Preface
A ciphe r i s a schem e fo r creatin g code d messages . Th e purpos e o f usin g
a ciphe r i s to exchang e informatio n securely . Throughou t history , man y dif -
ferent codin g scheme s have been devised . Thos e discusse d i n this boo k hav e
a mathematica l basis . On e of the oldes t an d simples t mathematica l system s
was use d b y Juliu s Caesar . Thi s i s where w e will begi n ou r study . Buildin g
on that simpl e system , w e will then conside r mor e complicate d schemes , ul -
timately endin g wit h th e RS A cipher , whic h i s use d t o provid e securit y fo r
the Internet . I n additio n t o developin g variou s encryptio n scheme s an d th e
underlying mathematics , thi s boo k ha s severa l othe r goals . On e i s to intro -
duce th e reade r t o numbe r theory , th e are a o f mathematic s tha t concern s
integers an d thei r properties . Consequently , som e mathematica l concept s
are presente d i n greate r detai l tha n i s needed t o understan d an d implemen t
a cipher. I n addition, proof s o f some theorems ar e included fo r thos e reader s
who ar e intereste d i n learnin g mor e abou t thi s aspec t o f mathematics .
The boo k i s structure d differentl y fro m mos t mathematic s texts . I t
does no t begi n wit h a mathematica l topic , bu t rathe r wit h a cipher . Th e
mathematics i s develope d onl y a s i t i s needed ; th e application s motivat e
the mathematics . Th e followin g convention s ar e use d throughout . Th e firs t
time tha t a term i s introduced, i t appear s i n italic typ e an d th e sentenc e o r
paragraph i n which it appear s contain s the definitio n o f the term. Messages ,
plain o r coded , ar e writte n i n capita l letter s i n a specia l font . Fo r example ,
EXAMPLE and TBQDHST . Occasionally, equation s o r phrase s containin g sym -
bols ar e se t of f fro m th e bod y o f th e text . Whe n i t i s necessar y t o refe r
to thes e statement s late r i n th e chapter , the y ar e numbere d consecutivel y
within th e chapter . Fo r example , i f w e were considerin g th e equatio n
(1) x + 5 = l l ,
we migh t sa y "solvin g fo r x i n (1) give s x = 6. " I t i s traditiona l i n math -
ematics t o mar k th e en d o f example s an d proof s wit h a specia l symbol . I n
this text , thes e ar e indicate d b y .
As i s typica l i n mathematic s textbooks , mos t chapter s en d wit h ex -
ercises. Man y o f thes e problem s ar e simila r t o solve d example s an d ar e
designed t o assis t th e reade r i n masterin g th e basi c material . Answer s t o
these routin e problem s ar e containe d i n Chapte r 26 . A fe w o f the exercise s
are one-of-a-kind , intende d t o challeng e th e intereste d reader .
vii
Previous Page Next Page