Error-correcting polynomials
(a) alice has a length 8 message to bob. there are 2 communication channels available. when n packets are fed through channel a, the channel will only deliver 5 packets (picked at random). similarly, channel b will only deliver 5 packets (picked at random), but it will also corrupt (change the value) of one of the delivered packets. each channel will only work if at least 10 packets are sent through it. using each of the 2 channels once, how can alice send the message to bob?
(b) alice wishes to send a message to bob as the coefficients of a degree 2 polynomial p. for a message [m1,m2,m3], she creates polynomial p = m1x^2 +m2x+m3 and sends 5 packets: (0,p(,p(,p(,p(,p( however, eve interferes and changes one of the values of a packet before it reaches bob. if bob receives(03), and knows alice’s encoding scheme and that eve changed one of the packets, can he still figure out what the original message was? if so find it as well as the x-value of the packet that eve changed, if not, explain why he can not. (work in mod 11.)
(c) alice decides that putting the message as the coefficients of a polynomial is too inefficient for long messages because the degree of the polynomial grows quite large. instead, she decides to encode the message as values in a degree 2 polynomial. for a 5 length message [m0,m1,m2,m3,m4], she creates a degree 2 polynomial p such that p(0) = m0,p(1) = m1,p(2) = m2,p(3) = m3,p(4) = m4. (alice makes sure to choose her message in such a way that it can be encoded in a polynomial of degree 2.) she then sends the length 5 message directly to bob as 5 packets: (0m4). eve again interfere and changes the value of a packet before it reaches bob. if bob receives(00) and knows alice’s encoding scheme and that eve changed one of the packets, can he still figure out what the original message was? if so find it as well as the x-value of the packet that eve changed, if not, explain why he can not. (work in mod 11.)
(d) after getting tired of decoding degree 2 polynomials, bob convinces alice to send messages using a degree 1 polynomial instead. to be on the safer side, alice decides to continue to send 5 points on the polynomial even though it is only degree 1. she encodes and sends a length 5 message in the same way as part (c) (except using a degree 1 polynomial). eve however, decides to change 2 of the packets. after eve interferes, bob receives (0,−,−, ,−,5). if alice sent (0,−,−,,3), (4,5), for what values of x will bob not be able to uniquely determine the alice’s message? (assume bob knows that eve changed 2 of the packets and work in mod 13.)