# Shaurya Baranwal - Homework 14 March 10, 2024

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Encode113G := proc(x, q) local a,b,c,d,eqs,i,S:

 n := nops(x) + 4:
 eqs := {seq(0 = add(x[i], i=1..n) + ((n+1)^j)*a + ((n+2)^j)b + ((n+3)^j)c + ((n+4)^j)d, j=0..3)}:
 S := msolve({eqs}, q):
 subs(S, [op(x), a,b,c,d]):

end:

Sy113G := proc(x, q) local i, r, n:

  n := nops(x):
  [seq(add(x[i]*i^(r-1),i=1..n) mod q, r=1..4)]:

end:

Decode113G:=proc(y,q) local S,eq,i,j,L,i1,a,b:

  S:=Sy113(y):
  if S=[0$4] then
    RETURN(y):
  fi:

  eq:=(S[2]^2-S[1]*S[3])*i^2+(S[1]*S[4]-S[2]*S[3])*i+ (S[3]^2-S[2]*S[4]) mod q:

  if eq=0 then
    i:=S[2]/S[1] mod q:
    a:=S[1]:
    RETURN(y-[0$(i-1),a,0$(nops(y)-i)] mod q):
  fi:

  L:=[]:

  for i1 from 0 to q-1 do
    if subs(i=i1,eq) mod q=0 then
      L:=[op(L),i1]:
    fi:
  od:

  i := L[1]:
  j := L[2]:

  b := (i*S[1]-S[2])/(i-j) mod q:
  a := S[1]-b mod q:
  y-[0$(i-1),a,0$(j-i-1),b,0$(nops(y)-j)] mod q:

end:

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