Reference / Paper · 1935
Friedman's coincidence paper as reissued by the Signal Corps: Vogel quintuple-disk and Schneider ciphers solved by coincidence counts; footnote 1 says it revises the 1922 Riverbank paper (87 pp scan, declassified 2014)
. - ____ ............ - .................... -- . ·- -· ___................... . . REF ID:A64722 fr 11 ___ C" ,, .. ........ ..........i_ ·.,; lJOl;.11~. ·: . .. .,a~ Register 1 OFFICE or TB~ (.'}J'.l;..:F SlGXAL Ol'l'ICBB .f •· THE INDEX OF COINCIDENCE AND ITS APPI. lCATIONS IN CRYPTANALYSIS ... TECHNICAL PAPER u,. WILLIA.::.f F. PlllEDl:l:AN Clalar.,, Slpal laMUa-Sec&lllll WarPINalullTlalalqDI.._ ~ ~ ~ · lJXITBD BTA.TBB GOVDKllBNT PRJJiTlliG Ol'RC& I L _ --· -·- eclassified and approved for release by NSA on 01-06-2014 pursuant to E.O. 13526 REF ID:A64722 .\ THE INDEX OP COINQDENCE AND ITS APPUCATIONS IN CRYPTANALYSIS 1 INTRODUCTION Frc11ut'ney tables in tho annlysis and solution of ciphBrB have• commonly been employed to uinkc u~~umptions of 1>ln.in-toxt cc111.iv11lcnls for tba clplacr lc~tcl'IS constituting a message. 'l'he ='ignifi<'..ilD<'.O of tho v11.rious 1>J11UK."B of tJ1c curves thomsclveis, .i.e., tbo crests and troughs nud t-h~ir r<•lntivo positions in such froqurncy t-o.bles, has boen rcco{."llizcd to some extent, but lnl",:•'~Y only iu connoction witb tho clatarmination o! two more or lOllS preliminaey points in tht>~r :inulys.is: (1) whether tho frequency di,.tribution approximates that of a substitution t·i1,lm· i11\"olving only ono alphabet or more thnn ono nlpli:a.bet; (2) whother this approximation rorr•'"J>0nd11o to that of a standard alpbabet, direct or revosed, or that of a mixed alphabet. I1 will be shown in this pnpor that tl10 frequency tables of cmtain types of ciphera have ,;,.,::1iiP. 1•lrnrnctcristics of a mathematical or r11tl1or sto.tistical nature, approaching more or leaa rl .. ..-,.f.v Lboso of ol'din11ry statistical curves. 'rlH!Ne cbaructoristics may bo used in the solution .,:· i.11rh l'i1>be.rs to tho exclusion of any analy:1is of tho frequencies of individual letters consti• ,,,; ..~ thll tables or curves, and t.ailAout a1&'1/ aanmpliou """"*-' q/ plaVHm m/,w1 Jor O&a ,· ·,,;,, 1· lttltr~. · lt hi truo that cipher systems admitting of such treatment are not very COD1ll10nly encount 1·:·«,l. Bu:; i1111SJ11uch u such systems are always of a complex naturo, which the ordinary 11u•thuds or cryptanalysis would find rather ho.filing, a description of a purely mathematical .• iinlysia th11t ma.y be applied to other cases similar to the onea herein described may be eon-=itl.~:~11 ,·nli111ble. In fact, it is posaible that the principles to be set forth may ftnd con11-id<'r:.uly ''i<l<!r applicn.tion in other phases of cryptanalyais thu.n is apparent at this time. 1'wo examples of such n treatment will be given in detail: One dealing with a.substitution <'ipl ..~r whcroin. a. series of messages employing as many a.a 12S random mixed sccondary.alpha; ... h c•1rn he solvod without assuming a plnin-t.e.-,:t vnlue for a single cipher letter; the other a muldplo atlJ>bo.bet, combined subatitution-tra.nsposition cipher, IOlved from a Bingle message oC Cnir fo11g1ih. 1 'liac pN11CDt paper 'WU prepared In 1928. It ia a revision of an carllor paper with the ume title, published in 192:! by the Riverbank LAboratoriea. Geneva, Ill. Tho author t.akCll thla opportunltJ to thank CoL George P•1hy11.11, or the Riverbank W.boratorles, for bis oourteay in granting pormimdon to pubUah t.bill revision • •\ltl11111icb bct.t.cr mct.hod11 liave beeu ulaboratod alnco tho revision wu prap&Nd, I~ baa boaD doomod of lnterod liiKt.oriclllly Lo publisl& 1.hia pa.per In it. 1028 form without change. (1) _,___""'"*.,.,_.,.,___,.,...,.._,_..,._.......... ~,.., _4_.C_"!"..,..P.Z"!l.0""1._,..411414'"''""14-.""!.~if .. '"""*"!"". · - - - .--.-~~-· •• ·--· •·· .... :. t. I >. t f r .r . ; REF ID:A64722 ( PABT I .THE VOGEL QUINTUPLE DISK CIPBBB. 1 This ci1>hor systcnn involves tho uso or Jivo superimposed disks bearing dissimilar ranclom mixt'd. 11.lphabcts. Tbase disks are 1nountcd upon a cinmlll.l" hnso plate, tile periphery of "·hich is divided into 26 tlegmcnls; ono of these is nuirk~ll "J>lu.in ", indicating tho &Ob'1llt'nt in lino with wbfob tho llUCC".ct;tUVO letters o( tl1e pltlfo tc.~t 111:1 fOliJld OD tbo fivo revolving disks 0.1"0 to bo brought Cor cJlciphm·nicnt. Tho ron111.foing 25 sr.gmt'Jlta or the hnso llln.to bear the numbcra from 1 to 25 in a. mixod seq ucncc; wbfol1 WL\ Im vc cullod the " N umorico.l key. " This key mo.y, howovcr, consist or less thlln 25 numbCl"S, in which cnsc one or more of tho segments of the base plato will remain blank. Tbe numb01'8 constituting tbe key are.written on the·base plato in a clockwise direction beginni!-11 immccliA~ly 11t tl10 ria;ht of the plain segment (fig. 1). · METHOD o:r ENCI.rmm~Bl'fT In the nccompanying example illustrating tho details of enciphorment it. will bo 10011 that tJ1e numerical key consi&ts of 21 numbers, looving blank, therefore, tho four segments immediately preceding the plo.in segment.