Boomerang attacks were first introduced by Wagner and allow an adversary to concatenate two high probability differential trails to attack a cipher. This is especially useful if there is a lack of long differentials with sufficient probabilities. The adversary can therefore decompose the encryption function $F$ in two subciphers $f$ and $g$ such that $F = f \circ g$. Then the adversary can search for high probability trails $\Delta \rightarrow \Delta^*$ with probability $p$ for $f$ and $\nabla \rightarrow \nabla^*$ with probability $q$ for $g$. The differential trails can then be combined in a chosen plaintext/adaptive chosen ciphertext attack to mount a boomerang distinguisher and then a key recovery attack based on this distinguisher to recover the secret key.

A boomerang distinguisher |

The basic attack works as follows:

- The adversary chooses a random plaintext $X_1$ and calculates $X_2 = X_1 \oplus \Delta$.
- The adversary requests the ciphertexts for $X_1$ and $X_2$ from an encryption oracle which are $Y_1 = F(X_1)$ and $Y_2 = F(X_2)$
- Calculate ciphertexts $Y_3 = Y_1 \oplus \nabla$ and $Y_4 = Y_2 \oplus \nabla$.
- Request the decryptions of $Y_3$ and $Y_4$ to obtain $X_3 = F^{-1}(Y_3)$ and $X_4 = F^{-1}(Y_4)$.
- If the difference between $X_3$ and $X_4$ is the same as between $X_1$ and $X_2$, namely $\Delta$ we obtain a correct quartett $(X_1, X_2, X_3, X_4)$.

Calculating a correct quartet requires an attacker to consider both plaintext pairs $(X_1, X_2)$ and $(X_3, X_4)$ and results in a total probability of (pq)^2.

For an attack to succeed, for the probability of the boomerang distinguisher it must hold that $(pq) > 2^{n/2}$. For N plaintext pairs, an adversary expects about $N\cdot(pq)^2$ correct quartets in an attack, while there are only $N\cdot2^{-n}$ (where n is the blocksize) correct quartets for an ideal primitive.

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