lecture 4(3) - Lecture 4: The mod function Daniel Frances c...

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Unformatted text preview: Lecture 4: The mod function Daniel Frances c 2012 Contents 1 Introduction 2 2 Centrality of mod arithmetic to Internet computations 2 2.1 Modular Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Modular Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Modular Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Modular Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.5 Modular Exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1 1 Introduction It turns out that the entire issue of polynomial versus exponential algorithms is fundamental to Encryption over the Internet. In the early days of the Internet, it was realized that to support commercial transactions over the Internet, governments needed to ensure that there were secure means for encrypting information. It used to be the case that encryption algorithms were shrouded in the highest level of secrecy. If your enemy knew your encryption algorithm then he would know how to decode it. Developers of encryption algorithms worked for top secret organizations, and their algorithms would never see the light of day. Then the approach changed, encryption algorithms became widely known. In fact currently used encryption algorithms were the winners of international contests among largely univer- sity academics, to see who could develop the most secure algorithms. The way these algorithms manage their secrecy, is to use a secret key . The key is simply a number. The key becomes somehow known to the intended reader of the message, but remains secret to the intruder. The algorithms are so good that without the key it is physically impossible to decipher the message. The way this is done is by ensuring that the algorithms to encode with the key are really fast, but the algorithms to decode with the key...
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This note was uploaded on 03/31/2012 for the course MIE 335 taught by Professor Frances during the Spring '12 term at University of Toronto- Toronto.

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lecture 4(3) - Lecture 4: The mod function Daniel Frances c...

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