•· .Assuming a sorias of messages, let us suppose the first three .begin as followa: 1. Prepare for bombnrdment at Harvey • 2. Enemy attack on Huntorstown • • • 3. Second Field Artillery Brigade • • • Revolving .the five ciphor disks successively, nnd thus bringing the first 5 letters of message 1, PREPA, in line with the pin.in segment, reading f1·om tl1e outer disk inward in the order 1-2-34-5, the cipher letters for tbis first set of 5 plo.in-te.'Ot letters ore then taken in the so.me order from the segments of tile. disk11-dir~tlyin lino with thatscgmentof tl1e base plate that b8ara the number 1. In this case it is the eighteenth segment after !iho plain, in a clockwise direction, and, as sllown in figure 1, the equivalent cipher lettors for this group are JIEKJR. The second sot of five plo.intext letters of mcsso.go 1, REFOR, arc then in a similar manner set in line under the plain segment, and their oquivalent ciphor lotters a.re takon from the segmont immediatoly following segment 1 of the 1111morical key, in o. clockwise direction, viz, segmont 13. The cipher letters in tllis co.so are VZQWH~ The third 1,rroup of lettors in lllt'SS11~ro 1 finds its cipher equivo..lonts at seg1nent • Whi:t! nn duty in tl1c Codo a11d Cipbor Scctiw1 of the Intc1ligonco Division of tho Gcntt.a.1 Stair, G.U.Q., A.E.F., Lt. C(Jl. F. Moonuan, Chief of ScctJ.ou, tun1cd over t.o 1.l1c \\"riter for 11t.udy a cipher ayatcm to;cc~hcr with a 11Cri;•s or 26 t.cat mcaagca aubmittod by Mr. E. J. Vngcl, Chief Clerk, who lwl ta.kou con11idorablo ln.t.ercat in ory1.t.or=·•l'hY and luul, ""'n. rcault of his atmlic11, dovi1md t.l1c ay11tmn J>l'C8lllltcd tor oxnmina.tlon. Tho writer \\"Orkcc! UJ•·•ll tho oi11hor durh1ic bill lcia11ro momenta, hut. the 1>rohlozn involved cu.111ldorablo laLbor nnd aolntion WBR DCIL CollUiJICtcd before bcinp; reliovod from clu1.y at. t1111t 11ta.t.iou. Tho 1111\ln 1>rl11oiplca ror aolntion, however, were .i.;t.:Lhln.hr.d ancl only 1.hc rlct.ailod work l'OD111;11cd t.o bo COIDJ>lot.cd. After an b1t.urvnl of more than a ye&&r, wbilo Dir.•,1 ur or t.l1c Cipher Dc1>u.rt.inont of tbc l\iverbank I.:ibora.t.urloa, Genova, Ill., tbc writer Lunwd his a.tt.c:1•tii1u u1icc ni.ore tu tl1ill cipl1or and auocoeclod bi aomplotcly anlving ~ problem b;y careyiug out thUllC princi1>•C1> ;.o their logical conclwiiOD. • It is l"COOIDmondod that the roader prepare a duplicate of frho aet ot diake In order tba.t he may moro rcadil;y follow the various steps in the analJ'aiL i l I i I (2) - I I ·--·-:-- ,,_ .... .,.__ _ _ _,,..._,....... -..---·-"'·'""'**"IJl'!ll,l.-!'11'!1!1111?'""1'"-l,P.,.)S"l!l+:f-:SP~i~41-.---• - - · . ..... _ _ _ _,_ _..._..,..._ _ REF ID:A64722 .- ., ·~ to~; fhr. iom•t.h, 11t R<';!UU'llt 8. 'fhr. firth ~roup of plnin-toxt lottan, howevor, wiJI take it.a c•iphcr <>q11h•nlent11 fr.n:1 tlu• :ir::it ~:-gmt>nt to t.he.right oC tho pin.in segment, inasmuch as the srgmt'nts inunooinr<'!,,· fl1Howi:ig s"t!mt'nt 8 :i.ro :1;:mk. Plo.in-text letters TATHA, tb.erofore, will bl" cncir>llrrl•d oi1 srg:ni'm O, Jmcomfog XONJE. 'l'J1is met.hod is co11tinued. inJike maDDar throug~ out ntl'SS:igc 1. IC mcssnb"C 1 conta1ins more tfom .21 groups of li Iottera,.the twenty-eecond.group will ~1ko its cipht'r cquivnlonts on Bqtmcnt. 1 ngnin; tho twmty-tbird on aegmcmt 13,.~ ao oil. · - ~ r. . I :· t. ·. ! ~ r i .. ": ! : FIC'C'Rll l . . . . . Pl'O<'r'''!ing now to 111~n1.~ 2, tho fi111t 5 pin.in-text letters, ENEMY, aro set up under tho plain :-tC'ftm"nt 1md tho cipht'r letters arc tn.ktm from l\l'.gmcnt 2, bocoming LTVBll. Tho aocond group nC .!> lrtlr.1"R i" t'n<'iphorcd on segment 1, tbt'i third grmip on sogmont 13, and ao on., throughout tho m~:-1n~r.. Tho flrst 5 lr.ttm-s of mcsango 3, SECON, n.ro encipliemd on segment 3, becoming YAPAC. 'l'h<' rirst g;:onp of cipbcr Iattm:s in " mC1MDgo is always to ho taken Crom tho se.ganenf. br.nrin1t tlu\ 11umhcr which coincides witl1 tho scrin.l or li.cccssion number of 'tho mC!'SNlge in iho ·day'f' ncth·ity. It '\\ill ho sC'.an, thero!ore, tbn.t no matter what repetitions occur in th~ plain-t.ext beginnings ol mossngcs, tbe cipher letter& will give no evide~~ o~ such repetitions, for e&C'li niesaage .has a diflerent starting point, determined, as said before, by the serial number of the meaaage in the ,. - - · - · - - . - · - _ ... __ ...... _ _ _ _ _ _ _ _ _ _ _..__...,....,..,... ... _,_,__,,,_,__,,__uc_a,..;uz;;1111,.""'·,..,.,'"'11,111>!,AA,...!""f.""' _ _ _• - - - - - · REF ID:A64722 .