Cryptography in an Algebraic Alphabet

8/2/2019 Cryptography in an Algebraic Alphabet

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Cryptography in an Algebraic Alphabet

Lester S. Hill

The American Mathematical Monthly, Vol. 36, No. 6. (Jun. - Jul., 1929), pp. 306-312.

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306 CRYPTOGRAPHY IK A N ALGEBRAIC ALPHABET [ J u n e , J u l y ,T h e following officers were e lec ted: Ch ai rma n, Professor C. H . Ashton ,Vice -Cha irm an , P rofe ssor Em m a H yd e , Sec r e t a ry-Treasure r , L . T. D o u g h e r t y .T h e morning session was a jo int meet ing wi th th e K ansas Assoc ia tion ofMathem a t i c s Teache r s , Miss M . Bird UTeimar ,of W ich ita , presiding. A t thissession, Professor J . 0. Hassler , of O klaho ma U nive rs i ty , spoke on t h e va lue ofm a them a t i ca l h i s to ry to the t eache r and to the pup i l , and P rofe ssor E . B .Stouffer , of th e Un iversi ty of K an sas , gave a repo rt of th e Math em atica l Con -gress a t Bologna in th e Sum me r of 1928. A t the jo int luncheon of t he two As-sociat ions, which fol lowed the morning session, Professor U . G. Mitchell , ofthe Univers i ty of K ansa s , gav e a very inte res t ing ta lk an d dem ons t ra t ion of' 'Ma them a t i c s and Poe t ry ." I n the a f t e rnoon , t he Kansas Sec t ion m e t insepa ra te sess ion, th e program consist ing of two p ap ers :1 . "Th e G am m a- func t ion , " by P rofe ssor A sh ton .2. ' (Some prop ert ies of E ule r 's phi-fu nction ," bj- Professor Riche r t . Ab stra cts of thes e papers fol low: 1 . J u s t two hun dred years ago, Euler in t roduced a new fun ct ion, which hasbeen th e sub jec t of m an y papers an d a few ent i re volumes. Near ly a hun dredyea r s af t er i t s in t roduc tion by Eule r , Legendre nam ed i t t h e Gam m a- func t ion .Com para t ive ly l i t t le has been w r i t ten ab ou t th is func t ion in th is cou nt ry , e i therin our books or in ou r journals. In th is expository pap er , i t is def ined by anintegra l , by inf ini te prod ucts , and by i t s d if fe rence equat ion , and some of i t spropert ies are discussed.2. I f m is any given posi t ive integer , the number of integers not greaterth an m an d pr im e to i t , is cal led E uler 's ph i-function of m , (or indicato r of m ),an d is deno ted b y q5(m). I t is well known th a t this function is of f re qu en t oc-curence in the theory of num bers . Th is paper dea ls wi th th e fund ame nta lpropert ies of t he phi-fun ction. LCCS T . DOUG HERTI ' , ecre ta ry

C R Y P T O G R A P H Y I N A N A L G E BR A IC A L P H A B E TB y LESTER S. HILL, Hunter College1. The Bi-Operational AlphabetL e t a o , a l , . . . , a z6deno te an y permu ta t ion of th e le t te r s of t he E ngl isha lph abe t ; an d l e t us a s soc ia t e th e l e t t e r a ( w i th the in t ege r i. We def ineopera t ions of mo dular addi t ion an d mul t ip l ica t ion (modulo 26) over the a lph a-b e t a s f ol lo w s: a i + a j = a , , a i a i = a t , w h e r e r is t h e r e m ai n d er o b t a in e d u p ondividing the in teger i + j by the in teger 26 and t i s the remainder obta ined ondividing ij by 26. T h e in teger s i a n d j m ay be th e sam e or d if fe ren t.I t i s easy t o ver i fy th e following sa l ient proposi tions concerning th e bi -ope ra tiona l a lphab e t t hus se t up :(1) If a ,p, 7 a r e any l e t t e r s of t he a lp hab e t , cr+P =P+cu, cup = p a , cu+ ( P + r )

= ( a + P )+ r , 4 P r ) = ( 4 r , u(P+r) = 4 + ~ .


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3079291 CRTPTOGR.LPHI- I N A N ALGEBRAIC ALPH-LBET(2) The re is exactly one "zero" l ett er, namely ao ,characterized by the fact

th at the equation a + a o = a is satisfied whatever be the letter denoted by cr.I t should be observed that , by our definition of multiplication, if a denotes anyletter of t he alphabet, we have: a a o= a o a= ao.(3) Given an y letter cr , we can find exactly one letter 6, dependent upon a ,such that a + p =ao. We call /3 the "negative" of cr , and write: 6 = -a. Evi-dent ly, if 6 = -a , then also a = -6.

