Allows for exact addition multiplication division exponentiation etc on

Allows for exact addition multiplication division

This preview shows page 14 - 23 out of 26 pages.

Allows for exact addition, multiplication, division, exponentiation, etc on computers! More in depth look at this: Math 113, Math 115 14 / 26
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Extended GCD Algorithm Goal: find ( d , a , b ) st gcd( x , y ) = d = ax + by Allows us to find inverses if gcd( x , y ) = 1! Recursive call on y , x mod y to get ( d , a , b ) Return ( d , b , a - ⌊ x y b ) 15 / 26
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Chinese Remainder Theorem Given coprime n 1 , n 2 , ..., n k , unique soln modulo N = i n i to system of equations x a i (mod n i ) Key is finding “basis” elements b i st b i 1 (mod n i ) b i 0 (mod n j ) for j ̸ = i 16 / 26
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Private Key Cryptography One-Time Pad: xor message w/random, shared pad Perfect security – but only for one message! 17 / 26
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Public Key Cryptography RSA: way to avoid logistical issues of OTP Private key: ( N = pq , d ) Public key: ( N , e = d - 1 (mod ( p - 1 )( q - 1 ))) Encryption: E ( m ) = m e (mod N ) Decryption: D ( c ) = c d (mod N ) Correctness: FLT + CRT Security: 18 / 26
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Polynomial Representations Two equiv representations of degree d polynomials: Coefficients ( c d x d + ... + c 1 x + c 0 ) Values ( ( x 1 , y 1 ) , ..., ( x d + 1 , y d + 1 ) ) Convert coefficients to values: evaluate polynomial Other direction: Lagrange interpolation 19 / 26
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Interpolation Interpretation Given points ( x 1 , y 1 ) , ..., ( x d + 1 , y d + 1 ) , want degree d poly through them Key is finding “basis” polys i ( x ) st i ( x i ) = 1 i ( x j ) = 0 for j ̸ = i Note similarity to proof of CRT! 20 / 26
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Error Correcting Codes Application of polys: fix transmission errors Reed-Solomon: interpolate poly through message P ( 1 ) = m 1 , P ( 2 ) = m 2 , ..., P ( n ) = m n Recover P means recover message! k erasures needs n + k packets k corruptions needs n + 2 k packets 21 / 26
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Countability Main idea: “same size” means “has bijection” Use N as point of comparison eg | N | = | Z | = | Q | = |{ 0 , 1 } * |
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  • Fall '19
  • Logic, Public-key cryptography, Mathematical proof, Discrete Math Review

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