keyboard for enterng nput letters (ether plantext or cphertext), a plugboard resemblng a mnature old telephone swtchboard, a system of rotors, and a 26-letter lampboard for dsplayng output letters. Pressng an nput letter on the keyboard sent an electrc current through the plugboard and rotors, where the encrypton or decrypton took place, and the current ended at the lampboard where a small bulb was llumnated to ndcate the output letter. The layout of letters on the keyboard and lampboard was smlar to the layout on a modern keyboard, and s shown n Fgure 4.1. Fgure 4.1 Layout of letters on an Engma keyboard. Pressng an nput letter on the keyboard on an Engma sent current desgnatng the letter frst to the plugboard. The plugboard was stuated on the front of an Engma, and had 26 open sockets, one to represent each possble letter. The plugboard sockets could ether be left open or connected n

3 crypttext 2017/4/4 9:57 page 47 # THE ENIGMA CIPHER MACHINE 47 pars by short cables. If a par of sockets was connected by a cable n the plugboard, then current desgnatng ether letter represented by the sockets would be converted at the plugboard to desgnate the other letter. If a socket was left open n the plugboard, then current desgnatng the letter represented by the socket would leave the plugboard stll desgnatng the same letter. Example 4.1 Consder an Engma plugboard wred wth the sockets representng M and Z connected, the sockets representng N and S connected, and all other sockets left open. Then, current desgnatng M wll be converted at the plugboard to desgnate Z. Smlarly, current desgnatng S wll be converted at the plugboard to desgnate N. On the other hand, current desgnatng E wll leave the plugboard stll desgnatng E. There were many d erent choces for whch plugboard sockets could be connected n an Engma, wth anywhere from zero to 13 cables used, and usually a very large number of possbltes for whch par of sockets could be connected by each cable. Varyng the number of cables would have maxmzed securty, but standard German operatng procedure was to use a fxed number of cables. Wth a fxed number of cables, 11 cables would have maxmzed securty (as we wll verfy n Secton 4.3), but for most of the war standard German operatng procedure was to use 10 cables. Each Engma provded for use n the feld came wth 12 cables, wth two held n reserve n case any of the 10 n use became faulty. After leavng the plugboard, current went through a system of rotors that was stuated n the back of an Engma. Each ndvdual rotor was a crcular dsk about the sze of a hockey puck. We wll call the flat sdes of a rotor the rght and left sdes, snce rotors could only be placed n an Engma standng on end wth each sde facng n a partcular drecton. Both flat sdes of a rotor contaned 26 contact ponts, one to represent each letter, wth the letters consdered n alphabetcal order around both sdes of the rotor clockwse (when the rotor s vewed from the rght). The contacts on the rght sde of a rotor were wred to the contact ponts on the left, but not necessarly straght across. The dea was that current could enter one sde of a rotor at one of the contact postons, representng a letter, and pass through and ext the rotor on the other sde at a d erent contact poston, representng a d erent letter. Example 4.2 Consder an Engma rotor wred wth the contacts connected as lsted n the followng table (.e., wth each rght contact lsted n the frst row wred to the left contact below t n the second row). Rght contact: Left contact: ABCDEFGHIJKLMNOPQRSTUVWXYZ EKMFLGDQVZNTOWYHXUSPAIBRCJ

