# Substitution–permutation network

**SP-network**, or **substitution–permutation network** (**SPN**), is a series of linked mathematical operations used in algorithms such as AES (Rijndael), , , , , , and .

Such a network takes a block of the and the as inputs, and applies several alternating “rounds” or “layers” of and to produce the block. The S-boxes and P-boxes transform of input bits into output bits. It is common for these transformations to be operations that are efficient to perform in hardware, such as (XOR) and . The key is introduced in each round, usually in the form of “round keys” derived from it. (In some designs, the S-boxes themselves depend on the key.)

is done by simply reversing the process (using the inverses of the S-boxes and P-boxes and applying the round keys in reversed order).

An **S-box** substitutes a small block of bits (the input of the S-box) by another block of bits (the output of the S-box). This substitution should be , to ensure invertibility (hence decryption). In particular, the length of the output should be the same as the length of the input (the picture on the right has S-boxes with 4 input and 4 output bits), which is different from S-boxes in general that could also change the length, as in , for example. An S-box is usually not simply a of the bits. Rather, a good S-box will have the property that changing one input bit will change about half of the output bits (or an ). It will also have the property that each output bit will depend on every input bit.

A **P-box** is a of all the bits: it takes the outputs of all the S-boxes of one round, permutes the bits, and feeds them into the S-boxes of the next round. A good P-box has the property that the output bits of any S-box are distributed to as many S-box inputs as possible.

At each round, the **round key** (obtained from the with some simple operations, for instance, using S-boxes and P-boxes) is combined using some group operation, typically .

A single typical S-box or a single P-box alone does not have much cryptographic strength: an S-box could be thought of as a , while a P-box could be thought of as a . However, a well-designed SP network with several alternating rounds of S- and P-boxes already satisfies **Shannon’s properties**:

- The reason for
**diffusion**is the following: If one changes one bit of the plaintext, then it is fed into an S-box, whose output will change at several bits, then all these changes are distributed by the P-box among several S-boxes, hence the outputs of all of these S-boxes are again changed at several bits, and so on. Doing several rounds, each bit changes several times back and forth, therefore, by the end, the ciphertext has changed completely, in a manner. In particular, for a randomly chosen input block, if one flips the*i*-th bit, then the probability that the*j*-th output bit will change is approximately a half, for any*i*and*j*, which is the . Vice versa, if one changes one bit of the ciphertext, then attempts to decrypt it, the result is a message completely different from the original plaintext—SP ciphers are not easily . - The reason for
**confusion**is exactly the same as for diffusion: changing one bit of the key changes several of the round keys, and every change in every round key over all the bits, changing the ciphertext in a very complex manner. - Even if an attacker somehow obtains one plaintext corresponding to one ciphertext—a , or worse, a or —the confusion and diffusion make it difficult for the attacker to recover the key.

Although a that uses S-boxes (such as ) is quite similar to SP networks, there are some differences that make either this or that more applicable in certain situations. For a given amount of , an SP network has more “inherent parallelism” and so — given a CPU with a large number of — can be computed faster than a Feistel network. CPUs with few execution units — such as most smart cards — cannot take advantage of this inherent parallelism. Also SP ciphers require S-boxes to be invertible (to perform decryption); Feistel inner functions have no such restriction and can be constructed as .