- 4 dny'i; neth·it~. Tl1is automu.t.ic prevention or initio.J ropotitions in the cipher tut is true, howe\·er, onl~· of n. number of messngcs equal to t.he ~ongth oi tho numerical koy; for, with a number of mt'lli11:tt:"'" ~r":i.tcr tbnn tho ll'ngtl1 of the ke~·, rho initial segment.a from which tho cipher letter& o.rt'! to be tnkcn bt'gin to repent. In this cnsc, mcangc 22 would noceseoril.y hn.ve the aa.me initial point as mcssngo 1; ml'SSoge 23, t·ho snino as message 2, and 80 on. Messages 43, 64, 85, ••• , would nll bt'gin \\it.h Bt'gment 1 ; messages 44, 65, 86, • . • with 1og111ent 2, and 80 on. Tbo sl'!r.rccy of tho &)'Rtom is dapondnnt upon o. frcc111ont cbo.ngeof alphabets and a still moro fmqucnt chnngo or numorico.l key, sinco it must ho n11&11mcd, as with n.11 cipher systems or devicoa, thnt soona1·. or later tho general mot.hod oC onciphonnent \\ill become known to the enemy. The only roliamco, tbareforc, Cor t.lio sn.foty of the mC888ges must bo placed in keeping the specific alphabets and tlie numerico.l key for a givon series of Me&Sftb"ell from the onemy. PmNCIPL:t:s oP SoLuTiox The following meun.gas 1 a.re 888umed to laave been intercepted within one day 11.nd t.b.erafore to bo in tho snme alpba.bets and koy: MESSAOE No. 1 f GIRIE IMNEE FEXVP HPVZR UKSEK llVQCI VXSFW GVART YBZKJ WVUPV XZCBD BDOLS GHINZ LJ'CTE KSLPY VPBYD WTRJK BDDFA AN.JD XGHED ERYVP YPWDJ DF'l'JV ZHTWB WXTMF OZDOJ MLVXK QNXVD MEss.a.oz No. 2 - ULJCY GXAEU DTEIL UZBRW GJZSS QLUOX PTFOO NWSHD BPTJO HQRYY YAIRZ KTEllP UAYYK ISRDZ VUVKW HXAYD YAGSll CURBZ LBXOV EBBPI Biil.CB UJWCF ZSLXV QFXUE llPZllK MQZZT KllURW EJVB Mt:&!AGlil No. 3 CBGSF VGABP HOZFV QQNSQ NQLQL DIGDI XCWAI QFJOQ MBMJB SEPSO DHREM ELKIP KXND QYBIH BHFDC GLYWC YGMMP EEXZH UBBSB SBONG URQKW YRAYU NYUCS LNDIV VNSXN WGVME YXPDF WGTZE KRLGU ZJFZJ I YLMKW 'l'YDEL MBBBAGB No. 4 UFHUJ LTMKJ PONFG RIUGG OZGWS UBNMW WGILB JNXTD BPREX MMWHB OBFVO TFGSJ SLXEH RTZ.MI LLUUX FIFIC PGSBA KRCAS DiKQV SLDKS NTESD QVQBN RZDMB JQLYH L'l'MXB BSVWL ILKYU NMFEB HB M.BSSAoE No. s v.::.:~Q Xl..'l'J~~ IZ'i!1'X ~'OC#IE 1 ...- - · · -·-- ....... _. __ - KZJJK DCVQD KSSXY TSUXE GRROH PXKZF ZKFMS VGDWU AWQWr' ZES1X GWCEM JPNVB GRJGB IBWOH YMOAP GXMSB OZZGK FVURN NJFGQ DTPLV STOID DWVLR TXTBH YJXXW BKOJQ FOUHO T EX.i.tEF 'J.'bCllO are 1.b.e act1181JD.1111aps1n1bmlt.t.ed by Mr. Vogel• - !'I'll,._..,_.,._.___...~~-·--------.. - · · - - · · · - - . . . . . - -........- · - - - - - . - - - · - -............. , .....'!!!'_,SQ ...."'"'"'""·-II'!" .....~ •. REF ID:A64722 - 5 MB&&AGl!l ~ o. 6 XGORF IWTVU VGQSW LNGQ'l' GCHAX DUEIQ NPCLU KQDIG OGHPI XCYIO IPJRI HlJIW) XAOWK BK3UH SKWWW XHEYZ JAKUK BEQMG YLllNC KYJJF SDSTU DCBUK OQUVA WHSD SUAEU BQAPY RllDJIW FWGSQ HYllRQ LPUVH DZU'l'W BFMO ' ME&SA.GJll No. 7 UFUCL HJYDY ZTHHA HWJWJ InlZI VZIJE QODUT llZPRX SWNFC DKXXR TOSVL llNHNO RZR'l'X QIPFN HOUNL P'GUVO TPOWI VHNCQ RTDCO LNTTll YXMXR TUIIG ZOJCU BTXJK KGUEJ llSJBS ESVWJ IKGSL APA MzssAG.E No. 8 ~ RPQHF XOHQN VTEGJ IAFEO BWUT VPSKO GXHRP OIHWF LBTKF QESIO KLXHR NCVNY XSQllC XVI.JC NWWVY UllHVB IPA.TI CFllZT DillQI IRUJI NSBWR HUYNA VPRXS SCUZB JHCDE '•WUHJV YXVNK AID.AA EQLVJ llYYTH GSEYA TJVSC WEGZ TFOHC HNPll . r MBSS.AGB No. 9 .- \ ..f. NYMPE YZYRZ AWLPP IMPBB VWAXZ QSVFC I'l"lll'l' KZNFC I»fSUA NNCMP SOJKI GDPQZ IIPllV ZOCBO UCKXR SEPEll OTZNM mm NDEUH YUEIP RFOHI QLQHG IJFRT UTNQC JFJJAF SQRWO DAJll'l' YKXXQ VRQUW ~.: t r '' l I M:USAGB No. 10 f· . .. XXHHC FYllFK QRBJV YVJID JBm' JBLXU ·KSOXR KNHJK UFRXL WELQJ QJKFI RLSSF BQJIR BZKYH JWJWP AYSKO UWJDX CQRVW ZVXXH HZHCW SVEYH NEANW G TEPDA l .f· .i J MESSA.Gil No. 11 .. BASFH VFSCG CHKSB ZOPRZ CZEVH OCADC SEYBZ MGSOZ XYUOC FFFVJ OZCJE AQSJY llZCllO TOXVll LGRXO XXCKK KEXYA VWONX GTCl'l' TGLOI IWORT JJVQB HYNK CllPSQ llVQGJ . i I { : .. : I ME88A.GB No. 12 t OMTXX WUXZE YEHOJ ALJCO EPLPJ RBCVX VVARW DYBDQ F'l'ZDA XEWG ROHUP OFSGQ PONLI RA IBP KACIB CllY.BB VKHEU XSPFG WKZTE KZYIZ ZXFJO HZIVG ESGOX YYCEO B • i. MEBBAOB No. 13 QNKIT FZHNC GJBSA JIQBX PFTAO RJLUD IKPKI THUWU EVGHU LJUFZ UBSllD VHMHZ UWVYQ DJPFY KE'l'llN CLEQS DQJNA GEYHC JBCPG RNNNH BSYZR STGBV E ~ . 'l ' Ci • ,, A:·"·-'''*'''.' ....., ....t ..f5!.i.WN •.. 40)2.• F. I 4 q _........ .. . . ·- REF ID:A64722 - () l\h:ssAc:.. M No. 1·1: CLYZK RINOK NBTAK ;:m:jBI WRZRX DQTAR AEKllY llOXLT DXZSI( ZP\YNI JWLI TPWR TPQGH RJZCQ XCNHU NOKllI FJW\'VR ZXROX PZUGG HMYNB DUHQZ BJDFA FJAYU RKHKB ZMKMI D z.mi\K YVi"ii;'L l\Kl]EF MESSAGE No. 15 OSULC QQDCX UUHTF :XDCAS WZAFW CBIMT FOQOK BEHQV SQAUC DGIAS KNRPI OPWEY WPRBI BAQRK PLLXQ UTI SE \'lCFUO JKQHY OIWVX KSMSX MBXOZ Ru1-IHF JYUJG DYNllA- CWULT PK\'lEE: DRDEM JMCFB WLHVX FIFZR STVJV NZFCU BII ME&&AGE No. 16 VHXVD RSRY:t PVNWQ QEDMJ LTKFN RMDGT DNllBM JDGVP ·JCJMND RRQDM STTAL GKPPO PZTCU BNCEE HARB KRGNU ZXCWQ VEZ'l'C DIPSG ZUFVH OZIGB TQXND HMHEW VHTL'l' MESsAG:iit No. i7 .- UMMOL HVVEQ GWTJI PQTNS AEFZH· TOFQH OOXNQ NPDDT QVSJR RAABX ZCVWR WJEG NMJHQ CHZUM TTSIU GHELW HUMFH LZHNH CJEDW AKHZA SDZIE HIH\9G LBUFZ DVPUB DCMXO NAYEY GQHID .. M.e:1:1sAOE No. 18 LALNH QUDUA ZBZUD VSJFE MEHX?I EUWZ'l' OKINO OOSIL TASEG OVQVX PMKKW BQRBI VGDGJ JDAH'N RZDI.A. WQAXB FBRLA AHJEP MO PUMEU HJQJP Z.SPPQ VZWDL HE:CDA LPJJS ZOJYB .. M Jo:Hti." (;)•; ~ o. l u YTWL TWEVD MBHKV YlUlAF BKIJC ZJSRN Url1.iNZ. OPRFC. RDONY YUSUK TFECD BMQVB YUJPZ BHLDY VVPllD IHTPI GNXBQ X.A.UAQ OUFVO GSMKB BAKIG CHCCB ~ OLDQA zzocv ROZ.A.II ZYNVQ WFONZ C'l"l'ES. IRWER GKETA v~wvv VWCZP. TWC:l'J FHFEH VNDCO llZVLK KFHPB VCYU WBQHM. UYJPT MESSAGE ~ o. :?O OBHOK UWRON AJDFH FRQMI ULOTG XXI EV HMAKV PVMAV OITKD LDQIW UYVWI JEJRQ MCUZP KGUDN ~PBF· TQVZP- IZJTU RBUFZ JUSIZ DCIGX QESJD LIZW l..OPME Y)IEXI HOIJK KUYHK AORUY LVD Mu&AGE ~o. 21 - O!f.XOW LTYIEW KFJHN ZMSKK GXFDL YLCPT YYYNQ CYYMA ZWRAD HG?IBI HHCGM FDGMN XIQDN NPLYQ NPJNZ OWFVV KMGVH KHCUC STNVZ CGXHV LZZXX SVTKG POEAC OJYQU MEULH KHYDD E'l'PDN QYZVU HIMGG RBHEI cuso .... ~·······-~ I PU. Z =· I "¥ c -·•'""'W"' • -~ REF ID:A64722 - 7 )ih:BSAr.E Xo. ~2 AQMQ\Y FFONY GVFZH YJOUN AE:;QG HTZPB YRNEI cxmF UKCTB BETEX. ZSL.ru< JJMIV JFCEU QFPMO TFSOQ S'l;'D""'"" ... r ;,! JTPSY DWIVZ QF".aJV XKF.SC IAONR JNPXG CLRIL CJCFI TWSZJ OJUGA HOGXW sxxsz HVRUC QEHQR LHIGG FVDSQ ·R.JCSH BK No. 23 L}!i<G? KUKNY WJROY XZHPK JJIHU LUVBU ZRUDA XXHRF MOAFT Xt:'\1VE YZAJE NFDEX. GDZGA JBZET XYHAL DGECA CDPYll B'l'llYK LCSMI YVSWW DQNAP KFPAO FIQTR KQQIE HHYCA GHEBF PFTYF DZAUO w MKSSACl1G 1 MUBA.GE No. 24 FIYGR HQMIP FAONR 'QCAVJ MFIAC ZURAN ULZFE YDLOT UZllJM SETNP XUWAI DGAllL QBTFK VYIGT FWVK y - 'l'KNDA LGPKS CWPSK VEKYG YOOHV IAUNO SRUEU RUWKE RIYIT UZHRQ tJLXFU :XSINC UFRNA APKQP KAXUF TQRGT JODEE XHRXE WIZWK IIZYY · HTAVV OTJEP VZXCL YFYOE ZUAPV QRYXB 'VHMWL IWORY OISIE . RESKE DDQYB RCWS UXWQX JNSSI· HCHXX ZllZKW :' ~ I . t I. t . t· :. ~· . it . I t · · 11 BSNWH LTTXN LPSJJ UTNIE BDWVG MZNGB M.VKNT ZWXXQ CACVC LNICD KIOZG DLRZV O'l'TTO Rrl'HL LSIJU SYBllC LPXGC RLE'l'I PAWD SMQVB WDPDB BYGMQ PVUGD QCVQ It mny bo of ndvnnto.go w bl'.gin the eluciclntio11 of tho principles of solution by translnting this <'~phPr into terms of the sliding of primruy nlphnbot.s against one another with tho consequent pmdur.tion of a multiplicity of soconda.ry alphabet.a. For example, by using ordinary sliding al~>hn.bots sucli o.s arc commonly used in cryptanaqsis, we may produco the sAme results 11s aro briven by the set of concentric diRks. l.it\t 111 use the n.Jpbabet.s of the illustrath·e di,;ks, mounted upon sliding strips in pnira, o.nd let us slide each pair of alpha.bots 8 fottPrs apart. Thus,.if we consider the uppor one of ca.ch pair of alphabet& in fic."Ul'O 2 11s tho plain-text alphabet and begin each alphabet arbitrarily with the lotter A, we have the following: .. I; i[ MBBBAGlt: No. 26 XFYKU NDBFZ KNDMF GBVIJ VDCYO GYZLI TRURC GBIWL ZAECE XQDYB FDBPB SESMA TRJTG PZJCM LSFXW ANHJO VWRIH MPMHG DURHG IJKIR ESOWP MJQEC GGJBF SGMFC DLZCS EMVZL GAMFQ ULGXC HQWVZ XZUAB GLHMH ZFTIQ YQXGD SEC CE CIJTI BUILZ DJBCW IHSAM RHCMJ RAQBJ VEASF TZCWL AUHKA MSTZZ XRLCH IOKZO EPLQU XJZKI ..' .t . 1: MEsa.t.Gl!l No. 2S KVKRA '! I QMOOK OKWHR llPXQI PIQMN ALWNK HZIKQ XQNUY. GQZNG ·.DFTFO YJODO XI RIX KDCBX UEQTU VPI WA NFWOH GDXY EKBMG llGETD WFKKZ PXXZL XFRWL FH'l'EG INNJI ETVUP QHTJY" OP FPETJ CDVNY LllKQU CDALX YY.DKR GWllQL FQMEJ KOBMS GWILM FGCVS NCZGB HUIRT FJYAZ YQNVO WSMSS CCllFY . •I .' I . i· I: ;~ , I . . i ~· '· i I . '.: .t .. ..' I }' ! I ~. !· .. iI. ..• ; ) . .t i t I· i. . ) REF ID:A64722 ) ) l'IOVHll I Plain text __ •• AUFQZERHYGWJOIDMNCTXSBLPKVAUFQZERHYGWJOID AUFQZERHYGWJOIDINCTXSBLPKV Jl'lain text. - - _ AOXGFZKYLPUKETDJVSWIRBNHCQAOXGFZMYLPUKETD AOXGFZMYLPUKITDJVSWIRBNHCQ Pia.in text. _- _ AWJBQIHVKPUFOGTNEDSZXCMLRYAWJBQIHVKPUFOGT AWJBQIHVKPUFOGTNEDSZXCMLRY Plain toxt. - - ' {apher ______ _ ACVXZWMTFNUIOSEHJRDLGKQBPYACVXZWMTFNUIOSE Plain text. - - - AETJUDKVZNWHXOGYKFRLQBPCISAETJUDKVZNWHXOG AETJUDKVZHWHXOGYKFBLQBPCIS 1{Qpher______ _ I 1 \apher ______ _ ! 3{Qphor ______ _ ·• i I {Cipher ______ _ , .,., j ACVXZWMTFNUIOSEH~RDLGKQBPY ... . . . .. REF ID:A64722 . ....... . Xofr now tlu1t thr fiMt set or a pliiin-lc•:.(. 11 1.tc•rs, PREPA, yiold.'\ the MtnC\ set. of 5 ciplu~r lrttrn;, MEKJR, thut. wr found on pit~ :? h;:L· lirii11~.:: t.hC' tliKkR. 'l'J1r. only tbing wl1i1•h t.l1e8C !ivP pnii'li of irulrprntlt•nl. RlidinJ? nlplrnhr.l!'I hn\"t' in r.ommnn in figure 2 is tbn fA<'f, t11"t oar.h pair irns ht.en Niel nr>nrt thl' An mo numbrr ur lt•tt1•?-,., \"il'-, 8 i if\\"(' f'oniiider tho 11J>Tll'!r olphaLflt in r.nch J>1iir n~ tho stnlionary alJ>bahot., thrn lh<' lowC1r 0110 llll';. h.-rn shift.r.d 8 intervals tut.hr rig:l1t, or IR 1 - - intrr\•nls to thl\ Jr.ft., or t.hr. 11111>rr nl1>l111hN.. ThiR rorrr.sponru. to t.bc poAition of muuhnr 1 in figure 1, for thl' Iott~ number ocr.upics tbr. ri~htcc.'llth sagmcnt to tho right of tbn plu.i11 l":e;:inent, or tbe ci1.d1t-h to f.ho left. The relative positions of t.hc numben in the numf.ricnl key, tberefore, CO?l't'Spond t-0 tl10 numbol'B of intervals tho primary nl1>ho.bRt.s in tl1e form of sliding strips would have to hr. displnced in order to produce th<' snmo rrsult.s as the disks. Now tho sliding against itsolr of a prim1ny sequenco contninfog 26 lott<'rs will give rise to a series of 26 secondary cipbor alphabets; 1 likewise, ea.ch primdl'J' concentric sequence will give rise to o. series of 25 