kaliski-math-computer-security-new

kaliski-math-computer-security-new - The Mathematics of...

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The Mathematics of Computer Security Burt Kaliski, RSA Security April 27, 2006
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Introduction Mathematics has many applications in computer security today, including: setting up encryption keys transferring messages securely The practical side of number theory Concepts to be covered here: Modular arithmetic Prime numbers Chinese Remainder Theorem No advanced math background assumed …
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Modular Arithmetic “Remainders-only” arithmetic Addition, subtraction, multiplication, division relative to a modulus n Numbers typically between 0 and n -1 Notation: x mod n means “remainder after dividing x by n x y (mod n ) means “ x and y have the same remainder mod n
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Modular Addition 5 + 5 = 10 (5 + 5) mod 10 = 0 ? ? ? ? ? ?
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Modular Subtraction (7 - 9) mod 10 = 8 (7 - 4) mod 10 = 3 5 - 5 = 0 (5 - 5) mod 10 = 0 ? ? ? ? ? ?
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Modular Multiplication 5 × 5 = 25 (5 × 5) mod 10 = 5 ? ? ? ? ? ?
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Modular Division proof: 3 × 9 mod 10 = 7 undefined: no x such that 4 × x mod 10 = 7 5 ÷ 5 = 1 undefined: more than one x such that 5 × x mod 10 = 5 ? ? ? ? ? ?
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Greatest Common Factor The greatest common factor (GCF) of two numbers is the largest number that evenly divides both Two numbers are relatively prime if GCF = 1 Examples: GCF (9, 10) = 1 GCF (4, 10) = 2 GCF (5, 10) = 5 Division by d mod n is defined only if GCF ( d , n ) = 1 The greatest common factor can be computed recursively: GCF ( a , b ) = GCF ( b , a mod b )
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This note was uploaded on 11/11/2011 for the course MATH 112 taught by Professor Jarvis during the Winter '08 term at BYU.

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kaliski-math-computer-security-new - The Mathematics of...

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