sss_n3_t1_aes.m

/**
@ This is for (3, 1) - SSS scheme
@ We use the irreducible polynomial in AES
@ Wei Cheng, 2021-12-11
**/

P<X>:=IrreduciblePolynomial(GF(2),8);
P<X>:= X^8 + X^4 + X^3 + X^1 + 1;  // AES
print "Irreducible Polynomial P<X> = ", P;
GF256<X> := ext<GF(2)|P>;

Element2Integer := function(e)
    local s, p, res;
    s := ElementToSequence(e);
    p := 1;
    res := 0;
    for si in s do
        if si eq 1 then
            res := res+p;
        end if;
        p :=2*p;    //bit by bit
    end for;
    return res;
end function;

Integer2Element := function( i )
    local p, res;
    p := 0;
    res :=0;
    for j in [0..7] do
        if (i mod 2) eq 1 then
            res := res + X^p;
        end if;
        p := p+1;
        i := Floor(i/2);
    end for;
    return res;
end function;

print "--------------------- Main program -----------------------";
print "--------------------- (3, 1)-SSS -----------------------";
n := 3;
t := 1;
for i:= 2 to 254 do
    for j:= i+1 to 255 do
        print "i, j = ", i, j;    
        x_1 := Integer2Element(1);
        x_i := Integer2Element(i);
        x_j := Integer2Element(j);
        G := Matrix(GF256, t, n, [ x_1, x_1, x_1]);   
        H := Matrix(GF256, t, n, [ x_1, x_i, x_j]);
        C := LinearCode(G);  
        D := LinearCode(H); 
        print "Dimension: ", Dimension(C + D);
        print "WD orig D (word): ", WeightDistribution(D);
        print "WD dual D (word): ", WeightDistribution(Dual(D)); 
        print "WD dual D  (bit): ", WeightDistribution(Dual(SubfieldRepresentationCode(D, GF(2))))[1..5];
    end for;
end for;

exit;