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Unformatted text preview: CSC405 Introduction to Computer Security Fall 2007 Number Theory and Public Key Cryptography Introduction RSA and ECC , which do encryption and digital signatures ElGamal and DSS , which do digital signatures DiffieHellman , which allows establishment of a shared secret but doesn’t have any algorithms that actually use the secret (it would be use, for instance, together with a secret key scheme in order to actually use the secret for something like encryption) Zero knowledge proof systems , which only do authentication The thing that all public key algorithms have in common is the concept of a pair of related quantities, one secret and one public, associated with each principal. Modular Arithmetic Modular arithmetic uses the nonnegative integers less than some positive integer n , performs ordinary arithmetic operations such as addition and multiplication, and then replaces the result with its remainder when divided by n . The result is said to be modulo n or mod n . When we write “ x mod n ”, we mean the remainder of x when divided by n . Applications 1 Modular arithmetic is referenced in number theory, group theory, ring theory, abstract algebra, cryptography, computer science, and the visual and musical arts. It is one of the foundations of number theory, touching on almost every aspect of its study, and provides key examples for group theory, ring theory and abstract algebra. In cryptography, modular arithmetic directly underpins public key systems such as RSA and DiffieHellman, as well as providing finite fields which underlie elliptic curves, and is used in and a variety of symmetric key algorithms including IDEA and RC4. In computer science, modular arithmetic is often applied in operations involving binary numbers and other fixedwidth, cyclic data structures. The modulo operation, as implemented in many programming languages and calculators, is an application of modular arithmetic that is often used in this context. In the visual arts, modular arithmetic can be used to create artistic patterns based on the multiplication and addition tables modulo n (see external link, below). In music, modular arithmetic is used in the consideration of the twelve tone equally tempered scale, where octave and enharmonic equivalency occurs (that is, pitches in a 1:2 or 2:1 ratio are equivalent, and Csharp is the same as Dflat). History Modular arithmetic was studied first by Carl Friedrich Gauss and was written about in his book Disquisitiones Arithmeticae in 1801. Modular Addition mod 10 addition 3 + 5 = 8, just like in regular arithmetic. The answer is already between 0 and 9. 1 http://en.wikipedia.org/wiki/Modular_arithmetic 2 7 + 6 = 13 in regular arithmetic, but the mod 10 answer is 3....
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This note was uploaded on 10/13/2008 for the course CSC 405 taught by Professor Carter during the Spring '08 term at N.C. State.
 Spring '08
 CARTER
 Algorithms, Computer Security

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