The S-Box Substitution

After the compressed key is XORed with the expanded block, the 48-bit result moves to a substitution operation. The substitutions are performed by eight substitution boxes, or S-boxes. Each S-box has a 6-bit input and a 4-bit output, and there are eight different S-boxes. (The total memory requirement for the eight DES S-boxes is 256 bytes.) The 48 bits are divided into eight 6-bit sub-blocks. Each separate block is operated on by a separate S-box: The first block is operated on by S-box 1, the second block is operated on by S-box 2, and so on. See Figure 12.4.

Each S-box is a table of 4 rows and 16 columns. Each entry in the box is a 4-bit number. The 6 input bits of the S-box specify under which row and column number to look for the output. Table 12.6 shows all eight S-boxes.

The input bits specify an entry in the S-box in a very particular manner. Consider an S-box input of 6 bits, labeled b₁ b₂ b₃ b₄ b₅ and b₆. Bits b₁ and b₆ are combined to form a 2-bit number, from 0 to 3, which corresponds to a row in the table. The middle 4 bits, b₂ through b₅ are combined to form a 4-bit number, from 0 to 15, which corresponds to a column in the table.

For example, assume that the input to the sixth S-box (i.e., bits 31 through 36 of the XOR function) is 110011. The first and last bits combine to form 11, which corresponds to row 3 of the sixth S-box. The middle 4 bits combine to form 1001, which corresponds to the column 9 of the same S-box. The entry under row 3, column 9 of S-box 6 is 14. (Remember to count rows and columns from 0 and not from 1.) The value 1110 is substituted for 110011.

Figure 12.3  Expansion permutation.

It is, of course, far easier to implement the S-boxes in software as 64-entry arrays. It takes some rearranging of the entries to do this, but that’s not hard. (Don’t just change the indexing without rearranging the entries. The S-boxes are designed very carefully.) However, this way of describing the S-boxes helps visualize how they work. Each S-box can be viewed as a substitution function on a 4-bit entry: b₂ through b₅ go in, and a 4-bit result comes out. Bits b₁ and b₆ come from neighboring blocks; they select one out of four substitution functions available in the particular S-box.

The S-box substitution is the critical step in DES. The algorithm’s other operations are linear and easy to analyze. The S-boxes are nonlinear and, more than anything else, give DES its security.

The result of this substitution phase is eight 4-bit blocks which are recombined into a single 32-bit block. This block moves to the next step: the P-box permutation.

The P-Box Permutation

The 32-bit output of the S-box substitution is permuted according to a P-box. This permutation maps each input bit to an output position; no bits are used twice and no bits are ignored. This is called a straight permutation or just a permutation. Table 12.7 shows the position to which each bit moves. For example, bit 21 moves to bit 4, while bit 4 moves to bit 31.

Figure 12.4  S-box substitution.