By Song Y. Yan

This e-book presents a superb creation to the classical uncomplicated quantity idea and the fashionable algorithmic quantity idea, and their purposes in computing and knowledge know-how, together with desktops layout, cryptography and community protection. during this moment version proofs of many theorems were supplied, additional additions and corrections have been made.

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**Extra resources for Number Theory for Computing**

2. five Hash features ...................................... three. 2. 6 errors Detection and Correction equipment ................ three. 2. 7 Random quantity iteration .......................... three. three Cryptography and knowledge safeguard ....................... three. three. 1 creation ......................................... three. three. 2 Secret-Key Cryptography .......... , ................... three. three. three information/ complex Encryption commonplace (DES/ AES) ........ three. three. four Public-Key Cryptography ............................. three. three. five Discrete Logarithm dependent Cryptosystems ................ three. three. 6 RSA Public-Key Cryptosystem ......................... three. three. 7 Quadratic Residuosity Cryptosystems ................... three. three. eight Elliptic Curve Public-Key Cryptosystems ................ three. three. nine electronic Signatures .................................... three. three. 10 electronic Signature general (DSS) ...................... three. three. eleven Database protection .................................... three. three. 12 mystery Sharing ....................................... three. three. thirteen Internet/Web safety and digital trade ......... three. three. 14 Steganography ....................................... three. three. 15 Quantum Cryptography ............................... three. four Bibliographic Notes and additional analyzing ...................... Bibliography . ................................................. 415 Index ......................................................... 429 Notation All notation could be so simple as the character of the opemtions to which it really is utilized. CHARLES BABBAGE Notation (1791-1871) clarification set of common numbers: N = {1, 2, three, · · ·} set of integers (whole numbers): 7L = {0, ±n: n EN} set of optimistic integers: z+ = N set of confident integers more than 1: seventy one. ,>1 = {n: n E 7L and n > 1} set of rational numbers: ((] = { ~ : a, b E seventy one. , and b =1- zero} set of genuine numbers: lR={n+O. d1d2ds···:nEZ, diE{0,1,···,9} and no limitless series of 9's looks} c set of advanced numbers: C = {a + bi : a, b E lR and that i 7L/n7L additionally denoted by way of Zn, residue sessions modulo n; a hoop of integers; a box if n is fundamental (7L /n7L)* multiplicative workforce; the weather of this workforce are the weather in 7L / n7L which are fairly leading to n: (7L/n7L)* = {[a]n E 7Ljn7L: gcd(a,n) = 1}. = v=I} finite box with p parts, the place p is a primary quantity finite box with q = pk a chief energy (arbitrary) box ring Notation xviii g crew 191 order of crew g Bernoulli numbers: ( n ~1 ) En + ... + ( Fermat numbers: Fn = 22 n n ~1 + 1, ) Et + Eo = zero n 2': zero Mersenne primes: Mp = 2P- 1 is fundamental every time pis best sq. root of x kth root of x asymptotic equality approximate equality 00 infinity ===} implication ¢:::::} equivalence D clean image; finish of facts u house Prob chance degree lSI cardinality of set S E member of c c right subset subset binary operations EB binary operation (addition); specific or (XOR) eight binary operation (multiplication) f(x) "' g(x) f(x) and g(x) are asymptotically equivalent (9,*) (9,*) and (Ji,*) are isomorphic j_ ~ (Ji,*) undefined encryption key decryption key encryption strategy C = Eek (M), the place M is the plaintext decryption procedure M = D ak (C), the place C is the ciphertext Notation XlX f(x) functionality of x J-1 inverse off binomial coefficient integration logarithmic necessary: Li(x) = Li(x) sum: X1 i=l n TI r lndt t }2 + X2 + · · · + Xn product: x1x2 · · · Xn :ri i=l n!