Cryptanalysis of ForkAES

Abstract

Forkciphers are a new kind of primitive proposed recently by Andreeva et al. for efficient encryption and authentication of small messages. They fork the middle state of a cipher and encrypt it twice under two smaller independent permutations. Thus, forkciphers produce two output blocks in one primitive call.

Andreeva et al. proposed ForkAES, a tweakable AES-based forkcipher that splits the state after five out of ten rounds. While their authenticated encrypted schemes were accompanied by proofs, the security discussion for ForkAES was not provided, and founded on existing results on the AES and KIASU-BC. Forkciphers provide a unique interface called reconstruction queries that use one ciphertext block as input and compute the respective other ciphertext block. Thus, they deserve a careful security analysis.

This work fosters the understanding of the security of ForkAES with three contributions: (1) We observe that security in reconstruction queries differs strongly from the existing results on the AES. This allows to attack nine out of ten rounds with differential, impossible-differential and yoyo attacks. (2) We observe that some forkcipher modes may lack the interface of reconstruction queries, so that attackers must use encryption queries. We show that nine rounds can still be attacked with rectangle and impossible-differential attacks. (3) We present forgery attacks on the AE modes proposed by Andreeva et al. with nine-round ForkAES.

A Previous Yoyo Game

The yoyo game was introduced by Biham et al. for the cryptanalysis of Skipjack [3]. Recently, Rønjom et al. [20] reported a deterministic distinguisher for two generic Substitution-Permutation (SP) rounds. This result has been applied to eight-round ForkAES to perform a key-recovery attack. Let us look at some definitions originally introduced in [20]. Let $F:\mathbb {F}^n_{q} \rightarrow \mathbb {F}^n_{q}$ be a generic permutation where $q=2^k$. Then, F is given by $F = S \circ L \circ S \circ L \circ S$, where S is a concatenation of n parallel S-Boxes on n individual words from $\mathbb {F}_q$ and L denotes the linear layer over $\mathbb {F}^n_{q}$. A vector of words $\alpha =(\alpha _0,\alpha _1,\cdots ,\alpha _{n-1}) \in \mathbb {F}^n_q$ forms the states. The Zero-difference Pattern is defined as:

Definition 1

(Zero-difference Pattern [20]). Let, $\alpha \in \mathbb {F}_{q}^{n}$ for $q=2^{k}$. The Zero-difference Pattern for $\alpha$ is $\nu (\alpha )=(z_0,z_1,...,z_{n-1})$, where $\nu (\alpha )$ takes values in $\mathbb {F}_{2}^{n}$ and $z_i=1$ if $\alpha _i=0$ or $z_i=0$ otherwise.

The weight $wt(\nu (\alpha ))$ refers to the number of active words in $\alpha$. The Yoyo game depends then on the swapping of words among the texts. The following definition describes the swapping mechanism.

Definition 2

(Word Swapping [20]). Let, $\alpha ,\beta \in \mathbb {F}_{q}^{n}$ be two states and $v \in \mathbb {F}_{2}^{n}$ be a vector, then $\rho ^{v}(\alpha ,\beta )$ is a new state in $\mathbb {F}_{q}^{n}$ created from $\alpha , \beta$ by swapping components among them. The i-th component of $\rho ^{v}(\alpha ,\beta ) = \alpha _i$ if $v_i = 1$ and $\rho ^{v}(\alpha ,\beta ) = \beta _i$ otherwise.

Yoyo Distinguisher for Two Generic SP Rounds. Two generic SP rounds can be written as $G_{2} = L \circ S \circ L \circ S$ where the final L layer can be omitted since it does not affect the security. Also, the substitution layers do not have to be equal. After modification, $G_{2} = S_1 \circ L \circ S_2$. The deterministic distinguisher for two generic SP rounds is described by the following theorem.

Theorem 1

(The Yoyo Game [20]). Let, $p^0,p^1 \in \mathbb {F}_{q}^{n}$, $c^0=G_{2}(p^0)$ and $c^1=G_{2}(p^1)$. For any vector $v \in \mathbb {F}_{2}^{n}$, $c^{'0}=\rho ^{v}(c^0,c^1)$ and $c^{'1}=\rho ^{v}(c^1,c^0)$. Then

$\nu (G_2^{-1}(c^{'0}) \oplus G_2^{-1}(c^{'1})) =\nu (p^{'0} \oplus p^{'1}) =\nu (p^{0} \oplus p^{1})$.