Bl'COndnry a.lpbohet45, If the numericol koy consists of 26 numbers, all these socondory alphabot.s will be employed; if it consist.a of less than 25 :q~mbcrs, then a correspondingly docrenscd number of sr.condllJ'ics will be employed. · Since en.ch primary sequence can givo rit1a to o. set of 25 secondaries, the total number of poBBible sl'<.'ondnry alphabets in the wl1ole syst<'.nl is 125; but if the numerical key coDSists of l<'l!S tbn.n 25 numb<'l"S, t.hen the total number of sccondnrfos will be leBB than 126 by exact multiples or 5, since the absence of on(\ or n1oro numbers from the key aflects all five primary concentric Requenccs. For example, if tl1e key consists of 21 numbers, t-hon thero will be involved 21 XS, or 105 secondary alpha.bets. In a rnessoge of exactly 105 lcttcrs, then, each Jett.er \\ill be l'ncipharcd Ly o. different scccmd11ry nlphnb~t. 1£ tho n1essaltf' contains more than 105 lettera, them nil the lcttrrs after tho lOoth will ho t'ncipht,rct·I by the some socondnry alpbabet.s u at tlae begiuning or t-hc mes11ngo and in tho snmo SC\C[Uenco. In tlw l\xplnnntion of the motbod of onr.i1>bem1en1. it was made clear that the substitution pror0Cdltl in o. l'C.'-1t11ln.r mo.nncr, taking successive groups of 5 letters; tho cipher equivalents a.re f.akcn from tl1e sucoossive 8'\gJllf'.nts, protm<'ding in o. clockwise direction from any given initial BC'.gmont. It follows, thorofore, tlaat in a single long message whoroin the complete encipherment requires tJ1e pnssing fluougb of this soquence of segJIICDt.s JDOre tho.n .one tllno, there exist periodic or cyciic phenomr.na of a type found in various cipbers, due to the presence of a definite or regular CJJele. Jn this case, tl1e longtli of tliis cyclo in terms of groups of 5 fottera corresponds exactly \\ith thr length of tl10 numerical koy; it.a lcJ1gtb in terms of in<lividua.1 lottera is five times the lcngtl. of the koy. For tbo snke of clarity, wo sball refer to this cycle whon stated in terms of lot.tors as the pariod. Thus, wit.la a koy of 21 numbers, the length of the cycle is 21 groups, and the length of the period is 105 letters. If IL message conaist.s of 316 letters, for example, tl1e letters would pass through three complete cycles; tho lat, 106th, and 211Lh lotter& would be encipl1ered in exactly similar positions, and therefore by exactly the same seccmdary alphabet.. The 2d, lO';'t.b, and 212th Iett-en would likewise be onciphered by the same secondary alphabet, but of cou1"Sc not the same a.a the preceding secondo.ry alphabet. With a key of 23 nomben, the length of tl1e cycle is 23 groups, t.he length of the period, 115 letters; the lat; 116th, and 231at Iott.era would be enciphered by the same secondary alpha.bet; the .2d, 117th and 232d lot.tors by a dift'erent secondary, and so on. If we roprosen.t. the length of the period by "• then the lat, (n+l)th, (2n+l)th, (3n+l)th, ••• lotton fall in the same seoonda.ry alphabet.; tl1e 2d, (r.+2)th, (2n.+2)tJ1, (3n.+2)th, • • • lett:era fall in another secondary alphabet; and so on. If o. mossnge be longer than the period, therefore, it will follow that the lat, 2d, 3d1 • • • nth secondary alphabet.a must contain ropetitiona of cipher letter&, representing . ·- . r : ,. . i .• \ ..; I. I I :~ :£ Ii I :~ ..I i I. :· i.. I The t\\'Cllf.)'-eixth aeco.ndarJ" alpll&bfi aolocidOI with the normal alphabet, liDae euh plalD-teG letter would bo rcprcsentad by ltlolf In tba\ IGCODdarJ alphabet. CZ:" I.,••= ·---··- ......-.--·· -- ·- ...____ "" :· ~ REF ID:A64722 iO .· rc•1wtiii1mll 1 f pl.1in-lc•xt. lc~H1·r:~, rur lh1•:-:11 1&:·1·011cl:11·:.· ::lph:1hnt.K, 111,! 11rtc•r alJ nnly ~inglo mi:tr.cl 1•iplll'r :1 ipii:il ·•·11'1, n 1111 1111• l"l'j!l'ti Iion or hi;.:h-fr1•1p1r111•;-.• l•·I i ••I~ in ur1li11m")' p)nin. t.nxL is ll llt'C'.CH.C1A'1"3' 1•h:1i":h'i1•ri:-d1· 1\i" II n :1I1>lmh1•; i.·;1) l:!l•~!l:il~~··:-:. ~w·li fl'I'' . i '. fom1 ,, m he• l!\"irlunc•l'rl U)' I l\pctiticmA in :h1 1·ipli1•r :c•xl 111 11, :!11, ;~,, i11t1•r\·1!!s, 111:d iill',\" m:ry hC' mv!d In dc1f.c1rmiun tha 1<·nglh oC tho J>1 ri1\1f. l·:x;ll'l I,\· how Ihil4 il4 done• will pr1'!11•ut '.v hr 1lr1uonlif-rnt.c?d. But. I 1111 1lc•tc·rmin:11 inn nf I h1• l1•11;.tl h of I Ill' jn•riucl i!: only IL lili~ht itl.r.p forwn.rd in thC' 111111Jy11iM. Jt. i!'I I-rue• f.lmf it. will ;..-i,·c• 111'1I.Im11•11:,?lh or flu• urmu•rfoul kny, hut t.luLt iR 1111. Wlmt. wa must kranw IU'XL j14 I.In• "'''llll"llC'IC' n( 111llnbr~, 01" rn.thr.r, lh1• ITlllllVl' ]'O:-if.inllS of tJ10 rmrnhcl'll in thi!' kny. 1 1 1 W<• mu:r :1~r.c•rt11in thiA hy lurthl'r sr.rutin.r uf tlw tl1corctfoal o.ncl actun.l results of the mot.hod or 1'1\l'l)lhl'l'llll'Rt·. Jt is often thr. C1U1r. With VllriOUK r.iphl.'l'A thn.t tbe mr.t.