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Unformatted text preview: CS70: Satish Rao: Lecture 12. Outline. 1. Polynomials 2. Secret Sharing 3. Polynomial Interpolation 4. Finite Fields 5. Erasure Coding Secret Sharing. Secret Sharing. Share secret among k people. Secret Sharing. Share secret among k people. Secrecy: Any k 1 knows nothing. Roubustness: Any k knows secret. Minimality: minimize storage. Polynomials A polynomial P ( x ) = a d x d + a d 1 x d 1 ··· + a . is specified by coefficients a d ,... a . 1 A field is a set of elements with addition and multiplication operations, with inverses. GF ( p ) = ( { ,..., p 1 } , + ( mod p ) , * ( mod p )) . Polynomials A polynomial P ( x ) = a d x d + a d 1 x d 1 ··· + a . is specified by coefficients a d ,... a . P ( x ) contains point ( a , b ) if b = P ( a ) . 1 A field is a set of elements with addition and multiplication operations, with inverses. GF ( p ) = ( { ,..., p 1 } , + ( mod p ) , * ( mod p )) . Polynomials A polynomial P ( x ) = a d x d + a d 1 x d 1 ··· + a . is specified by coefficients a d ,... a . P ( x ) contains point ( a , b ) if b = P ( a ) . Polynomials over reals : a 1 ,..., a d ∈ ℜ , use x in ℜ . 1 A field is a set of elements with addition and multiplication operations, with inverses. GF ( p ) = ( { ,..., p 1 } , + ( mod p ) , * ( mod p )) . Polynomials A polynomial P ( x ) = a d x d + a d 1 x d 1 ··· + a . is specified by coefficients a d ,... a . P ( x ) contains point ( a , b ) if b = P ( a ) . Polynomials over reals : a 1 ,..., a d ∈ ℜ , use x in ℜ . Polynomials P ( x ) with arithmetic modulo p : 1 a i ∈{ ,..., p 1 } and P ( x ) = a d x d + a d 1 x d 1 ··· + a ( mod p ) , for x ∈{ ,..., p 1 } . 1 A field is a set of elements with addition and multiplication operations, with inverses. GF ( p ) = ( { ,..., p 1 } , + ( mod p ) , * ( mod p )) . Polynomial: P ( x ) = a d x 4 + ··· + a Line: P ( x ) = a 1 x + a Polynomial: P ( x ) = a d x 4 + ··· + a Line: P ( x ) = a 1 x + a = mx + b Polynomial: P ( x ) = a d x 4 + ··· + a Line: P ( x ) = a 1 x + a = mx + b x P ( x ) Polynomial: P ( x ) = a d x 4 + ··· + a Line: P ( x ) = a 1 x + a = mx + b x P ( x ) P ( x ) = . 5 x + Polynomial: P ( x ) = a d x 4 + ··· + a Line: P ( x ) = a 1 x + a = mx + b x P ( x ) P ( x ) = . 5 x + P ( x ) = 1 x + 3 Polynomial: P ( x ) = a d x 4 + ··· + a Line: P ( x ) = a 1 x + a = mx + b x P ( x ) Parabola: P ( x ) = a 2 x 2 + a 1 x + a Polynomial: P ( x ) = a d x 4 + ··· + a Line: P ( x ) = a 1 x + a = mx + b x P ( x ) Parabola: P ( x ) = a 2 x 2 + a 1 x + a = ax 2 + bx + c Polynomial: P ( x ) = a d x 4 + ··· + a Line: P ( x ) = a 1 x + a = mx + b x P ( x ) P ( x ) = . 5 x 2 x + . 1 Parabola: P ( x ) = a 2 x 2 + a 1 x + a = ax 2 + bx + c Polynomial: P ( x ) = a d x 4 + ··· + a Line: P ( x ) = a 1 x + a = mx + b x P ( x ) P ( x ) = . 5 x 2 x + . 1 P ( x ) = . 3 x 2 + 1 x + . 1 Parabola: P ( x ) = a 2 x 2 + a 1 x + a = ax 2 + bx + c Polynomial: P ( x ) = a d x 4 + ··· + a ( mod p ) x P ( x ) Polynomial: P ( x ) = a d x 4 + ··· + a ( mod p ) x P ( x ) 3 x + 1 ( mod 5 ) Polynomial:...
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This note was uploaded on 02/29/2012 for the course COMPSCI 70 taught by Professor Rau during the Fall '11 term at University of California, Berkeley.
 Fall '11
 Rau

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