2
$\begingroup$

I came in contact with an algorithm that deals with SPN in an example, first I'd like to give a definition of what it is:

Let $l, m, $ and $N$ be positive integers, let $\pi_s: \{0,1\}^l \to \{0,1\}^l$ be a permutation, and let $\pi_p:\{1,...,lm\} \to \{1,...,lm\}$ be a permutation. Let $P = C = \{0,1\}^{lm}$ and $K \subseteq (\{0,1\}^{lm})^{N+1}$ consist of all possible key schedules that could be derived from an initial key $K$ using the key scheduling algorithm. For a key schedule $(K^1, ..., K^{N+1})$, we encrypt the plaintext $x$ using the a known algorithm (that I couldn't type):

So, I'd like to work on the following example:

Suppose $l = m = N = 4$. Let $\pi_s$ be defined as follows (with input $z$), and output (written in hexadecimal notation)$\pi_s$, ($0 \leftrightarrow(0,0,0,0)$, ..., $9 \leftrightarrow(1,0,0,1), A \leftrightarrow(1,0,1,0)$, and so on; and let $\pi_p$ be defined as:

$\pi(1)=1$, $\pi(2)=5$, $\pi(3)=9$, $\pi(4)=13$, $\pi(5)=2$, $\pi(6)=6$, $\pi(7)=10$, $\pi(8)=14$, $\pi(9)=3$, $\pi(10)=7$, $\pi(11)=11$, $\pi(12)=15$, $\pi(13)=4$, $\pi(14)=8$, $\pi(15)=12$, $\pi(16)=16$.

Suppose the key is $K = 0011 $ $1010$ $1001$ $0100$ $1101$ $0110$ $0011$ $1111$, with plaintext $x = 0010$ $0110$ $1011$ $0111$, then how to apply line by line (in the algorithm)? In addition I'd like to understand, for example, we attribute $w^{r-1} \oplus K^r $ to $u^r$, why $v_{<i>}\leftarrow \pi_s(u^r_{<i>})$?

Given that $v_{<i>} = (x_{{(i-1)}{l-1}}, ..., x_{il})$,

$\endgroup$
0

1 Answer 1

1
$\begingroup$

The question is read as;

We have Substitution–Permutation Network (SPN)

  • a block cipher with block size $lm$
  • Round key addition with $K^r$
  • $\pi_s$ is the diffusion part and it is S-box of input-output size $l$ and this is valid since SPN requires invertible S-boxes, also the sub-index also indicates this.
  • $\pi_p$ is the permutation for the confusion step with size $lm$.

A single round line by line (some parts not calculated since we don't know the permutation)

 [0010 0110 1011 0111]  : w^r-1 as the round input
 [0011 1010 1001 0100]  : X-or with round key K^r   
 [0001 1100 0010 0011]  : X-or result
 [Sbox Sbox Sbox Sbox]  : Apply the Sbox for each block i.e. \pi_x
 [ Permute to Confuse]  : Apply \pi_p for confussion

There is no key schedule defined so, we cannot apply more than two rounds or 1 round as AES did ( first x-or with the key than round ends with a subkey x-or)

In addition I'd like to understand, for example, we attribute $w^{r-1} \oplus K^r $ to $u^r$, why $v_{<i>}\leftarrow \pi_s(u^r_{<i>})$?

Given that $v_{<i>} = (x_{{(i-1)}{l-1}}, ..., x_{il})$,

  • $w^{r-1} \oplus K^r $ is probable the input before the key addition. We can assume it is the input to the round.
  • $u^r$ is the output of the key x-or.
  • $v_{<i>}\leftarrow \pi_s(u^r_{<i>})$; if we carefully look at the indexes this is diving the block size into $l$ sized block for the input to the $\pi_s$. In the example it has size 4 and we have call of 4 $\pi_s$ in a round.
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.