}1od of enciplumncnt is t•xr1•llr:1t in pri:wi)llr, 11.nd will yiC'ld 1>rnrt.ir.clly i11clr.t~i11hrr:1hlo Jn"8R1~gca ,wbon tl1e ml'Ml1geu1rc ,-eor,,· il'w in numb1~r, but tho \Ven.kncasos in t.hr. mothod aro quickly disclOBOd whon it is used for rrgnlor trnlfic r.mch n.s that noc01Sa.ry in milital'J' cryptography, whero many messages nre to be Rrnt cocl1 dny in the an.me kq. In tho cipher under examination, the weakness ia introduced by thr fnrt thnt. t.bo initial segment for on.ch ml'SBIJgo ol tl1e day's activity is determined _by. t11e 11eria1.l nmnhl r of tho messoge. 1 Now t.l1oro nro as m.1ny iuitin.l sogm.ent.s for each numorieal key as .there nro numbc:& in that koy. Once the sturting i1oint is determined, u.11 the messages pass througl1 the s11me cycle; diliercnt n1eaaagcs inercly begin o.t dHrerent point.sin the cycle. Now, since the numbl'.rs applying to these sto.rting points constitute tho sequence of numbers in the koy, the succcs3ive initial segments cqnstitute a seril'S or BMJ.uonce which, when properly reoonstruc~, will give us the sequence of numbers in tlie koy • .Alter tho numerical key has been reconstrl!cted, wei are yet a long way from solution, for we are still confronted by the more complex probll'lU of .reconatructing1 or solving, tho cipher alphabets. · We have 10 far analyzed tho solution of t.he problem into t.he following three atepa or.~=· .. 1. The dotcrmino.tion of tho lcingth nf the period. ·· 2. Tho reconstruction of the nummirnl key. · · ·: · 3. Tho rooonstruction of the cipher n.lpl1abota. ..... ··"' .. Lot us 11rocced, tberofore, to perform oo.rJ1 stop. 1. The determination of ihe length of ihe perlod.-lt was explained abovo how the cipher syrStem \\"ill result in tho production of repetitions in the cipher text at definite intervals dependent upon thofongtl1 of the period. The fmt, (n.+ l)tb, (2n+l)th, ... letters fall in the same secondary ol1>l11J.beoi; the second, (n+2)th, (2n.+2)th, •.. hitters fall in another seconcla"ry alphabet; and iSO on. If tbere a.re repetitions in the plain text a.t n intervals o.pn.rt, there will be corresponding repetitions in tho cipher text. Thero would ho involved. hero onJ.y a slightly modified caso of the ordin:uy process of factoring the intervo.ls between repetitions in the cipher text, oa n.pplil'd in tl~e solution of typical periodic mult.iple-alpl1:i.bet ciphers. Thus, in .this case, if it happens that the first, second, and third letters of a message, and also the (n+l)th1 (n+2)th, and Cn+3)th1 are the l<'t.tl'rs THE, tlien there must be a repetition of the initial trigraph .of the cipher tmt, representing THE, at a distance of n letters. But in o. cipher involving so many alphabets as t.llls one, the repetition of trigraphs and polygro.phs would naturally. be rather infrequent, ucept in a very long message. IIo\\•over, the paucity of trigraphs and polygrn.pha, and even of digraphs, need not prove to be o. great obstnclo, for the repetitions of individuo.l lcttel"IJ may ho used with great 11cc1U"acy for the ~me purpose, viz, t110 determi110.tion or the length of the period. The method is bo.sed 1 1 Howu\·er, were the i11iti&l ilepncmt-li detemiined h~ aO.ule other manner, the fbial reauna would be the aame, a11d the cipher could be aoived by a alight.modiflcat1on of method. Evan JI the inltlal 18pleDt.ll ,.._._aubject to no law, Cho cipher could •till be eolved by the method hereinatter act forth, with llOID8 mocWlca.tloDll. . . ..........- - - - - - - -.....- - -.....-~---------------- ·-·-~·--·-······-·----=---..,...-- REF ID:A64722 .- 11 tJU' l'Oll!llflll'fion or whnt \\"O }uw·o l 1ll1(•1) 0. ••'f'nblo of coincidcmr.eu, whicb WUI mow US m:1th1•m:1til•:1ll~· the• llll'"t pruhnhll• ll'UJ?th ur lhl' ]Wrind. Wo inay 88 Wl'll UHO tl10 text of our i;,•r:l'" or m,·,.."1~:;:,·~ tn illui<lrn(O tJic ]JMl'l'I'~. Ii we• u":::mnl' thr 1111ml'1irnl key' to conl'iRt of 20 11umhcra, then tho le.ngtlt of the period would be: l 00 lc·I :c·r~. l..l't 11~ Wl"itl' t-hn lnnl?<!Rt JJll't\MRgf' or our sarir.a--vhc, mCllisogo 26-in l'.Xnctly 111pori111pnsc•d linc•11 l'.Olltnining 100 Jr.tll'rs CnrJ1, lllH] tlum rnn.ko R l'.OUnt of tho racurranC<'ll, or DlO?O 11rr111·1•tl'ly, t-ha r.oinr.icll'nr.c...,' of fof.toai. within 1.ho indhidunl columm tJ1us formed. :\ot1~ tho rapct.itfon of tho laU.t!r Fin tJH' Mceontl column. Tbis fact is indica.ted by placing 3 C')ICi'k Jll•l1'k in tho tabu)otion of coint"Jdt'D('Cll. Wl1ero A Jotter appears three times Within t)10 i-::tml\ rolumn (B, in column S), tbrca check mll.l'ks n.ro recorded, for thoro we have a coinaicll'nco bct\\"Cl'll tho first nnd sr.cond, scconcl o.nd tbird, and fll'Bt n.nd tl1ird oecurren.ces. Where a letter O.IJJ><'tLl"R four tiincs in tho Mme colun1n, six chock marks are recorded. Tho number of coincid<'Jl~l'" fur r.acb cnsn l".Orl"l'sponds to tbo munber o! combinn.tions of two things that can be made from a totnl of n. tbings.1 ·· · •• W"c note tho.ton tbe o.ssumption of a period of JOO letters there is a total of' 39 coincidences. :Now, if° tho period is really 100 Iott.era in length, then the repetitions of lettera Within columns a.re not mere coincidences brouglit about