Preface A cipher i s a scheme for creatin g code d messages . Th e purpos e o f usin g a ciphe r i s to exchange information securely . Throughou t history , man y dif- ferent codin g schemes have been devised. Thos e discussed i n this book hav e a mathematical basis . On e of the oldest an d simplest mathematica l system s was used b y Juliu s Caesar . Thi s i s where w e will begin 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 used 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 presented i n greater detai l than i s needed t o understan d an d implemen t a cipher. I n addition, proof s of some theorems are included for those 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 that 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 contains the definition o f the term. Messages , plain o r coded , ar e written 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 we were considerin g th e equatio n (1) x + 5 = ll , 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 few o f the exercise s are one-of-a-kind , intende d t o challeng e th e intereste d reader . vii
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