(4) Given any letters a ,6 we can find exactly one letter y such tha t a + y =P .We wri te : y = p - a . I t is obvious that 6 - a= P+ ( - a ) ; and also that if/?-a =ao , hen /3 = a .

(5) Distinguishing the twelve let ter s, a l l a3, a&, i l as, al l, al5, a17, a19, a21,a23,az5,with subscripts prime to 26, as "pr imary" letters , we make this assertion,easily proved: If a is any primary letter a nd 6 is any lett er, there is exactly oneletter y for which cry = P . We write: y =P/a . Each primary letter a has theLLrecipr~cal"l/a, where al is the "unit" letter; and the reciprocal is likewiseprimary. If a is primary, we shall call the "fraction" p/a ((admissible." Atable of the letters represented by t he twelve particular admissible fractionsal /a enables us, when used with th e formula P/cr =/3(al/a), to find immediatelythe letter represented by a ny admissible fraction.

(6) In an y algebraic sum of terms, we may clearly omit terms of which t heletter no is a facto r; and we need not wr ite th e letter al explicitly as a factor inany product.

For the limited purposes of th e present paper it will not be necessary todefine exponential notations, etc.

Let th e letters of th e alphabe t be associated with integers as follows:a b c d e f g h i j k l m5 23 2 20 10 15 8 4 18 25 0 16 13

or, in another convenient formulat ion:

13 14 15 16 17 18 19 20 21 22 23 24 25m m f l v i q d u x b ' t j

I t will be seen that

a n = z , hm = k , cr = S , etc.


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308 CRYPTOGRAPHY IN A N ALGEBRAIC ALPHABET [ J u n e , J u l y ,T h e ze ro l e t t e r is k , a nd the un i t l e t te r is p. T h e p r im a r y l e t te r s a r e : a b f jn o p g u v y z .

Since this part icular a lphabet wil l be used several t imes, in the i l lustra t ionof fur th er deve lopm ents , we appen d t he fo llowing table of nega t ives an d re -ciprocals :Let te r : a b c d e f g h i j K L m n o p q r s t u v w x y zNegative : u o t r l y i x g p k e ? ~ z q b j n d w c a z s f vReciprocal : u v n j f z p Y a b q o

T h e s o l u t i o n of t h e e q u at io n z + c r = t i s c r = t - z , o r c r = t + ( - z ) = t + v = f .T h e sys t e m of tw o l i ne a r e qu a t ion s : ocr+up =x, ncr+iP = q ha s t h e so lu t ioncr = u , /3 = o, which m ay be obta ine d by th e fami l ia r meth od of e l imina t ion orby fo rmu la (see Sect ion 4 ) .

3. Concerning D eterminants i n the Bi-Operat ional AlphabetT h e d e te r m i n an t 1 1i l a12 . . . Q i n

where the a i j den ote le t te rs of th e b i -opera t iona l a lphab e t de f ined in Sec t ion 1 ,ha s t he sa me de f in i t i on , a nd the sa me p rope r t i e s , a s t he c o r re spond ing e x -press ion in ordin ary a lgebra -except th a t addi t ions an d mul t ip l ica t ions a re ,of cou rse, effected in th e mod ula r sense.We note expl ic i t ly these proper t ies :I. L e t 6 de no te t he va lue of t h e n - th o rde r de t e rm ina n t D ; le t l l l i i de no teth e va lue of t h e de te rm inant of order n - 1 ob ta ine d f rom D by s t r i k ing ou t t herow an d column in which t he e lement a i j l i e s ; an d le t Ai j = & Mii , t he pos i ti veor neg a t ive sign be ing used according a s the in teger i + j s e ve n o r odd . Th e n

each of t he sum s

h a s t h e v a l u e 6 if i = j ,a n d th e va lue a . if i# j.11. T h e va lue of D is not chang ed: (1) if rows and colum ns a re in te rch ang ed;or (2 ) if t o each e lemen t of a n y row (column ) is adde d cr t ime s t he c o r re spond inge lement of an othe r row (colum n) , where cr de no te s a ny l e t t e r of t h e a lpha be t .111. T h e val ue of D is changed o nly in sign (1) if two rows (colu mn s) ar ein t e rc ha nge d ; o r ( 2 ) if t he s igns of a l l e lements in an y row (colum n) are cha nge d.IV . T h e value of D is no t changed if t he e lem ents of a n y row (colu mn ) ar e


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3099291 C R Y P T O G R ~ ~ P H T A L P H ~ ~ B I S TN A N A L G E B R A I Cmultiplied by any primary letter /3 and the elements of ano the r row (column)by the reciprocal, al / /3 ,of p .

\Ve shall call D a "primary determinant" if it s value is a primary lette r.\Ye shall not have to deal, in this paper, with determinants t ha t ar e not p rimary.

L E R ~ A :y m ean s of properties 11and 111,we may obviously convert thedeterminant of n-th order ;

in to a variety of n- th order de term ina nts , all of w hic h have the va lz~ e , where cr isa n y assigned letter of the alphab et.1 In .I,, all elements, except those of the prin-cipal diagonal, are equal t o t he zero letter a o ;and all elements of t ha t diagonal,except the last, are equal to the unit letter a l .

We have only to make cr a primary le tter if we wish to set up with gr ea tease a wide variety of n-th order primary determinants.

4. Nornzal T ransfo rma tions and Polygraphic Cipher Syste msTh e determinant D ,of n -t h order , which was writ ten out in Section 3, fixes

the linear transformation with coefficients a i , :

\T7e call D the det ermi nan t of t he transformation T ; and we say that T is a"normal" transformatioil if its determinan t is primary.TIIEOKEXI:A nornzal transfornzation T has a n unique inverse T-l, of whichthe equations are :

where D denotes the valz ~e f the det erm ina nt of T, and D,denotes the valz~eof thedeternzinant obtained therefrom by replacing a,;by y , ( j= 1 , 2, . . . , n ) . Moreover,T is th e inverse of T-1. The values of th e determinant s of T and T-I arereciprocals (see Section I ) , and therefore T-l is a normal transformation.

B y m e a n s o f 11 ,111 , IV , an y assigned de termina nt which is o f th e n t h order and whose va lueis an y pr imary le t te r oi can be ob ta ined f r om ,,I,.


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310 CRYPTOGRAPHY IN AN ALGEBRAIC ALPHABET [June, July,Given any pair of inverse normal transformations T and T-1 in our bi-

operational alphabet, we have a device which may be applied (1) to convertan y message sequence of n letters into a corresponding cipher sequence of nletters, and (2) t o convert t he cipher sequence back into t he message sequencefrom which it came. In othe r words, we have all the appar atus of an extra-ordinari ly effective polygrafihic (n-graphic) cipher systenz. We ma y regardxlxz . . . x, as the message sequence, and dete rmine th e cipher sequence ylyz. . y, by means of T , using T-1 for dec iphe rment; or we ma y encipher withT-1 and decipher wi th T, trea ting ylyz . . . y,, as the message sequence andxlxz . . . x, as the cipher sequence. In either case, we begin by writing themessage in sequences of n lette rs, as will be illust rated in Section 5.

A polygraphic cipher consisting of th e inverse normal transformat ion ofthe literal sequences xi, y i( i= 1, 2, . . . , n) may suitably be called a linearcipher of order n, an d designated as a C,.

5. Illustration of Linear CifilzersLet us employ the particular bi-operational alphabet considered in Section 2.EXAMPLE: TOconstruct and apply a cipher of t yp e C3.Selecting any primary letter, say y,we can immediately obtain from

a host of different primary dete rminants all of which have the value y, as pointedout in the LEMMAn Section 3. One of these is th e determinantYz

kx

Ok

of the normal transformation,r n y

of which the inverse,(Ti') = ~ Y Izyz f dy3,

xz = vyl f fzyz4- qya,X 3 = fy1 + qyz + xy3,

is easily found . I t will be observed t hat the values of the determinants of T Iand TI-1 are y and q respectively, and th at these letter s are reciprocals.

Let the message to be enciphered consist of t he word Mississippi. Writ ing


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19291 CRYPTOGRAPHY IN A N ALGEBRAIC ALPHABET 311this message in 3-letter sequences, and filling the last sequence with any pre-arranged letter, say k, we have:

v z i s s i s s i p p i k .Substituting m i s for xlxax3 in TI, we find b q t for yly2y3, thus converting themessage sequence m i s into the cipher sequence b q t . Proceeding in like man-ner with t he other message sequences, we obtain as th e enciphered form of ourmessage: b q t s e i a e p y f c. We should probably send it in the customaryfive-letter grouping : bqtse ia ep y fc.

T o decipher, we substitu te b q t for ylyzy3 in T I - I obtaining m i s for ~ 1 x 2 ~ 3 .Proceeding in the same way with the other cipher sequences, we regain t heentire original message.

