Improved quantum attack on Type-1 Generalized Feistel Schemes and Its application to CAST-256

Generalized Feistel Schemes (GFS) are important components of symmetric ciphers, which have been extensively researched in classical setting. However, the security evaluations of GFS in quantum setting are rather scanty. In this paper, we give more improved polynomial-time quantum distinguishers on Type-1 GFS in quantum chosen-plaintext attack (qCPA) setting and quantum chosen-ciphertext attack (qCCA) setting. In qCPA setting, we give new quantum polynomial-time distinguishers on $(3d-3)$-round Type-1 GFS with branches $d\geq3$, which gain $d-2$ more rounds than the previous distinguishers. Hence, we could get better key-recovery attacks, whose time complexities gain a factor of $2^{\frac{(d-2)n}{2}}$. In qCCA setting, we get $(3d-3)$-round quantum distinguishers on Type-1 GFS, which gain $d-1$ more rounds than the previous distinguishers. In addition, we give some quantum attacks on CAST-256 block cipher. We find 12-round and 13-round polynomial-time quantum distinguishers in qCPA and qCCA settings, respectively, while the best previous one is only 7 rounds. Hence, we could derive quantum key-recovery attack on 19-round CAST-256. While the best previous quantum key-recovery attack is on 16 rounds. When comparing our quantum attacks with classical attacks, our result also reaches 16 rounds on CAST-256 with 128-bit key under a competitive complexity

Quantum Distinguishing Attacks against Type-1 Generalized Feistel Ciphers

A generalized Feistel cipher is one of the methods to construct block ciphers, and it has several variants. Dong, Li, and Wang showed quantum distinguishing attacks against the $(2d-1)$-round Type-1 generalized Feistel cipher with quantum chosen-plaintext attacks, where $d\ge 3$, and they also showed key recovery attacks [Dong, Li, Wang. Sci China Inf Sci, 2019, 62(2): 022501]. In this paper, we show a polynomial time quantum distinguishing attack against the $(3d-3)$-round version, i.e., we improve the number of rounds by $(d-2)$. We also show a quantum distinguishing attack against the $(d^2-d+1)$-round version in the quantum chosen-ciphertext setting. We apply these quantum distinguishing attacks to obtain key recovery attacks against Type-1 generalized Feistel ciphers

Quantum cryptanalysis on some Generalized Feistel Schemes

Post-quantum cryptography has attracted much attention from worldwide cryptologists. In ISIT 2010, Kuwakado and Morii gave a quantum distinguisher with polynomial time against 3-round Feistel networks. However, generalized Feistel schemes (GFS) have not been systematically investigated against quantum attacks. In this paper, we study the quantum distinguishers about some generalized Feistel schemes. For $d$-branch Type-1 GFS (CAST256-like Feistel structure), we introduce ($2d-1$)-round quantum distinguishers with polynomial time. For $2d$-branch Type-2 GFS (RC6/CLEFIA-like Feistel structure), we give ($2d+1$)-round quantum distinguishers with polynomial time. Classically, Moriai and Vaudenay proved that a 7-round $4$-branch Type-1 GFS and 5-round $4$-branch Type-2 GFS are secure pseudo-random permutations. Obviously, they are no longer secure in quantum setting. Using the above quantum distinguishers, we introduce generic quantum key-recovery attacks by applying the combination of Simon\u27s and Grover\u27s algorithms recently proposed by Leander and May. We denote $n$ as the bit length of a branch. For $(d^2-d+2)$-round Type-1 GFS with $d$ branches, the time complexity is $2^{(\frac{1}{2}d^2-\frac{3}{2}d+2)\cdot \frac{n}{2}}$, which is better than the quantum brute force search (Grover search) by a factor $2^{(\frac{1}{4}d^2+\frac{1}{4}d)n}$. For $4d$-round Type-2 GFS with $2d$ branches, the time complexity is $2^{{\frac{d^2 n}{2}}}$, which is better than the quantum brute force search by a factor $2^{{\frac{3d^2 n}{2}}}$

Quantum Attacks on Some Feistel Block Ciphers

Post-quantum cryptography has attracted much attention from worldwide cryptologists. However, most research works are related to public-key cryptosystem due to Shor\u27s attack on RSA and ECC ciphers. At CRYPTO 2016, Kaplan et al. showed that many secret-key (symmetric) systems could be broken using a quantum period finding algorithm, which encouraged researchers to evaluate symmetric systems against quantum attackers. In this paper, we continue to study symmetric ciphers against quantum attackers. First, we convert the classical advanced slide attacks (introduced by Biryukov and Wagner) to a quantum one, that gains an exponential speed-up in time complexity. Thus, we could break 2/4K-Feistel and 2/4K-DES in polynomial time. Second, we give a new quantum key-recovery attack on full-round GOST, which is a Russian standard, with $2^{114.8}$ quantum queries of the encryption process, faster than a quantum brute-force search attack by a factor of $2^{13.2}$

