# Cryptography Engineering: Exercise 3.7

Exercise 3.7 from Cryptography Engineering:

Describe an example system that uses DES but is insecure because of the DES complementation property. Specifically, describe the system, and then present an attack against that system; the attack should utilize the DES complementation property.

### Complementation Property of DES

The complementation property of DES says that $\overline{E_k(m)}=E_\overline k(\overline m)$, where $E_k(m)$ is the value of $m$ DES encrypted with key $k$.

### Example System

A multi-user file system where each file is DES encrypted with a single key, $k$.

### Attack

The attacker sends two 64-bit blocks to be written to the file system, $b$ and $\overline b$. The file system writes out two encrypted blocks to physical storage, $E_k(b)$ and $E_k(\overline b)$.

For each key candidate $x$, compute $E_x(\overline b)$. This is a single cryptographic operation. If the value is equal to $E_k(\overline b)$, then $x=k$ (i.e., the key was found).

Otherwise, compute the complement of the previous crypgraphic operation, namely $\overline {E_x(\overline b)}$, which by the complementation property is $E_\overline x(b)$. If $E_\overline x(b)=E_k(b)$, then $\overline x=k$ (i.e., the key was found).

The attack is thus able to check 2 key candidates for each cryptographic operation, reducing the attack on the $2^{56}$ possible DES keys in half to $2^{55}$.

The main idea for this answer is taken from fgrieu’s answer on Cryptography Stack Exchange.