4 crypttext 2017/4/4 9:57 page 48 #56 48 CHAPTER 4. THE ENIGMA MACHINE Then, current desgnatng M that enters the rotor on the rght wll ext the rotor on the left desgnatng O. Smlarly, current desgnatng M that enters the rotor on the left wll ext the rotor on the rght desgnatng C. Also, current desgnatng S that enters the rotor on ether sde wll ext the rotor on the other sde stll desgnatng S. Wehrmacht Engmas could accommodate three rotors placed sde-by-sde, whle Kregsmarne M4 Engmas could accommodate four rotors. Although rotors could only be stuated wth each sde facng n a partcular drecton, current could pass through the rotors n ether drecton. The reason for ths s that whle current always ntally passed through the rotors from rght to left, to the left of the rotor slots was a reflector whch sent the current back through the rotors from left to rght. In addton, the reflector was tself lke half a rotor n the sense that on ts rght sde there were 26 contact ponts, one to represent each possble letter, but on ts left sde there were no contacts. The contacts on the rght sde of a reflector were wred to each other n 13 pars. Unlke plugboard sockets, reflector contacts were always fully connected. Also, unlke rotor contacts, reflector contacts could not be connected n a way such that a letter was connected to tself. Example 4.3 Consder an Engma reflector wred wth the contacts connected as lsted n the followng table (.e., wth each contact lsted n the frst row wred to the contact below t n the second row). Contact: A B C D E F G I J K M T V Pared contact: Y R U H Q S L P X N O Z W Then, current desgnatng M that enters the reflector wll ext the reflector desgnatng O. Smlarly, current desgnatng O that enters the reflector wll ext the reflector desgnatng M. There were many d erent choces for how rotor and reflector contacts could be connected n an Engma, but because rotors and reflectors had to be hard-wred and changng the wrng was a very d cult thng to do, rotors and reflectors wth only a very small number of d erent wrngs were ever produced and used n the feld. Rotors wth only fve d erent wrngs were produced for Wehrmacht Engmas. These rotors were labeled wth the Roman numerals I V, and the contacts connected n each are lsted n Table 4.1 on page 49. Of these fve rotors, three were used at a tme n Wehrmacht Engmas. Any three could be used, and they could be arranged n any order. Recall that whle Wehrmacht Engmas could only accommodate three rotors, Kregsmarne M4 Engmas could accommodate four. Kregsmarne

5 crypttext 2017/4/4 9:57 page 49 # THE ENIGMA CIPHER MACHINE 49 Rght contact: I left contact: II left contact: III left contact: IV left contact: V left contact: Table 4.1 ABCDEFGHIJKLMNOPQRSTUVWXYZ EKMFLGDQVZNTOWYHXUSPAIBRCJ AJDKSIRUXBLHWTMCQGZNPYFVOE BDFHJLCPRTXVZNYEIWGAKMUSQO ESOVPZJAYQUIRHXLNFTGKDCMWB VZBRGITYUPSDNHLXAWMJQOFECK Contacts connected n Wehrmacht rotors. M4 Engmas could actually hold four rotors n the same space that Wehrmacht Engmas had for three. Ths was accomplshed by usng a thnner reflector, whch allowed for a thnner fourth rotor to be nserted between the leftmost full-sze rotor and the reflector. For the three full-sze rotors n Kregsmarne M4 Engmas, any of the Wehrmacht rotors I V could be used, as well as any of three addtonal rotors wth d erent wrngs. These three addtonal full-sze rotors were labeled wth the Roman numerals VI VIII, and the contacts connected n each are lsted n Table 4.2. Rght contact: VI left contact: VII left contact: VIII left contact: ABCDEFGHIJKLMNOPQRSTUVWXYZ JPGVOUMFYQBENHZRDKASXLICTW NZJHGRCXMYSWBOUFAIVLPEKQDT FKQHTLXOCBJSPDZRAMEWNIUYGV Table 4.2 Contacts connected n addtonal Kregsmarne M4 rotors. The eght full-sze rotors I VIII were too wde to ft nto the space avalable for the thnner fourth rotor n Kregsmarne M4 Engmas. For ths thnner fourth rotor, rotors wth only two d erent wrngs were produced. These rotors were labeled wth the Greek letters (beta) and (gamma), and the contacts connected n each are lsted n Table 4.3. Rght contact: left contact: left contact: ABCDEFGHIJKLMNOPQRSTUVWXYZ LEYJVCNIXWPBQMDRTAKZGFUHOS FSOKANUERHMBTIYCWLQPZXVGJD Table 4.3 Contacts connected n thnner Kregsmarne M4 rotors. Reflectors wth only two d erent wrngs were produced for Wehrmacht Engmas. These reflectors were labeled wth the letters B and C, and the contacts connected n each are lsted n Table 4.4 on page 50. Because Kregsmarne M4 Engmas were modfed to hold four rotors nstead of three, reflectors produced for Wehrmacht Engmas were too wde to ft n them. As a result, d erent thnner reflectors had to be produced for Kregsmarne M4 Engmas. Reflectors wth only two d erent wrngs

