the day of Question to send a message containing h Packets I want - PDF document

CS 70 July 9,2020 the day of Question to send a message containing h Packets I want chambers am using corrupts K of the Packets The network I don't know which k Packets are corrupted we is fixed regardless of the length of the K message the I need is minimum number of Packets what recover the original message to send to should I send redundant Packets

Error correcting codes messaging through an Unreliable channel Today composed of packets messages are Errors I Erasure errors or dropped packets 1 Lost packet drops codes Erasure Tolerate Corrupted Packets 2 Error correction codes errors Tolerate in packet the Algebraic Polynomials correcting codse Error combinatorial Graph Their Redundancy Example O I I 2 I 2 2 3 3 3 x x x I I t jamazz z.gg z.gg Receive Il Receive 11,2

Erasure Errors original message Received message I 4 5 2 2 3 I 3 4 5 4 0 4 4 I 3 0 I f Ugh L2 Dog 3,0 14,3 15,4 fest g 3,0 g 15,4 In general channel that loses Ig Packets Packet message send solution Packets can more we g need Kel copies for each Redundancy of hu packet Axckel Total packets so Can we do better Polynomials 9 yes gChgmTJ Original messages he points tgmDgU2sm27 of degreen T Dex points n on PLA Remember any n Points sufficient is to reconstruct PCD Evaluate Pix Points new on mm The received message has K hit K nm Reconstruct PCD n received Packets the using open is PG g The message

Problem I want to send a message with n Packets bassChannel loses k Packets Channel and Recore you send nek Packets Question can the message Erasure coding scheme message m mm been bits Dfm f 2b l has Each Packet P Z 2b he k P GF Finite Field P F z construct Pex of degree h 1 using 1 l f i kn Pei s Mei 2 Send the message f PCD y p Chak any received Reconstruct Pex n Packets from 3 Pcm 4 Recover the message 9 pug

Error correction corrupts k Packets Noisychanned challenge finding which Packets are corrupt h Packets hi g Mn On noisy channel problem communicate that corrupts Packets K Reed Solomon Code a Pol nomial Pax of degree n l I that make encodes message g Pays Mn Pll mis a PC 2 Send Pcl E7y 3 rn f Recieved Values Vz 2K Properties Points for her Pci ri least at vs a degree h 1 Polynomial that PCN is x of received Point Thx is unique contains n k why is Pcx unique polynomial a degree ma Act is Assume Proof of Vj for where neck out Li trek

for Qui ri times nek total Points contained for Pci srj ne k timed Q and P 2hr2m by choose from Total number of Points 2K to h Points Qeigs Pci ri a sp least at At n and PCN are degree n Dex Brute force Algorithm For each subset of rink Points haka n of n 1 Polynomial Fit degree X Check if with K of the total Pains h consistent If output yes QCD a subset of spu her points where For Ies method will reconstruct PCD Unique degree in 1 that firs n Ponits Qty with consistent points risk Q ex QCD Pex

n 3g K 5 I Example he 2k s 6g if so B Received r 3 g ng 3 2 Find PLA contains Az X that 9 8 hek 1 4 Points 3 3 X AD Azra i g I 2 A Zeta pass 0 Haze 5am as 3 consistent solution wrong and solve Assume no Points Assume Point 2 is wrong and solve the solution contains exercise In general fpwsamxh tf T e.ie with he2KM 2K men p og Pu f a Pas E ziaisz mod P I O K of these equations h L not 214 2 mia Vm are Pcha Correct Is How to find the errors

Try all combinations number of ways to choose 2K n out of K n 2K ie n exponentials counting to find Packers efficiently How the n e f Pu Aj i i o modP Iai pass r2 of them µ i i e n l not satisfied i are mla m pcneu7 ro Iff Ritfrj multiply equation i by o Idea All equations satisfied are to multiply by 0 which don't know this one we will another polynomial use we yn 2K Ig 2g Assume errors at e g ez geek are a Error locator polynomial Define EU x e x en K 0 I E Cei I

Ea r Ell Efi Pci E 2 Pass aiEu iso E mod P s c h l ICx sj a Elnek Im Elma Penny E miao D E K I s XK ibk.it ebo ECX to Ier Eek ooh Ka Unknown unknowns h ELM P ibi hack equations and we have he Kunknowns h 2K nonlinear equation nah Scary Quy Define.se Oink Ya a's so EEuationsiqIriEiTfn blineardiQ.h equations Nazis 4k unknowns Unknowns f K E Unknowns a a 2k n g Eu To summarize nah Ee b j O J CD Epa I 2 PEW K l h K l 2iais Rz ED bj 2 KrodP i k l j K I ink 1 Z bj M miaj a QCne2K m 0 j Eenerry msmig2k

M MT2k 9h14 Ks Example Rs 3 Bssr Lsd Vgs Ksl ExPx i zx2eaH Ady 4 Elbe ri X boo I Qli Roi Then Eli 3 CI bo A31 aztq ao Ea imam I 014 bo Azt2aze4a 1A 60374927 Saipa 3315 bro Az 63 5 gbz arbs Az Ao Ig XI 6 36 QIN 5 so Kong division X E Cx z T PcDsxz Pays Qexis Pex E Cx Berlekamp Welch Error correction Message mug Mn 3 Sender 1 Formdegree n 1 polynomial Pix where Pci smii n 2 Send Pch 2K Pcs

Receiver g hi y Vn zig Hg Receive s Eci R I Zoo solve QLE herk equations to find QQ and Etty P EU Q Rx IE ex 3 Compute 4 compute Pu Pcn The solution always earst since the solution is this way constructed Question what if the hem equations not Independent when there less than Kerrors are Assam there is another solution QingEtx Q fQf Do we have PlX s f Eli Q ti we have Elif Sidhe 2K s ri Efi 4 Q j I

s Q'Li E Ii Eti this nhk Qu are equal hezk at E'ex Q 9 degree ha ECM are a h K l K KQED ECM QUI E Ex divide BY EIDE'Cx7 P x EED E Cx Summary Any del points a unique degree d Polynomial Any del points give the Polynomial back Recover information can lose any k Erasure tolerance Nek Secret sharing n people any K recover Recover from corruptions Send more information neck K errors are correct nek one degree n 1 Polynomial consistent only can fix k bad equations by multiplying by error Polynomial a Polynomial is a polynomial A Polynomial times he 2k Coefficients in all he 2k Correct equations