Quantum Speed-Up for Multidimensional (Zero Correlation) Linear Distinguishers

This paper shows how to achieve a quantum speed-up for multidimensional (zero correlation) linear distinguishers. A previous work by Kaplan et al. has already shown a quantum quadratic speed-up for one-dimensional linear distinguishers. However, classical linear cryptanalysis often exploits multidimensional approximations to achieve more efficient attacks, and in fact it is highly non-trivial whether Kaplan et al.\u27s technique can be extended into the multidimensional case. To remedy this, we investigate a new quantum technique to speed-up multidimensional linear distinguishers. Firstly, we observe that there is a close relationship between the subroutine of Simon\u27s algorithm and linear correlations via Fourier transform. Specifically, a slightly modified version of Simon\u27s subroutine, which we call Correlation Extraction Algorithm (CEA), can be used to speed-up multidimensional linear distinguishers. CEA also leads to a speed-up for multidimensional zero correlation distinguishers, as well as some integral distinguishers through the correspondence of zero correlation and integral properties shown by Bogdanov et al.~and Sun et al. Furthermore, we observe possibility of a more than quadratic speed-ups for some special types of integral distinguishers when multiple integral properties exist. Especially, we show a single-query distinguisher on a 4-bit cell SPN cipher with the same integral property as 2.5-round AES. Our attacks are the first to observe such a speed-up for classical cryptanalytic techniques without relying on hidden periods or shifts. By replacing the Hadamard transform in CEA with the general quantum Fourier transform, our technique also speeds-up generalized linear distinguishers on an arbitrary finite abelian group

Statistical cryptanalysis of block ciphers

Since the development of cryptology in the industrial and academic worlds in the seventies, public knowledge and expertise have grown in a tremendous way, notably because of the increasing, nowadays almost ubiquitous, presence of electronic communication means in our lives. Block ciphers are inevitable building blocks of the security of various electronic systems. Recently, many advances have been published in the field of public-key cryptography, being in the understanding of involved security models or in the mathematical security proofs applied to precise cryptosystems. Unfortunately, this is still not the case in the world of symmetric-key cryptography and the current state of knowledge is far from reaching such a goal. However, block and stream ciphers tend to counterbalance this lack of "provable security" by other advantages, like high data throughput and ease of implementation. In the first part of this thesis, we would like to add a (small) stone to the wall of provable security of block ciphers with the (theoretical and experimental) statistical analysis of the mechanisms behind Matsui's linear cryptanalysis as well as more abstract models of attacks. For this purpose, we consider the underlying problem as a statistical hypothesis testing problem and we make a heavy use of the Neyman-Pearson paradigm. Then, we generalize the concept of linear distinguisher and we discuss the power of such a generalization. Furthermore, we introduce the concept of sequential distinguisher, based on sequential sampling, and of aggregate distinguishers, which allows to build sub-optimal but efficient distinguishers. Finally, we propose new attacks against reduced-round version of the block cipher IDEA. In the second part, we propose the design of a new family of block ciphers named FOX. First, we study the efficiency of optimal diffusive components when implemented on low-cost architectures, and we present several new constructions of MDS matrices; then, we precisely describe FOX and we discuss its security regarding linear and differential cryptanalysis, integral attacks, and algebraic attacks. Finally, various implementation issues are considered

Quantum Cryptanalysis on Contracting Feistel Structures and Observation on Related-key Settings

In this paper we show several quantum chosen-plaintext attacks (qCPAs) on contracting Feistel structures. In the classical setting, a $d$-branch $r$-round contracting Feistel structure can be shown to be PRP-secure when $d$ is even and $r \geq 2d-1$, meaning it is secure against polynomial-time chosen-plaintext attacks. We propose a polynomial-time qCPA distinguisher on the $d$-branch $(2d-1)$-round contracting Feistel structure, which solves an open problem by Dong et al. In addition, we show a polynomial-time qCPA that recovers the keys of the $d$-branch $r$-round contracting Feistel structure when each round function $F^{(i)}_{k_i}$ has the form $F^{(i)}_{k_i}(x) = F_i(x \oplus k_i)$ for a public random function $F_i$. This is applicable to the Chinese block cipher standard {\texttt{SM4}}, which is a special case where $d=4$. Finally, in addition to quantum attacks under single-key setting, we also show related-key quantum attacks on balanced Feistel structures in the model that adversaries can only control part of the key difference in quantum superposition. Our related-key attacks on balanced Feistel structures can easily be extended to ones on contracting Feistel structures

The need for polymorphic encryption algorithms: A review paper

Current symmetric ciphers including the Advanced Encryption Standard (AES) are deterministic and open. Using standard ciphers is necessary for interoperability. However, it gives the potential opponent significant leverage, as it facilitates all the knowledge and time he needs to design effective attacks. In this review paper, we highlight prominent contributions in the field of symmetric encryption. Furthermore, we shed light on some contributions that aim at mitigating potential threats when using standard symmetric ciphers. Furthermore, we highlight the need for more practical contributions in the direction of polymorphic or multishape ciphers