6 crypttext 2017/4/4 9:57 page 50 #58 50 CHAPTER 4. THE ENIGMA MACHINE Contact: A B C D E F G I J K M T V B pared contact: Y R U H Q S L P X N O Z W Contact: A B C D E G H K L M N Q S C pared contact: F V P J I O Y R Z X W T U Table 4.4 Contacts connected n Wehrmacht reflectors. were produced for Kregsmarne M4 Engmas. These thnner reflectors were also labeled wth the letters B and C, and the contacts connected n each are lsted n Table 4.5. Contact: A B C D F G H I L M R S T B pared contact: E N K Q U Y W J O P X Z V Contact: A B C E F G H I L P Q S U C pared contact: R D O J N T K V M W Z X Y Table 4.5 Contacts connected n Kregsmarne M4 reflectors. As we have noted, pressng an nput letter on an Engma keyboard sent an electrc current through the plugboard and rotors, and the current ended at the lampboard where a small bulb was llumnated to ndcate the output letter. To be more precse, pressng an nput letter on the keyboard sent an electrc current frst through the plugboard, then through the rotors (ether three or four dependng on the Engma model) from rght to left, through the reflector, back through the rotors from left to rght, and then through the plugboard a second tme. After leavng the plugboard for the second tme, the current went to the lampboard where a bulb was llumnated to ndcate the output letter. What we have presented so far already gves a very large number of possble confguratons for an Engma. However, we are not done. Before a rotor was placed n an Engma, t could be rotated nto any of 26 possble orentatons. The orentaton of a rotor n an Engma dctates the path that current follows through the rotor. Example 4.4 Consder an Engma rotor rotated from ts orgnal orentaton (for whch current desgnatng a letter wll travel through the rotor along the path we orgnally ndcated n Tables 4.1, 4.2, and 4.3) fve postons counterclockwse (when the rotor s vewed from the rght). Then, current desgnatng B enterng the rotor wll not travel through the rotor along the path we orgnally ndcated for current desgnatng B, but rather along the path we orgnally ndcated for current desgnatng G, snce the contact for G wll have rotated nto the poston orgnally occuped by the contact for B. Smlarly, current desgnatng any letter enterng the rotor

7 crypttext 2017/4/4 9:57 page 51 # THE ENIGMA CIPHER MACHINE 51 wll travel through the rotor along the path we orgnally ndcated for current desgnatng the letter fve postons later n the alphabet, wrappng from the end of the alphabet to the start f necessary. For nstance, current desgnatng X enterng the rotor wll travel through the rotor along the path we orgnally ndcated for current desgnatng C. To assst Engma operators wth orentng rotors correctly, etched n a rng around the edge of each rotor were the letters A Z (or sometmes the numbers 01 26), lsted n order clockwse (when the rotor was vewed from the rght). For each rotor slot n an Engma, a small wndow was cut to show the letter (or number) at a partcular locaton on the rng. We wll call ths letter the wndow letter. The wndow letter for a rotor ndcates the orentaton of the rotor. A number called the rotor o set also ndcates the orentaton of a rotor n an Engma. The rotor o set for a rotor s a whole number between 0 and 25, wth 0 meanng the rotor s n ts orgnal orentaton (for whch the wndow letter wll be A), 1 meanng the rotor has been rotated 1 poston counterclockwse (when vewed from the rght), 2 meanng the rotor has been rotated 2 postons counterclockwse, and so on, through 25 meanng the rotor has been rotated 25 postons counterclockwse. There s no need to consder rotor o sets larger than 25, snce rotatng a rotor 26 postons counterclockwse wll take the rotor back to ts orgnal orentaton wth rotor o set 0. The etched rng around the edge of an Engma rotor was also movable, and could be rotated nto any of 26 d erent postons whle the wred part of the rotor was held fxed. Ths s a complcaton, because rotatng the rng changes the wndow letter wthout changng the rotor o set. A number called the rng settng ndcates the poston of the rng on a rotor. The rng settng for a rotor s a whole number between 1 and 26, wth 1 meanng the rng s n ts orgnal poston (for whch wth rotor o set 0 the wndow letter s A), 2 meanng the rng has been rotated 1 poston counterclockwse (when the rotor s vewed from the rght), 3 meanng the rng has been rotated 2 postons counterclockwse, and so on, through 26 meanng the rng has been rotated 25 postons counterclockwse. There s no need to consder rng settngs larger than 26, snce rotatng a rng 26 postons counterclockwse wll take the rng back to ts orgnal poston wth rng settng 1. For a rotor n an Engma for whch the rotor and ts rng have been rotated some number of postons, the wndow letter gves a number of postons t can be assumed that the rotor and ts rng have been rotated n total. Example 4.5 Consder a rotor n an Engma for whch the rotor and ts rng have been rotated some number of postons. If the wndow letter s

