#Do not post homework
#Ramesh Balaji,3/10/2024,hw14

# Part 1: generalize procedures encode113, rv113, and syl113
#Adapted Code from Joseph Koutsoutis
Encode113G := proc(x,q) local unk, eqns,a,b,c,d,eq1,eq2,eq3,eq4,i,S:
 eq1 := 0 = add(x[i], i=1..nops(x)) + add(unk[i], i=1..4);
 eq2 := 0 = add(i * x[i], i=1..nops(x)) + add((nops(x)+i)^1 * unk[i], i=1..4);
    print(eq2);
 eq3 := 0 = add(i^2 * x[i], i=1..nops(x)) + add((nops(x)+i)^2 * unk[i], i=1..4);
 eq4 := 0 = add(i^3 * x[i], i=1..nops(x)) + add((nops(x)+i)^3 * unk[i], i=1..4);
 S := msolve({eq1,eq2,eq3,eq4}, q):
    print(S);
 subs(S, [op(x), seq(unk[i],i=1..4)]):
end:

Sy113G := proc(x, q) local i,r:
    [seq(add(x[i]*i^(r-1),i=1..nops(x)) 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 is the list of roots of eq (mod q)
L:=[]:
for i1 from 0 to 10 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: