Linear approximation table of AES S-Box

Question

I am trying to create linear approximation table of AES SBox to better understand linear cryptanalysis, I have followed the formula in this paper (page 7 of pdf file) to be able to generate the linear approximation table of AES S-Box, specifically that is $$\frac{\#\{x\in R|x \cdot t_x=B(x)\cdot t_y\}}{2^8} -\frac{1}{2}.$$ This is equivalent to $$\frac{\#\{x\in R|x \cdot t_x=B(x)\cdot t_y\}}{2^8} -\frac{2^7}{2^8},$$ so for each pair $ t_x $, $ t_y $ of the linear approximation table will be calculated as $ x\in R|x \cdot t_x=B(x)\cdot t_y\ - 2^7 $ (with # symbol denotes cardinality). Once I got the formula I wrote the following code:

import java.io.BufferedWriter;
import java.io.File;
import java.io.FileWriter;
import java.io.IOException;

public class LinearProbabilityV2 {
    public static void main(String[] args) throws IOException {
        File file=new File("Result_Linear_approximation_table.txt");
        FileWriter fw=new FileWriter(file);
        BufferedWriter bw=new BufferedWriter(fw);
        int SBox[] = { 0x63, 0x7c, 0x77, 0x7b, 0xf2, 0x6b, 0x6f, 0xc5, 0x30, 0x01, 0x67, 0x2b, 0xfe, 0xd7, 0xab, 0x76,
                        0xca, 0x82, 0xc9, 0x7d, 0xfa, 0x59, 0x47, 0xf0, 0xad, 0xd4, 0xa2, 0xaf, 0x9c, 0xa4, 0x72, 
                        0xc0, 0xb7, 0xfd, 0x93, 0x26, 0x36, 0x3f, 0xf7, 0xcc, 0x34, 0xa5, 0xe5, 0xf1, 0x71, 0xd8, 
                        0x31, 0x15, 0x04, 0xc7, 0x23, 0xc3, 0x18, 0x96, 0x05, 0x9a, 0x07, 0x12, 0x80, 0xe2, 0xeb, 
                        0x27, 0xb2, 0x75, 0x09, 0x83, 0x2c, 0x1a, 0x1b, 0x6e, 0x5a, 0xa0, 0x52, 0x3b, 0xd6, 0xb3, 
                        0x29, 0xe3, 0x2f, 0x84, 0x53, 0xd1, 0x00, 0xed, 0x20, 0xfc, 0xb1, 0x5b, 0x6a, 0xcb, 0xbe, 
                        0x39, 0x4a, 0x4c, 0x58, 0xcf, 0xd0, 0xef, 0xaa, 0xfb, 0x43, 0x4d, 0x33, 0x85, 0x45, 0xf9, 
                        0x02, 0x7f, 0x50, 0x3c, 0x9f, 0xa8, 0x51, 0xa3, 0x40, 0x8f, 0x92, 0x9d, 0x38, 0xf5, 0xbc, 
                        0xb6, 0xda, 0x21, 0x10, 0xff, 0xf3, 0xd2, 0xcd, 0x0c, 0x13, 0xec, 0x5f, 0x97, 0x44, 0x17, 
                        0xc4, 0xa7, 0x7e, 0x3d, 0x64, 0x5d, 0x19, 0x73, 0x60, 0x81, 0x4f, 0xdc, 0x22, 0x2a, 0x90, 
                        0x88, 0x46, 0xee, 0xb8, 0x14, 0xde, 0x5e, 0x0b, 0xdb, 0xe0, 0x32, 0x3a, 0x0a, 0x49, 0x06, 
                        0x24, 0x5c, 0xc2, 0xd3, 0xac, 0x62, 0x91, 0x95, 0xe4, 0x79, 0xe7, 0xc8, 0x37, 0x6d, 0x8d, 
                        0xd5, 0x4e, 0xa9, 0x6c, 0x56, 0xf4, 0xea, 0x65, 0x7a, 0xae, 0x08, 0xba, 0x78, 0x25, 0x2e, 
                        0x1c, 0xa6, 0xb4, 0xc6, 0xe8, 0xdd, 0x74, 0x1f, 0x4b, 0xbd, 0x8b, 0x8a, 0x70, 0x3e, 0xb5, 
                        0x66, 0x48, 0x03, 0xf6, 0x0e, 0x61, 0x35, 0x57, 0xb9, 0x86, 0xc1, 0x1d, 0x9e, 0xe1, 0xf8, 
                        0x98, 0x11, 0x69, 0xd9, 0x8e, 0x94, 0x9b, 0x1e, 0x87, 0xe9, 0xce, 0x55, 0x28, 0xdf, 0x8c, 
                        0xa1, 0x89, 0x0d, 0xbf, 0xe6, 0x42, 0x68, 0x41, 0x99, 0x2d, 0x0f, 0xb0, 0x54, 0xbb, 0x16};
        for (int k = 0; k < 256; k++) { // input mask
            for (int j = 0; j < 256; j++) { // output mask
                int count = 0;
                for (int i = 0; i < 256; i++) {
//                  Calculating the right side: S(x) AND O_y (O_y là output mask)
                    int result_side_right = SBox[i] & j;
//                  Calculating the left side" x AND I_y (I_x là input mask)
                    int result_side_left = i & k;
                    if (result_side_left==result_side_right) {
                        count += 1;
                    }
                }
                bw.write((count -128)+" ");
            }
            bw.write("\n");
        }
        bw.close();
    }
}

where int result_side_right = SBox[i] & j to calculate $ B(x)\cdot t_y $, int result_side_left = i & k to calculate $ x \cdot t_x $ and count is the result corresponding to each pair $ t_x, t_y $. The output from the above program will look like this, but according to sagemath the result should look like this. Did I misunderstand something here?

Answer 1

At issue here is the fact that SBox[i] & j does not compute $B(x)\cdot t_y$. The latter is a 0 or 1 value and the former is an 8-bit value. Instead you must calculate poppar(SBox[i] & j) where poppar is the population parity function that returns 0 if the input has an even number of bits set and 1 if it has an odd number of bits set. This is equal to the $\mathbf F_2$ sum of the masked bits and hence the dot product.