8 crypttext 2017/4/4 9:57 page 52 #60 52 CHAPTER 4. THE ENIGMA MACHINE U, then t can be assumed that the rotor and ts rng have been rotated 20 postons counterclockwse (when the rotor s vewed from the rght) n total, snce U s 20 postons after A n the alphabet. As a result, f the rng settng s known to be 4 (.e., f the rng s known to have been rotated 3 postons counterclockwse), then the rotor o set wll be 17 (.e., t can be assumed that the rotor has been rotated 17 postons counterclockwse). Put another way, the rotor o set can be obtaned by subtractng the rng settng from the number of the poston of the wndow letter n the alphabet. Snce U s the 21st letter n the alphabet and the rng settng s 4, the rotor o set s 21 4 = 17. Example 4.6 Consder a rotor n an Engma wth rng settng 12 and wndow letter Q. Snce Q s the 17th letter n the alphabet, the rotor o set s = 5. For a rotor n an Engma, when the rng settng s subtracted from the number of the poston of the wndow letter n the alphabet, t s possble to obtan a negatve result. We show how to account for ths n the next example. Example 4.7 Consder a rotor n an Engma wth rng settng 22 and wndow letter D. Snce D s the 4th letter n the alphabet, subtractng the rng settng from the number of the poston of the wndow letter n the alphabet gves a rotor o set of 4 22 = 18. Ths means t can be assumed that the rotor has been rotated negatve 18 postons counterclockwse (when the rotor s vewed from the rght). However, recall that rotor o sets are whole numbers between 0 and 25, and so cannot be negatve. Ths negatve result s easy to remedy, though, snce rotatng a rotor negatve 18 postons counterclockwse gves the same orentaton as rotatng the rotor postve 8 postons counterclockwse. Thus, the rotor o set s actually 8, a number that can be obtaned by smply addng 26 to 18. That s, f subtractng the rng settng from the number of the poston of the wndow letter n the alphabet gves a negatve result, the rotor o set can be obtaned by addng 26 to the result. Example 4.8 Consder a rotor n an Engma wth rng settng 16 and wndow letter K. Snce K s the 11th letter n the alphabet, subtractng the rng settng from the number of the poston of the wndow letter n the alphabet gves = 5. Thus, the rotor o set s = 21. As we have noted, the rotor o set for an Engma rotor dctates the path that current follows through the rotor. Example 4.9 Consder Engma rotor V wth rotor o set 5, and current desgnatng B enterng the rotor on the rght. Wth rotor o set 5, current

9 crypttext 2017/4/4 9:57 page 53 # THE ENIGMA CIPHER MACHINE 53 desgnatng B enterng the rotor wll not travel through the rotor along the path we orgnally ndcated for current desgnatng B, but rather along the path we orgnally ndcated for current desgnatng G, snce the contact for G wll have rotated nto the poston orgnally occuped by the contact for B. Note that B s the 2nd letter n the alphabet, and f we add the rotor o set to 2, the result s = 7, whch corresponds to the fact that G s the 7th letter n the alphabet. Accordng to Table 4.1, current desgnatng G that entered rotor V on the rght would ext the rotor on the left desgnatng T. However, when the current enters the rotor t s supposed to desgnate B, not G. Snce T s the 20th letter n the alphabet, we must only subtract the rotor o set from 20 to obtan 20 5 = 15. Snce the 15th letter n the alphabet s O, the current wll actually ext the rotor on the left desgnatng O. Example 4.10 Consder Engma rotor IIIwth rotor o set 17, and current desgnatng G enterng the rotor on the rght. Snce G s the 7th letter n the alphabet, and = 24, the current wll travel through the rotor along the path we orgnally ndcated for current desgnatng the 24th letter n the alphabet, X. Accordng to Table 4.1, current desgnatng X that entered rotor III on the rght would ext the rotor on the left desgnatng S. SnceS s the 19th letter n the alphabet, and = 2, the current wll actually ext the rotor on the left desgnatng the 2nd letter n the alphabet, B. For a rotor n an Engma, when the rotor o set s added to or subtracted from the number of the poston of a letter n the alphabet, t s possble to obtan a result that s larger than 26 or smaller than 0. We show how to account for ths n the next example. Example 4.11 Consder Engma rotor IIIwth rotor o set 17, and current desgnatng Y enterng the rotor on the left. Snce Y s the 25th letter n the alphabet, and = 42, the current wll travel through the rotor along the path we orgnally ndcated for current desgnatng the 42nd letter n the alphabet. To remedy the fact that 42 s outsde the range of allowed results (.e., from 1 through 26), we can subtract 26 from 42 to obtan = 16. Thus, the current wll travel through the rotor along the path we orgnally ndcated for current desgnatng the 16th letter n the alphabet, P. Accordng to Table 4.1, current desgnatng P that entered rotor III on the left would ext the rotor on the rght desgnatng H. Snce H s the 8th letter n the alphabet, and 8 17 = 9, the current wll ext the rotor on the rght desgnatng the 9th letter n the alphabet. To remedy the fact that 9 s outsde the range of allowed results, we can add 26 to 9 to obtan = 17. Thus, the current wll actually ext the rotor on the rght desgnatng the 17th letter n the alphabet, Q.

