International Association for Cryptologic Research

International Association
for Cryptologic Research

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Papers from Transaction on Symmetric Cryptology 2018

Year
Venue
Title
2018
TOSC
Towards Low Energy Stream Ciphers 📺
Energy optimization is an important design aspect of lightweight cryptography. Since low energy ciphers drain less battery, they are invaluable components of devices that operate on a tight energy budget such as handheld devices or RFID tags. At Asiacrypt 2015, Banik et al. presented the block cipher family Midori which was designed to optimize the energy consumed per encryption and which reduces the energy consumption by more than 30% compared to previous block ciphers. However, if one has to encrypt/decrypt longer streams of data, i.e. for bulk data encryption/decryption, it is expected that a stream cipher should perform even better than block ciphers in terms of energy required to encrypt. In this paper, we address the question of designing low energy stream ciphers. To this end, we analyze for common stream cipher design components their impact on the energy consumption. Based on this, we give arguments why indeed stream ciphers allow for encrypting long data streams with less energy than block ciphers and validate our findings by implementations. Afterwards, we use the analysis results to identify energy minimizing design principles for stream ciphers.
2018
TOSC
SUNDAE: Small Universal Deterministic Authenticated Encryption for the Internet of Things 📺
Lightweight cryptography was developed in response to the increasing need to secure devices for the Internet of Things. After significant research effort, many new block ciphers have been designed targeting lightweight settings, optimizing efficiency metrics which conventional block ciphers did not. However, block ciphers must be used in modes of operation to achieve more advanced security goals such as data confidentiality and authenticity, a research area given relatively little attention in the lightweight setting. We introduce a new authenticated encryption (AE) mode of operation, SUNDAE, specially targeted for constrained environments. SUNDAE is smaller than other known lightweight modes in implementation area, such as CLOC, JAMBU, and COFB, however unlike these modes, SUNDAE is designed as a deterministic authenticated encryption (DAE) scheme, meaning it provides maximal security in settings where proper randomness is hard to generate, or secure storage must be minimized due to expense. Unlike other DAE schemes, such as GCM-SIV, SUNDAE can be implemented efficiently on both constrained devices, as well as the servers communicating with those devices. We prove SUNDAE secure relative to its underlying block cipher, and provide an extensive implementation study, with results in both software and hardware, demonstrating that SUNDAE offers improved compactness and power consumption in hardware compared to other lightweight AE modes, while simultaneously offering comparable performance to GCM-SIV on parallel high-end platforms.
2018
TOSC
The design of Xoodoo and Xoofff 📺
This paper presents Xoodoo, a 48-byte cryptographic permutation with excellent propagation properties. Its design approach is inspired by Keccak-p, while it is dimensioned like Gimli for efficiency on low-end processors. The structure consists of three planes of 128 bits each, which interact per 3-bit columns through mixing and nonlinear operations, and which otherwise move as three independent rigid objects. We analyze its differential and linear propagation properties and, in particular, prove lower bounds on the weight of trails using the tree search-based technique of Mella et al. (ToSC 2017). Xoodoo’s primary target application is in the Farfalle construction that we instantiate for the doubly-extendable cryptographic keyed (or deck) function Xoofff. Combining a relatively narrow permutation with the parallelism of Farfalle results in very efficient schemes on a wide range of platforms, from low-end devices to high-end processors with vector instructions.
