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lec-12 - CS70 Satish Rao Lecture 12 Outline 1 Polynomials 2...

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CS70: Satish Rao: Lecture 12. Outline. 1. Polynomials 2. Secret Sharing 3. Polynomial Interpolation 4. Finite Fields 5. Erasure Coding
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Secret Sharing.
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Secret Sharing. Share secret among k people.
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Secret Sharing. Share secret among k people. Secrecy: Any k - 1 knows nothing. Roubustness: Any k knows secret. Minimality: minimize storage.
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Polynomials A polynomial P ( x ) = a d x d + a d - 1 x d - 1 ··· + a 0 . is specified by coefficients a d ,... a 0 . 1 A field is a set of elements with addition and multiplication operations, with inverses. GF ( p ) = ( { 0 ,..., p - 1 } , + ( mod p ) , * ( mod p )) .
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Polynomials A polynomial P ( x ) = a d x d + a d - 1 x d - 1 ··· + a 0 . is specified by coefficients a d ,... a 0 . 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 ) = ( { 0 ,..., p - 1 } , + ( mod p ) , * ( mod p )) .
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Polynomials A polynomial P ( x ) = a d x d + a d - 1 x d - 1 ··· + a 0 . is specified by coefficients a d ,... a 0 . P ( x ) contains point ( a , b ) if b = P ( a ) . Polynomials over reals : a 1 ,..., a d , use x in . 1 A field is a set of elements with addition and multiplication operations, with inverses. GF ( p ) = ( { 0 ,..., p - 1 } , + ( mod p ) , * ( mod p )) .
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Polynomials A polynomial P ( x ) = a d x d + a d - 1 x d - 1 ··· + a 0 . is specified by coefficients a d ,... a 0 . P ( x ) contains point ( a , b ) if b = P ( a ) . Polynomials over reals : a 1 ,..., a d , use x in . Polynomials P ( x ) with arithmetic modulo p : 1 a i ∈ { 0 ,..., p - 1 } and P ( x ) = a d x d + a d - 1 x d - 1 ··· + a 0 ( mod p ) , for x ∈ { 0 ,..., p - 1 } . 1 A field is a set of elements with addition and multiplication operations, with inverses. GF ( p ) = ( { 0 ,..., p - 1 } , + ( mod p ) , * ( mod p )) .
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Polynomial: P ( x ) = a d x 4 + ··· + a 0 Line: P ( x ) = a 1 x + a 0
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Polynomial: P ( x ) = a d x 4 + ··· + a 0 Line: P ( x ) = a 1 x + a 0 = mx + b
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Polynomial: P ( x ) = a d x 4 + ··· + a 0 Line: P ( x ) = a 1 x + a 0 = mx + b x P ( x )
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Polynomial: P ( x ) = a d x 4 + ··· + a 0 Line: P ( x ) = a 1 x + a 0 = mx + b x P ( x ) P ( x ) = . 5 x + 0
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Polynomial: P ( x ) = a d x 4 + ··· + a 0 Line: P ( x ) = a 1 x + a 0 = mx + b x P ( x ) P ( x ) = . 5 x + 0 P ( x ) = - 1 x + 3
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Polynomial: P ( x ) = a d x 4 + ··· + a 0 Line: P ( x ) = a 1 x + a 0 = mx + b x P ( x ) Parabola: P ( x ) = a 2 x 2 + a 1 x + a 0
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Polynomial: P ( x ) = a d x 4 + ··· + a 0 Line: P ( x ) = a 1 x + a 0 = mx + b x P ( x ) Parabola: P ( x ) = a 2 x 2 + a 1 x + a 0 = ax 2 + bx + c
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Polynomial: P ( x ) = a d x 4 + ··· + a 0 Line: P ( x ) = a 1 x + a 0 = mx + b x P ( x ) P ( x ) = 0 . 5 x 2 - x + 0 . 1 Parabola: P ( x ) = a 2 x 2 + a 1 x + a 0 = ax 2 + bx + c
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Polynomial: P ( x ) = a d x 4 + ··· + a 0 Line: P ( x ) = a 1 x + a 0 = mx + b x P ( x ) P ( x ) = 0 . 5 x 2 - x + 0 . 1 P ( x ) = - . 3 x 2 + 1 x + . 1 Parabola: P ( x ) = a 2 x 2 + a 1 x + a 0 = ax 2 + bx + c
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Polynomial: P ( x ) = a d x 4 + ··· + a 0 ( mod p ) x P ( x )
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Polynomial: P ( x ) = a d x 4 + ··· + a 0 ( mod p ) x P ( x ) 3 x + 1 ( mod 5 )
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Polynomial: P ( x ) = a d x 4 + ··· + a 0 ( mod p ) x P ( x ) 3 x + 1 ( mod 5 ) x + 2 ( mod 5 ) Finding an intersection. x + 2 3 x + 1 ( mod 5 ) = 2 x 1 ( mod 5 )
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