10 crypttext 2017/4/4 9:57 page 54 #62 54 CHAPTER 4. THE ENIGMA MACHINE Example 4.12 Consder Engma rotor V wth rotor o set 5, and current desgnatng M enterng the rotor on the left. Snce M s the 13th letter n the alphabet, and = 18, the current wll travel through the rotor along the path we orgnally ndcated for current desgnatng the 18th letter n the alphabet, R. Accordng to Table 4.1, current desgnatng R that entered rotor V on the left would ext the rotor on the rght desgnatng D. Subtractng the rotor o set from the number of the poston of D n the alphabet gves 4 5= 1. Snce = 25, the current wll actually ext the rotor on the rght desgnatng the 25th letter n the alphabet, Y. The varous rotor o sets and rng settngs ncrease the number of possble confguratons for an Engma to an astronomcally large number. Even so, everythng we have presented so far would have ultmately made for nothng more than a glorfed substtuton cpher had t not been for one fnal feature that we have not yet mentoned the rotors revolved wthn the machne durng the actual encrypton and decrypton processes. Encryptng and decryptng messages wth an Engma was done one letter at a tme, and each tme an nput letter was pressed on the keyboard, the rghtmost rotor would mmedately (before the current reached the rotors) rotate one poston counterclockwse (when the rotor was vewed from the rght). In addton, for each Engma rotor I VIII, there was ether one or two notches on the rng around the rotor. Snce each notch was on the rng, ts poston n the rotor slot at any tme could be dentfed solely by the wndow letter. For each notch, there was one partcular poston n the rotor slot, dentfed by a wndow letter called the notch letter, for whch f the rotor rotated one poston counterclockwse, the notch would cause the rotor to the left, f t were one of the rotors I VIII, to also rotate one poston counterclockwse. That s, for each notch on the rng on the rghtmost rotor, once every 26 tmes the rotor rotated one poston counterclockwse the notch would cause the mddle full-sze rotor to also rotate one poston counterclockwse, and for each notch on the rng on the mddle full-sze rotor, once every 26 tmes the rotor rotated one poston counterclockwse the notch would cause the leftmost full-sze rotor to also rotate one poston counterclockwse. Addtonally, for the mddle full-sze rotor only, f a notch letter was showng n the wndow when an nput letter was pressed, the mddle full-sze rotor would tself rotate one poston counterclockwse, regardless of whether a notch on the rng on the rghtmost rotor would have caused t to rotate. To clarfy, durng the actual encrypton and decrypton processes, only Engma rotors rotated, not the rngs around the rotors. Once a rng had been set n the ntal confguraton of the machne, ts locaton around ts rotor was fxed, and durng the encrypton and decrypton processes,

11 crypttext 2017/4/4 9:57 page 55 # THE ENIGMA CIPHER MACHINE 55 the rotaton of the rotor alone changed the wndow letter. Also, only the full-sze rotors I V used n Wehrmacht Engmas and I VIII used for the rghtmost three rotors n Kregsmarne M4 Engmas rotated. The thnner rotors and used for the leftmost rotor n Kregsmarne M4 Engmas never rotated, although they could be set n the ntal confguraton of the machne wth a nonzero rotor o set. On the other hand, n both Engma models the reflectors B and C, whch also never rotated, were always set wth a zero o set. Fnally, after an nput letter was pressed on the keyboard, all rotaton of the rotors occurred before the current reached the rotors, and no addtonal rotaton occurred untl the next nput letter was pressed. The notch letters for each of the Engma rotors I VIII are lsted n Table 4.6. Rotor Notch letters Rotor I VIII to left rotates when wndow letter changes from! to I Q Q! R II E E! F III V V! W IV J J! K V Z Z! A VI M, Z M! N, Z! A VII M, Z M! N, Z! A VIII M, Z M! N, Z! A Table 4.6 Notch letters for Engma rotors I VIII. Example 4.13 Consder a Wehrmacht Engma ntally confgured wth rotors V, III, and I, n order from left to rght, and correspondng wndow letters QUO. Then, durng an actual encrypton or decrypton wth ths ntal confguraton, the wndow letters would change accordng to the followng sequence: QUO! QUP! QUQ! QVR! RWS! RWT! RWU!... Example 4.14 Consder a Kregsmarne M4 Engma ntally confgured wth rotors, IV, VII, and VI, n order from left to rght, and correspondng wndow letters DJYL. Then, durng an actual encrypton or decrypton wth ths ntal confguraton, the wndow letters would change accordng to the followng sequence: DJYL! DJYM! DJZN! DKAO! DKAP! DKAQ! DKAR!... We have now presented all of the detals of the operaton of an Engma, and are ready to see the full encrypton process. Example 4.15 Consder a Wehrmacht Engma ntally confgured wth the plugboard wred wth the sockets representng M and Z connected, the

12 crypttext 2017/4/4 9:57 page 56 #64 56 CHAPTER 4. THE ENIGMA MACHINE sockets representng N and S connected, and all other sockets left open, rotors V, III, and I, n order from left to rght, wth correspondng rng settngs 12, 4, and 8 and ntal wndow letters QUO, and reflector B. We wll determne the result of usng an Engma wth ths ntal confguraton to encrypt the plantext ENIGMA. Thefrstthngwedospressthekey for the ntal nput plantext letter E on the keyboard. Ths changes the wndow letters to QUP, and sends current desgnatng E to the plugboard. Snce the socket representng E n the plugboard s open, the current leaves the plugboard stll desgnatng E. Next, the current goes to the rotors. The rotor o sets are gven n the followng table. (We determned these rotor o sets for rotors III and V n Examples 4.5 and 4.6.) Rotor Rng settng Wndow letter Rotor o set I 8 P 16 8=8 III 4 U 21 4=17 V 12 Q = 5 The followng bulleted tems follow the current through the system of rotors: Frst, current desgnatng E enters rotor I on the rght. Snce E s the 5th letter n the alphabet, and = 13, the current wll travel through the rotor along the path we orgnally ndcated for current desgnatng the 13th letter n the alphabet, M. Accordng to Table 4.1 on page 49, current desgnatng M that entered rotor I on the rght would ext the rotor on the left desgnatng O. Snce O s the 15th letter n the alphabet, and 15 8 = 7, the current wll actually ext the rotor on the left desgnatng the 7th letter n the alphabet, G. Next, current desgnatng G enters rotor III on the rght. In Example 4.10, we determned that ths current wll ext rotor III on the left desgnatng B. Next, current desgnatng B enters rotor V on the rght. In Example 4.9, we determned that ths current wll ext rotor V on the left desgnatng O. Next, current desgnatng O enters reflector B. Accordng to Table 4.4 on page 50, current desgnatng O that enters reflector B wll ext the reflector desgnatng M. Next, current desgnatng M enters rotor V on the left. In Example 4.12, we determned that ths current wll ext rotor V on the rght desgnatng Y. Next, current desgnatng Y enters rotor III on the left. In Example 4.11, we determned that ths current wll ext rotor III on the rght desgnatng Q.

13 crypttext 2017/4/4 9:57 page 57 # THE ENIGMA CIPHER MACHINE 57 Next, current desgnatng Q enters rotor I on the left. Snce Q s the 17th letter n the alphabet, and = 25, the current wll travel through the rotor along the path we orgnally ndcated for current desgnatng the 25th letter n the alphabet, Y. Accordng to Table 4.1 on page 49, current desgnatng Y that entered rotor I on the left would ext the rotor on the rght desgnatng O. Snce O s the 15th letter n the alphabet, and 15 8 = 7, the current wll actually ext the rotor on the rght desgnatng the 7th letter n the alphabet, G. Now through the system of rotors, current desgnatng G goes back to the plugboard. Snce the socket representng G n the plugboard s open, the current leaves the plugboard stll desgnatng G. The current then goes to the lampboard, where a bulb s llumnated to ndcate G as the ntal output cphertext letter. The next thng we do s press the key for the second nput plantext letter N on the keyboard. Ths changes the wndow letters to QUQ (whch changes the rotor o set for the rghtmost rotor), and begns the journey through the machne for current ntally desgnatng N. A summary of the full encrypton process for the complete plantext ENIGMA s shown n Table 4.7 on page 58. The last lne n Table 4.7 shows the complete cphertext, GVYWHF. Fnally, for current travelng through an Engma, snce the current went through the plugboard at the start of ts journey and then agan at the end, and through the rotors from rght to left before gong through the reflector and then agan from left to rght after, and reflector contacts were always connected n pars, the machne would always produce nput/output letters n pars. That s, for example, for dentcal confguratons of an Engma, f enterng nput letter E yelded output letter G, then enterng nput letter G would yeld output letter E. 1 What s mportant about ths s that for a cphertext formed usng an Engma, the cphertext could be decrypted by ntally confgurng the machne dentcally to how t had been ntally confgured durng the encrypton of the message, and then nputtng the cphertext letters. That s, for example, for a Wehrmacht Engma ntally confgured dentcally to the ntal confguraton of the machne n Example 4.15, f the cphertext letters GVYWHF were nput, the resultng output would be the plantext letters ENIGMA. In the feld durng World War II, a cphertext formed usng an Engma was decrypted by ntally confgurng a d erent Engma dentcally to how 1 Ths s not volated by the nput letters E and G n Example 4.15, snce the confguratons of the machne (specfcally, the wndow letters, and thus the rotor o sets) were not dentcal for these nput letters.

14 crypttext 2017/4/4 9:57 page 58 #66 58 CHAPTER 4. THE ENIGMA MACHINE Summary of encrypton of plantext ENIGMA usng a Wehrmacht Engma wth plugboard connectons MZ and NS, left-to-rghtrotorsv,iii,iwthrngsettngs12,4,8 and ntal wndow letters QUO, and reflector B. Input letters E N I G M A Wndow letters QUP QUQ QVR RWS RWT RWU Rotor I o set Rotor III o set Rotor V o set Plugboard E S I G Z A Add rotor I o set M B S R L N Rotor I from rght O K S U T W Subtract rotor I o set G B I J H J Add rotor III o set X S A C A C Rotor III from rght S G B F B F Subtract rotor III o set B P J M I M Add rotor V o set G U O S O S Rotor V from rght T Q L M L M Subtract rotor V o set O L G G F G Reflector B M G L L S L Add rotor V o set R L Q R Y R Rotor V from left D O U D H D Subtract rotor V o set Y J P X B X Add rotor III o set P A H Q U Q Rotor III from left H T D Y W Y Subtract rotor III o set Q C L F D F Add rotor I o set Y L V Q P S Rotor I from left O E I H T S Subtract rotor I o set G V Y W H F Plugboard G V Y W H F Output letters G V Y W H F Table 4.7 Summary of encrypton usng a Wehrmacht Engma.

15 crypttext 2017/4/4 9:57 page 59 # THE ENIGMA CIPHER MACHINE 59 the machne used to encrypt the message had been ntally confgured, and then nputtng the cphertext letters. Thus, the key for an Engma cpher was the complete ntal confguraton of the machne used to encrypt the message, ncludng how the plugboard had been wred, whch rotors had been used n order wth rng settngs and ntal wndow letters, and whch reflector had been used. Despte ths, from a techncal perspectve the Engma was not d cult for operators to use n the feld, snce they dd not have to understand the encrypton or decrypton processes, but only how to confgure the machne. 4.1 Exercses 1. Fnd one or more mages of the followng. (a) Full vew of a three-rotor Engma (b) Full vew of a four-rotor Engma (c) Engma n use n the feld (d) Engma keyboard (e) Engma lampboard (f) Engma plugboard wth some or all sockets empty (g) Engma plugboard wth some or all sockets connected (h) Engma rotor () Engma rotor taken apart to show wrng (j) Engma reflector (k) Engma reflector taken apart to show wrng (l) Engma rotor rng etched wth letters (m) Engma rotor rng etched wth numbers (n) Engma wth top panel removed to show rotors n slots (o) Engma wth wndow letters (or numbers) showng through wndows (p) Engma rotor rng showng notch(es) (q) Dagram of the flow of current through an Engma

