# OK to post

# PART 1: 113 PROCEDURES:
# -----------------------

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