by chance superimposition of idcntico.l letters but are actual recurrences in the restricted sense of being tl10 resultant.a of the encipherment of similar plnin-tcxt letters by the same secondary nlpl111bet. But there is no way of determining from this single tabulation whether the assumption of a period of 100 letters is correct or not, and tl1crefore we do not know whether the repetitions in this cue are recurrences or coincidences. This we can determine, bo\\·ever, by a comparison of "ta.6ulation.s of coincidences made upon · \"orious assumptions of length of period. Theoreticnlly, tl1e correct uaumption should yield a lligbcr total or coincidences than the incorrect assumptions, beco.use the recumm.ce of highfrequency plnin-text letters (in English, E, T, 0, A, N, I, R, S, H, D) is to be expected, and the nmnbcr of such causally produced repotitions should certainly be greater than the number of ?CJ>Ctitions produced by mero cJ1n.nco in tl10 supcrimpoaition.1 Ll\t t!!'I proccoo, therefore, to make n. to.bfo of coincidonco for the "VO.rious probable lengths of pcrioci, fo~t trnnsmbing mmiso.go26 into lim•t1cor1'CSponding to hypotheses of 105, 110, 115, 120, and 1:.!5 l<•ttcrs.' Be!oro doing so, however, wo find it necessary to introduce a few remarks upo11 the dl'sii-nbility of using o. slight correction footor for this table. UJIOll .i- 1 .. . i i i r 1 We c!r:.w t.be dU.tim,t.iun bct.\t"COJl r«:urrmat111nJ coincidt!nt:U on tho grounda that tlae former term lhould, nncl wil1 l•cl\' be ui.,•cl to indicate rcpotit.io11• of lotton in t.lao ci1Jlaor toxt oaWlllll)' relatod to eaoll other by being eJ1l'ipl1.·r111c111a CJC idcnUcal 1>lain-toxt lottera by idcnticnl 1Ll11ba.bcta; whCl"OIUI t.ho latter term lndioatea :repetition• 1111t •':11:..;:11!,:.- n.•luk-d t.o c:icb o1hcr in tbia mmmOI' bnt simply t.ho runlt. or chance.· A repetition may tlaoreforo bo l•ii lll ~a "'"1:: l"l'llCC •Ir a N>inciclM1eo. Tho 1>roCCM11 ur factoring in ordinary multl11le-a.lpha.boti oJpben of tho poriodio t.n.c 1:1 ... i·•r i1111mr1>osc thc 1m1uar11tlc111 nnd olaMUicn.ticm of rc1>0t.itlo1111 int.otliatwokinde. UntilprovedothendA, 1111 r''i" :iti1111M 111u11L be ccmMidr.l'\!ll coinuick111c1111. : •rtU" (urnmlu. ill: c ..... ,1(n-l)/2. 1'lntH, \\"hr.11 ""'"5, tho 11umber of ooinald,onoaa 11110; whon a-8, the .. 0 ! ~- 1.iUluh1••• izic IJi. ; ;-lim:1• t.hi.11 lllL)lllr \\"RS writt.r.n, II. further af.ndy of tho COllCCpt. ot OOlnoidcD0811 hall mado it. pouiblu f.o predict, witia :. fair .!1•grcc of accnnLcy, j1111t. 11ow many coim1ldcnoua Hhould bo oxpcotoi:l for eorraat and lnaorroot . .umpt io:i,,;, 'l"h• · au:d hc1111Lt.ica.l and atu.tiatiaa.1 analylll?ll arc given in dot.ail lo W. F. Friedman, Analuna of a Mcrcla1.111ico- l;:lrr.ir,.rtll C·ru11tugruph, Section VI; 8. I<:nllb&ck, Btat.iaiit:al Mnhod1 in G'rn*&nalraia, Sectio.a. VII (Technical l>ubl!.·.:li•·.:.:, S. l. S., 1034). It reBulta from theao matbcrua.t.iaaJ. atudleathat the ratloofthenumberofactual coi.nc:i.:,,nc.'.; to tl1e total nnmbor of pouiblo colnaidencea ill .038 for an lnoorreet. aua and .088 for a correct one. Tilis ks. 1wicrl~ cJiminatea tho nec81Bl.t.y for tabulat.io1111 oorreaponding to eYff17 polllble cue and givea a reliable 1nuan11 ui c!cter111i11ing the aorrect auumpt.ion aa BOOn aa It Ill made. (See Not.ea 1 and 8 on paga 18 and 14 rcspcctivdy.) .. I • " " - ' " " " " " • ~ " ' " " ,,.. _ _ _ ,_ _ ' - • Tll.- :1.cssago need not be writ.tan out mon tll&D once if long atrlpa of ezau.aeotlaD paper are uaed, writing a llnu r,n c:.cl1 strip. Ea.ah line ahould contain 121 letten, and the Y&rlou atdpa aan 'then be anaupd to bring the prope•· lcttora into nperlmpallitlcm aooordiDg to each h.Jpot.bellll ID tum. REF ID:A64122 12 (. Jtnr n•111Jy nr1~umto coinpnriROD, t.ho tnLnl11 or coincidcnco obtained for tho various l1ypothesns · aahoulcl he 1•,1rrl'r-h•d iri ordl'r to muke propl'r nllm\·nnco tor tho diframnCC'a in totals clno B1Jlely to th." ,·:r::i:1t1,111 in lbo nun1bl'!' ol lot~ra in ~at~h column \\'lion trai1scribad aceonling to cucl1 · hyp~·d1r.z~ ~- 1 l!~rom n. c1·yptogrnphic point .of \"il'\\", a. totnl ol 100 coinoidCDces ia an o.rra11gcmnnt who..."t! t!it"::o a."'O 6 lette.."I in each colun1n repr~ent:s c slightly greater degree of ooincidence tl~an in ll.n Mre.ngcmont of the aame massage also yielding 100 coin.cidencea, where there aro 7 lettera in most or fr..e columlis; there ia 1688 opporturjt.y for comcidencea to be produced in the former oase. We should, therefore, reduce all the tobls of coinciclence to some common ba&ia. The recso:ci::.g \"'.-e !~vo followed in the estabHEbmcnt of n correction lactor to be applied is as !ollon: '..\::.:-3311~1:. Z6 cont.nins e:mct.ly 639 lett~""S. 'lb~.il tra.nscribed into lines of 100, 105, ••• , 126 lettfll.·s, the rolumns in e&eh of theae :live ret-u~s have the following number of letter&: I T .A.BU I Pcrlllll det&el'I) SO a<.