2018
TOSC
2018
TOSC
Multivariate Profiling of Hulls for Linear Cryptanalysis
Extensions of linear cryptanalysis making use of multiple approximations, such as multiple and multidimensional linear cryptanalysis, are an important tool in symmetric-key cryptanalysis, among others being responsible for the best known attacks on ciphers such as Serpent and present. At CRYPTO 2015, Huang et al. provided a refined analysis of the key-dependent capacity leading to a refined key equivalence hypothesis, however at the cost of additional assumptions. Their analysis was extended by Blondeau and Nyberg to also cover an updated wrong key randomization hypothesis, using similar assumptions. However, a recent result by Nyberg shows the equivalence of linear dependence and statistical dependence of linear approximations, which essentially invalidates a crucial assumption on which all these multidimensional models are based. In this paper, we develop a model for linear cryptanalysis using multiple linearly independent approximations which takes key-dependence into account and complies with Nyberg’s result. Our model considers an arbitrary multivariate joint distribution of the correlations, and in particular avoids any assumptions regarding normality. The analysis of this distribution is then tailored to concrete ciphers in a practically feasible way by combining a signal/noise decomposition approach for the linear hulls with a profiling of the actual multivariate distribution of the signal correlations for a large number of keys, thereby entirely avoiding assumptions regarding the shape of this distribution. As an application of our model, we provide an attack on 26 rounds of present which is faster and requires less data than previous attacks, while using more realistic assumptions and far fewer approximations. We successfully extend the attack to present the first 27-round attack which takes key-dependence into account.
2018
TOSC
New Yoyo Tricks with AES-based Permutations 📺
In Asiacrypt 2017, Rønjom et al. reported some interesting generic properties of SPNs, leading to what they call the Yoyo trick, and applied it to find the most efficient distinguishers on AES. In this work, we explore the Yoyo idea in distinguishing public permutations for the first time. We introduce the notion of nested zero difference pattern which extends the Yoyo idea and helps to compose it using improbable and impossible differential strategies to penetrate higher number of rounds. We devise a novel inside-out application of Yoyo which enables us to start the Yoyo game from an internal round. As an application, we investigate the AES-based public permutation AESQ used inside the authenticated cipher PAEQ. We achieve the first deterministic distinguisher of AESQ up to 8 rounds and the first 9-round distinguisher of AESQ that start from the first round with a practical complexity of around 226. We manage to augment Yoyo with improbable and impossible differentials leading to distinguishers on 9, 10, 12 rounds with complexities of about 22, 228, 2126 respectively. Further, with impossible differentials and a bi-directional Yoyo strategy, we obtain a 16-round impossible differential distinguisher with a complexity of 2126. Our results outperform all previous records on AESQ by a substantial margin. As another application, we apply the proposed strategies on AES in the known-key setting leading to one of the best 8-round known-key distinguisher with a complexity of 230. Finally, this work amplifies the scope of the Yoyo technique as a generic cryptanalysis tool.
2018
TOSC
Clustering Related-Tweak Characteristics: Application to MANTIS-6 📺
The TWEAKEY/STK construction is an increasingly popular approach for designing tweakable block ciphers that notably uses a linear tweakey schedule. Several recent attacks have analyzed the implications of this approach for differential cryptanalysis and other attacks that can take advantage of related tweakeys. We generalize the clustering approach of a recent differential attack on the tweakable block cipher MANTIS5 and describe a tool for efficiently finding and evaluating such clusters. More specifically, we consider the set of all differential characteristics compatible with a given truncated characteristic, tweak difference, and optional constraints for the differential. We refer to this set as a semi-truncated characteristic and estimate its probability by analyzing the distribution of compatible differences at each step. We apply this approach to find a semi-truncated differential characteristic for MANTIS6 with probability about 2−67.73 and derive a key-recovery attack with a complexity of about 255.09 chosen-plaintext queries and 255.52 computations. The data-time product is 2110.61 << 2126.