EXAMPLE T Ocons truc t and apply a cipher of typ e C4.: Choosing any primary let ter, say j, we may construct from

I K k K j Ian enormous number of different primary determinan ts all of which hav e th evalue j . One of these is th e de ter minan t of the normal transformation :

yl = g2t.l + rxz + 2x3 + a x 4 ,y2 = r x l + z x z + a23 + e24,( T z )y3 = ax1 + gx2 + hx3 + 2x4,y4 = ex1 + r x z + yx 3 + h x 4 ,

th e inverse of which is easily found to b e:

~4 = j y l + C Y Z + 2y3 + j y 4 .We note tha t t he determinants of T Zand T z - I have the reciprocal values j andj, the letter j being its own reciprocal.Let the message t o be enciphered be Delay opevat ions. Write it in the form:

d e l a y o p e r a t i o n s u ,filling the last sequence with any prearranged letter, say u . Substitutingd e 1 a for x1x2x3x4in T z , we find j c o w as the corresponding cipher sequencey l y 2y 3y 4 .Proceeding in this manner , we find th e enciphered form of our messageto be: j c o w z l v b d v l e q m x c .T o decipher, we substit ute j c o w for y1y2y3y4in T z - l ,etc.


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312 LIFE INSURANCE A C T U A R Y A N D HIS M A T H E M A T I C S [ J u n e , J u l y ,6. Concluding R e m a r ks

A gr ea t ma ny o ther c ryp tographic co ns t ruc t ions can , of course , be der ivedfrom th e a lgebr a , by no mean s fu lly deve loped in th i s p aper , of th e b i -opera -t iona l a lpha be t . T h e purpose of th e pape r , how ever , will have been accom-plished if th e s ingle constru ct ion described se rves to em pha size suffic iently th ecircum stance th a t se ts which fa i l to possess in ful l th e char acte r of a lgeb raicfie lds m ay st i l l adm it a large measur e of a m usi ng , an d possibly useful , a lge-bra ic man ipula t ion . I t need hard ly be sa id th a t if fu ll -f ledged f in i te a lgebra icf ie lds a re emp loyed, th e oppor tuni t ie s of th e c ryp togra phe r a re grea t ly extended ;he then has a t h is d isposa l a pe rfec t ly sm ooth a lgeb ra and i t s a ssocia ted geome-t ri es . T he wr i te r hope s t o subm i t a fu r the r c ommunic a t ion on t h i s sub j e c t .B ut th e num ber of m ark s in a f ini te f ie ld is necessari ly e i ther a pr ime or a powerof a pr ime . If our a lp hab e t is to be conver ted in to a f in i te f ie ld , th e bes t th a tc a n be done i s t o om i t one l e t t e r , s a y j , to o bta in a f ie ld of twenty-f ive ma rk s ;or to ad jo in an addi t iona l sym bol so th a t a f ie ld of twenty-seven m ark s is ava i l -ab le . T h e b i -opera t iona l a lph abe t1 of tw enty-s ix le t te rs , and th e fur the r de -ve lopm ent of i t s a lgeb ra , should the re fore be of some imp ortan ce in c ryp to-g ra phy .If polygraph ic c iphers based upo n n ormal t ransform at ions ( l inear ciphers)prove t o be of real interes t , we shal l indica te a surpris in g w ay in which th esec iphers m ay be m anipu la ted eas i ly and quickly , even for fa i r ly la rge va lues ofn ( sa y n = 8 , 9 , or l o ) , a nd t hu s ma de e ffec ti ve in a d i s t i nc tl y p ra c t i c al s e nse .I t should be remarked th a t a c ipher of t yp e C in wh ic h n > 4 , a lt h o u gh e a syto use , is extrao rdina ri ly difficult to "b rea k," offering very high resis tance toth e me thod s of cryptan alysis .

T H E L I F E I N S U R A N C E A C TU A RY A N D H I S M A T H E M A T I C S 2B y R A Y M O N D 1'. C A R P E N T E R , M e t r op o l it a n L if e In s u r a nc e C o.

I t is e s timated t h a t the am ou nt of l i fe insurance in force in Uni ted S ta te scompanies a t the end of 1928 is abo ut $95,000,000,000. T h e asse t s a re abo ut$16,000,000,000an d th e premiu m col lections in 1928 were over $3,000,000,000.T h e em ploy ed personn el of a l ife insu ran ce co m pa ny consist of t h e fieldor agency force an d th e home office force . An im po rtan t pa r t of t he homeoffice force is th e ac tuar ia l d ep ar tm en t .T h e a c tua ry ha s a wide ra nge of du t i es . He m us t be r e a sona b ly fa mi li a rwi th th e work of a ll depa r tm en ts of th e com pan y, and is somet imes ca lled i t s" technica l" ma n. His two main du t ies a re , f i rs t , th e ca lcula t ion of th e premium s1 Th e b i -operat ional a lphabet employed in th is paper i s an exam ple of a " r ing ." See th eBullet in of th e Nation al Resea rch Council , Report on Algebraic Numbers, p. 59.Thi s paper was read by invi ta t ion before the Mathe mat ical Associa t ion of America at NewYork C ity on Dec. 29, 1928.

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Cryptography in an Algebraic Alphabet

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