i PC x 0.5K 2 f d Z i 2 PCX X 2 o s PIX 8 4 x 2 - PDF document

CS 70 July 8,2020 of the day go Question a gift someone gives to Yining Khalidand Amin a locked chest with The gift is a long Password in How should some information about the we share such that with the instructors least two at Password of the three instructors need to share their to figure out the Password 2 information

CS 7A July 8,2020 Polynomials secret sharing a secret among n People share I knows nothing Any K Secrecy Roubustnessor Any k knows secret the Efficient storage minimize solution Polynomials a unique.fi T T The idea Points make Two This lecture a unique degree d del Points make Polynonia Polynomials Pix ad XD t Xt Ao a na a degree d polynomial which Pen is is specified by coefficients Ey qi TED if and only if PCN contains chgy Passy

Polynomials over reads ai aadg HEIR ao polynomial Enamples 112 d L a 5 Put i PC x 0.5K 2 f d Z i 2 PCX X 2 o s PIX 8 4 x 2 properties of Polynomials zero Polynomial of degree d Property 1 A non d roots most has at zeros Pcx if and only if 0 root X is a Property 2 del Given Pairs ix oil is a unique the Xi all there with distinct e most d Polynomial that such degree of at Pai 2 Ef b for gold i proof of Property I review polynomial division we need to First

a polynomial degree d If we have we can divide a polynomial of degree Ed using long by division pay q'ex 947 two remainder the quotient Hx the LX is less than for rcx The degree aux Is the degree of 2 3 Examples Divide 4 by 2 ex 3 ME 2 4 3k z 4 3 2 14 12 7 0t 9 27 9cm rep 9 9 SO Is 442 3 p 4 9 2 2 s Now prove property 1 we can zero Polynomial of degree d Property 1 A non roots was at most if andonly if t

A root Lemmata has PIE iffy Pex a Quy has remainder CX Paku a a w proof Pcx know Q rex x a we w P X QQ x a e T.tk eE.oIoicF ro a Quy Pens ex Rages If r o E K of with distinct d degree pen Lemma 2 roots written be y ad can as au az X ad X Az x Ai P LX se org By Induction proof d on PIXIE d o Base case some d70 Pays a ex a g for ex ad Iff of degree d1 Induction step PIX a del with roots AZ as ans g Pla lemma D CX By X ads dei d tailitdtl Cai ads Rai o Qiao

it del t e o Q Cai d x adj from Q ca C ex ad IH s s C CA ab ex ads X ad Pex to prove Property 2 Use Lemma l and 2 of degree d For Pcx CK Ad pays C Ct ai of I fl ed a Bai peas to d of degree at Pen has most d roots Proof of Property Ze del Property 2 Given Pairs ix 51,1 no there distinct is unique with all the Xi a d Polynomial that of at most such degrees Lor Phil Ti EE 1,2 delle 2 1 Enistence Uniqueness steps a Two

1 Enistence a unique line 2 Points t 31 cz find the uingue Pex for Pcxj aix Ao a't Aosz 1 3 A Pcs apse 4 2A 4 easy Pez Pcx X 2 quadratic function 3 Points A Unique D for Find the pix 1,7 unique 2,13 6,371 xk pcx3s Et 3 7 ME 7 aze.a.si b IEEE'ated 4 A Daze 2am as s 27 General In del points s Ltd Hdd t.HN MHz Pcx ao solve f Idr y as adxdd.it e Ao Y del r

exist solution the Does prove Lagr ange the enistense use To interpolation of values x Delta polynomial for set glide if a ai x X Xj and xjfXi Bien if o to interpolate it useful is why a polynomial us YD Passing slXdi9YdrD2o_yyzqjcontainsHbY contains CxzgYz lgsumo.ver z a i 97dm XD contains YdAdult Pen TH ex Xj Pex s Y g pcxzs.sk DefinefiCDs y.gg i Xoxo 1 Y del PlXdm X Xj Xj ski O

Pens 5 bicx G Pe 9 I 193 AKA l I 37 2 r r Cx Dcx Sz CA 2 1712 3

Uniqueness At most one Polynomial of degree d Points dal hits Remember property 1 non zero Polynomial Any of degree d et roots has at most contradition Proof of uniqueness proof by Polynomials PCN differen Assume two del points hit the Q CN t d PLA Rex Qui Defin reminded 0 Yi Yi Pcxi Qui Rcti g del FIEF b R has roots del

Polynomials fields on Finite what if XE 4 or XEIN still hold property 1 and Does 2 For Polynomial interpolation and proof add subtract property land 2 can of we multiply and divide Rexcepto numbers two Subtracting XEN S natural number be a not may XEZ.ro division of numbers two an integer not may be if what modul op arithmetic Polynomials with AdXdxad_gXd P I.a.xaolmedpjAdg.aagAoEf0gao.gP l2gXEf g.r GP M Prime number P is and multiplication Addition subtraction ode allowed are 9cdcxgP sl the if so Division all f sa RM of for inverse exist

property land 2 hold if the Hence Cmod Pj PCH for of are coefficient P prime a denoted by finite field This is over a CGaliors Field or G Fcp Fp addition The field with of elements is set multiplication and subtraction inverses the result of operators on the elements Note field is in The fields of at the element Enamples ftp.qypc 3qft2CmId5 3 PO ii if a mods Pem il x t 2 i of Pdx and Rza Intersection I 0 s bas x z e mods 3Xt2 Ntl 3 n mods Exec 1 Et X 12 2 E mods X

5 52 apes 5 PCM mod 5 4th X T tareded3 d 2 y Pi 2x AR Pz in 2 Points are given If DX X 7 z s five are Polynomials there with degree s Pz that Pass through 2 p Application Sharing Secret Secret among n People Share I Any K knows nothing Secrecy Roubustnessor Any knows secret the k Efficient storage minimize of n scheme O Shamir's K out as Efo se secret I and random 1 Choose Sg air a g am µ I 1 Let 2 PLA ak.gr s as where cross Ith ei Pci Points mad P with Share 3 I Eflg oh

1h Points the Result gives Any Polynomial The Ad secret s PED sao secret so K shares Puy reconstruct With of P values are K L shares with any for Pto possible