Mathematical Foundations of Public Key Crytogtaphy.In Mathcmatical Foundutions of Public Kcy Cryptography. the authors prcscnt the rcsults of more t11an 20 ycars of rcscarch and Lcaching cxpcricnce to hcip studcnts bridge the gap benveen math theory and crypto practice The book provides a theoretical structure of fundamcntal numbcr thcory und algcbra knowicdge supporting puhlic-kcy cryptography.Rather than simply combining number theory and modern algebra: this textbook ftaturcs the intcrclisciplinary charactcristics of cryptography-rcvcaling the intcgrations of mathcmaLical thcorics and puhlic-kcy cryptographic applications incorporating the complexity theory of algoritluns throughoiit, it introduces the basic number theoretic ancl algcbraic algorithms and thor compicxitics to prcwicic a prcliminary uncicrstanding of the applications of mathcmatical thcorics in cryptographic algorithms.SuppLying a seamless assimiLation of cryptography and mathematics. the book incluclcs covcrage of' cicmcntary numbcr thcory; algcbraic structure and attributcs of' group, ring, and fiLid; cryptography-rclatcd computing compicxity and basic algo- rithms, as weLl as lattice and fundamental methods of Lattice cryptanalysis.The tcxt consists of 11 chaptcrs. Basic thcory ancl tools of' cicmcntary numbcr thcory, such as congrucnccs, primitwe roots, rcsiduc classcs. and continucd fractions, are covered in Chapters l-6 The basic concepts of abstract aLgebra are introduced in Chaptcrs 7-9, whcre thrce basic algcbraic structurcs of'groups, rings. and ficids und thcir propcrtics arc cxplaincd.Chapter 10 is about computational complexities of several related mathematical algorithms, mcl harcl probicms such as intcgcr fzictorization md discrcte logurithm Chaptcr il prcscnts the hasics of lattice thcory and the lattice hasis rcduction algorithm-the LLL algorithm and its application in the cryptanalysis ofthe RSA algo-rithm.