16 crypttext 2017/4/4 9:57 page 60 #68 60 CHAPTER 4. THE ENIGMA MACHINE 2. Consder an Engma rotor wred wth the contacts connected as lsted n the followng table. Rght contact: Left contact: ABCDEFGHIJKLMNOPQRSTUVWXYZ ESOVPZJAYQUIRHXLNFTGKDCMWB (a) Suppose current desgnatng B enters the rotor on the rght. Fnd the letter desgnated by the current when t exts the rotor on the left. (b) Suppose current desgnatng I enters the rotor on the rght. Fnd the letter desgnated by the current when t exts the rotor on the left. (c) Suppose current desgnatng D enters the rotor on the left. Fnd the letter desgnated by the current when t exts the rotor on the rght. (d) Suppose current desgnatng T enters the rotor on the left. Fnd the letter desgnated by the current when t exts the rotor on the rght. 3. Consder an Engma reflector wred wth the contacts connected as lsted n the followng table. Contact: A B C E F G H I L P Q S U Pared contact: R D O J N T K V M W Z X Y (a) Suppose current desgnatng D enters the reflector. Fnd the letter desgnated by the current when t exts the reflector. (b) Suppose current desgnatng N enters the reflector. Fnd the letter desgnated by the current when t exts the reflector. 4. Fnd the rotor o set for the followng Engma rotors. (a) A rotor wth rng settng 7 and wndow letter J (b) A rotor wth rng settng 2 and wndow letter B (c) A rotor wth rng settng 22 and wndow letter E (d) A rotor wth rng settng 16 and wndow letter L 5. Consder Engma rotor IV wth rotor o set 4. (a) Suppose current desgnatng E enters the rotor on the rght. Fnd the letter desgnated by the current when t exts the rotor on the left.

17 crypttext 2017/4/4 9:57 page 61 # THE ENIGMA CIPHER MACHINE 61 (b) Suppose current desgnatng P enters the rotor on the left. Fnd the letter desgnated by the current when t exts the rotor on the rght. 6. Consder Engma rotor II wth rotor o set 9. (a) Suppose current desgnatng L enters the rotor on the left. Fnd the letter desgnated by the current when t exts the rotor on the rght. (b) Suppose current desgnatng W enters the rotor on the rght. Fnd the letter desgnated by the current when t exts the rotor on the left. 7. Consder Engma rotor VIII wth rotor o set 22. (a) Suppose current desgnatng P enters the rotor on the rght. Fnd the letter desgnated by the current when t exts the rotor on the left. (b) Suppose current desgnatng Y enters the rotor on the left. Fnd the letter desgnated by the current when t exts the rotor on the rght. 8. Consder a Kregsmarne M4 Engma ntally confgured wth rotors, II, VIII, and IV, n order from left to rght, and correspondng wndow letters BELI. Suppose ths Engma s used to encrypt a plantext. (a) Fnd the wndow letters after each of the frst and second plantext letters are entered. (b) Fnd the wndow letters after each of the thrd, fourth, ffth, and sxth plantext letters are entered. 9. Consder a Kregsmarne M4 Engma ntally confgured wth the plugboard wred wth the sockets representng G and Y connected, the sockets representng N and S connected, the sockets representng R and T connected, and all other sockets left open, rotors, II, VIII, and IV, n order from left to rght, wth correspondng rng settngs 2, 22, 16, and 7 and ntal wndow letters BELI, and reflector C. Determne the result of usng an Engma wth ths ntal confguraton to encrypt the plantext GERMAN, and fll n each of the entres n Table 4.8 on page 62 to summarze the full encrypton process. 10. Fnd some nformaton about the Amercan SIGABA cpher machne, and wrte a summary of your fndngs.

18 crypttext 2017/4/4 9:57 page 62 #70 62 CHAPTER 4. THE ENIGMA MACHINE Summary of encrypton of plantext GERMAN usng a Kregsmarne M4 Engma wth plugboard connectons GY, NS, andrt, left-to-rght rotors,ii,viii,ivwthrngsettngs2,22,16,7andntal wndow letters BELI, and reflector C. Input letters G E R M A N Wndow letters Rotor IV o set Rotor VIII o set Rotor II o set Rotor o set Plugboard Add rotor IV o set Rotor IV from rght Subtract rotor IV o set Add rotor VIII o set Rotor VIII from rght Subtract rotor VIII o set Add rotor II o set Rotor II from rght Subtract rotor II o set Add rotor o set Rotor from rght Subtract rotor o set Reflector C Add rotor o set Rotor from left Subtract rotor o set Add rotor II o set Rotor II from left Subtract rotor II o set Add rotor VIII o set Rotor VIII from left Subtract rotor VIII o set Add rotor IV o set Rotor IV from left Subtract rotor IV o set Plugboard Output letters Table 4.8 Summary of encrypton usng a Kregsmarne M4 Engma.