•lumna ui G lctten and 61" aolnm11a ol I lottora •• -· ------------------· -·-----···-------------14 oolt"JLtUI '>f-:! ~:l~f.ors :md 91 '1olumna of. I Jot.tcrs •••••••••••••••••••••••••••••••••••••• 99 cah:;.!~n&->! ;; ldt.1..1..-. and 11 cnlumn1 of' lot.ter1. _____________ ·---------·---------------------. 'ill ec•lur.mr. ·-·~ .) !cth.'TI and 36 colUlllllll or 4 lotters •• ··------------------------------------------ -Ii} c~·lu:~m;;1 ,,i:; lctt-cra a11d 61oolumnaol4 letit:n ••• --------------------------------····-------8\l cc:;i.1•·.ua o~.; !ctt:lrs nnd M aolumn1 of '.l lo~ten............................................... 100 lCIS 110 115 120 rr···---- l2i i..::s:i;:·'.:;~; °ii.At periect coincidence ca.11 occur in each column (all letteTB identical), then in a col·: ..:.i.n u! 5 lott.ers we can have 6X5/2-15 coincideDC88; in a. column of & letten, 5X4/2-10 eoinci·:i.-3m·,~s; ttnd in a column of 4 lettera, 4X3/2==6 coincidencea. Ii now ".-~ fmd the tot.al number of chances for coincidences for Nl.'h ul &lie arranpmeDCI given in t.i hle I, we ha.ve the following: TABia II t. l'triad 100 ms I I ! 110 111 I. 120 125 I - Totlll Comll&lllll• (lltlel'I) -- I I 196 I each---------···-- l,1, 120 14 colurm111 of 1& ohauee11 wa.eb, 91 uulumna uf 10 chances euh••·-----------1,018 99 culumm 10 elw.n.cea ea.oh, 11eolumnaor6 obanoee euh--------------79 oolu1DD1 of 10 uhauuea eaoh, 86 culumiia of 6 a'bamoem ea.oh•• .: •••••••••••• 1,008 9116 H colum11a of 10 cha.ncm each, 61 oolumna uC 6 ch&DCIS euh------·-------· 39 oolum111 10 olumcem each, 86 oolunm• uf 0 ohmacea eaeh. ______________ . 89 eol\unn1or115 chances each, 81aolul&lnaof10 ohanoea I uf of fl08 Clioosing for our buia of comparison tbe hypothe&ia of a period of 100 lotter&, the various proportions or clumcea for coiucideD.cell for each of the remaining· hypot.b.elm will conatitute comtetion facton to be applied in each case. Thay are u follow: · · · Tnr.s III ......... ... Ptrlud 01-b o.n.m 100 106 110. 116 1, 191 1, 120 1,0l6 1.00 L07 a.u-> 120 1,000 916 121 908 . -=. L 13. L 19 1. 21 LD --.1a • See footnote a on the preaedlq pap. Thia oomeUon futor la mm__,- If the number of aeWAI aoin· tenu of 'be total polllble numbs of oalvaldm- olde.aaea la red.aced to a pneatap •=- P ....... ~-~··--......-·--~:~;e~..~.~p~4~.4~.~~--~s?~'-.. '·••~.~-•~+•a~!~••WF A~*~»~.;qllll""'lo-z4~9"""'L•.•?~.A~o~.~>f•Aa.o~wlll!'llll"""'t4..,,•,'"'"""°~..........-.-.---.....-.--..........-:• REF ID:A64722 - 13 WC!' nro no\\· rcn&i:'\· to l'11tnhli!:ih tbo t.o.bles or coincidence for the various eypotheses. Space fllrbids till' nrtm1l ,1'.•monistrution of tbe Sl'\'t'rnl a.rrangomonta of messago 26 to correspond to thr ,-,ufou& h,rpothctirnl key ltn:t;b~-tl1ut shown in figure 3 is typical of them all. We slu1ll gh-e only tho final result in tabla IV. TABLK IV 1 i Pft'l••i I llellt'l$) 100 JO.ii 110 115 120 l:?S -.· - I I Col&u!ldalaaN DD tl&"h llJ110Lhllla 'l'alal ! , ...... CcmetlaD CGmlll8d IOtal I ! /HJ IHI IHI 1HI /HJ !HJ 1HI /Ill I IHI IHI IN! IHI IHI /HJ IHI !HJ !HJ 1HJ IHJ IHJ IHI IHI /HJ IHI IHI !HJ II IHI IHI IHI IHI IHI IHJ IHI IHI IHI IHI IHI IHI IHI IHJ IHJ !HJ IHI lH./ IHJ I I IHJ IHI IHJ IHI IHI IHI Ill ! ·- 39 . 4647 ' 60 36 33 LOO 1. 07 L 18 1. 19 1. 2li 1. 32 ···-...···-- I I 89. 0 48. 2 18.1 71.' 45..0 .41. 6 .. Th1•rr. j;:('l'ms to he no doubt but t1u1t thl°' pr.ritHi or 115 let.tors is cormct. Tho cyclo, thoroforl', ron"i:sb1 oC 115-:-u==23 groups, anti tlio nnml'1'ic~11.1 kl'y r.ontnins 23 11un1hon. Thi» means thut the two final el'gmcnta bcllr no numben.c, 1111d aro therefore blank segments. 2. The reconstruction of the numerical key.-i:foving ascertained. the length ol t.Jie period, u.nJ thus the langth of the numerioul kc~y, tho next ~tl"p is to recon='t.ruct. tho sequence of num!.Jcr& ron~ti1.i.1ting the key. As stated before, this procOSB is mado possible in this case hv thl' llll'thLHl of cucipherml'nt which iA such that all tho me11S&l(e& of the day's activity go tf11iu1gh l'xnctly the dlllllO cyolo, but tho succ1~vo nlCQlll~ begin at dii£e1'0nt initial pointa in thil'I \~yclo, 1111d thl'SC poin~ cflincido with foe rclativo i•ositions ol tho nmnbor& making up the Hl'C[UCnl!\1 af numhCl'R in tbe key. Wu do not know tho absoluto position of any numbers in the numerical key, but we may procerd fil'ld. to find tl1eir rol:l.tive positions, l"<\ganling tlie key in the nature of a continuous cycle or ch1dn. Later wo DlAY find the a.bsolutA positious of the numbers in tliil cycle, i.e., we sl1nll lu1.vo 1·cconsb·ucted tbe numerical key itself. Xow, us st.at.eel beCore, 11.ll mMRagea proceed througli the same cycle; it is only tlie initial puims tor the 11uit1:uigcs which uro different. Hence, if wo can deten11ino tho relative positions iu wliid1 n1c::;snf.."CS 1, 2, o.nd 3 sl10uld bo superimpoaocl in order to mako all throe messages coin<':••t' 111J l'\•gurds the port.ion of tho cycle through which they pass simultaneously, we shall t1m11 have dctcimincd tile rola.tive positions of tl10 numb8l'B