2018
TOSC
Cryptanalysis of Reduced round SKINNY Block Cipher 📺
SKINNY is a family of lightweight tweakable block ciphers designed to have the smallest hardware footprint. In this paper, we present zero-correlation linear approximations and the related-tweakey impossible differential characteristics for different versions of SKINNY .We utilize Mixed Integer Linear Programming (MILP) to search all zero-correlation linear distinguishers for all variants of SKINNY, where the longest distinguisher found reaches 10 rounds. Using a 9-round characteristic, we present 14 and 18-round zero correlation attacks on SKINNY-64-64 and SKINNY- 64-128, respectively. Also, for SKINNY-n-n and SKINNY-n-2n, we construct 13 and 15-round related-tweakey impossible differential characteristics, respectively. Utilizing these characteristics, we propose 23-round related-tweakey impossible differential cryptanalysis by applying the key recovery attack for SKINNY-n-2n and 19-round attack for SKINNY-n-n. To the best of our knowledge, the presented zero-correlation characteristics in this paper are the first attempt to investigate the security of SKINNY against this attack and the results on the related-tweakey impossible differential attack are the best reported ones.
2018
TOSC
Column Parity Mixers
We present column parity mixers (CPM), a generalization of the Θ mixing layer that is used in Keccak. Thanks to our description using matrix arithmetic, we can easily derive algebraic, diffusion, and mask propagation properties, leading to a surprising distinction between two types of CPMs. We compare CPMs to other popular types of mixing layers and argue that CPMs can be more efficient. While Keccak has a bit-oriented structure, we make the case that CPMs are also suitable for nibble- or byte-oriented designs. We outline a general substitution-permutation-network-based design strategy using a CPM, for which we show how one can attain strong bounds for differential and linear trails. We apply this strategy concretely to design a 256-bit permutation with an efficient inverse and strong trail bounds. Our permutation design uses a number of ideas that are of independent interest and allows a fast bitsliced implementation that compares quite well with other established ciphers and permutations.
2018
TOSC
Key Prediction Security of Keyed Sponges 📺
The keyed sponge is a well-accepted method for message authentication. It processes data at a certain rate by sequential evaluation of an underlying permutation. If the key size k is smaller than the rate, currently known bounds are tight, but if it exceeds the rate, state of the art only dictates security up to 2k/2. We take closer inspection at the key prediction security of the sponge and close the remaining gap in the existing security analysis: we confirm key security up to close to 2k, regardless of the rate. The result impacts all applications of the keyed sponge and duplex that process at a rate smaller than the key size, including the STROBE protocol framework, as well as the related constructions such as HMAC-SHA-3 and the sandwich sponge.
2018
TOSC
Mixture Differential Cryptanalysis: a New Approach to Distinguishers and Attacks on round-reduced AES 📺
At Eurocrypt 2017 the first secret-key distinguisher for 5-round AES - based on the “multiple-of-8” property - has been presented. Although it allows to distinguish a random permutation from an AES-like one, it seems rather hard to implement a key-recovery attack different than brute-force like using such a distinguisher. In this paper we introduce “Mixture Differential Cryptanalysis” on round-reduced AESlike ciphers, a way to translate the (complex) “multiple-of-8” 5-round distinguisher into a simpler and more convenient one (though, on a smaller number of rounds). Given a pair of chosen plaintexts, the idea is to construct new pairs of plaintexts by mixing the generating variables of the original pair of plaintexts. Here we theoretically prove that for 4-round AES the corresponding ciphertexts of the original pair of plaintexts lie in a particular subspace if and only if the corresponding pairs of ciphertexts of the new pairs of plaintexts have the same property. Such secret-key distinguisher - which is independent of the secret-key, of the details of the S-Box and of the MixColumns matrix (except for the branch number equal to 5) - can be used as starting point to set up new key-recovery attacks on round-reduced AES. Besides a theoretical explanation, we also provide a practical verification both of the distinguisher and of the attack.
2018
TOSC
Key Assignment Scheme with Authenticated Encryption 📺
The Key Assignment Scheme (KAS) is a well-studied cryptographic primitive used for hierarchical access control (HAC) in a multilevel organisation where the classes of people with higher privileges can access files of those with lower ones. Our first contribution is the formalization of a new cryptographic primitive, namely, KAS-AE that supports the aforementioned HAC solution with an additional authenticated encryption property. Next, we present three efficient KAS-AE schemes that solve the HAC and the associated authenticated encryption problem more efficiently – both with respect to time and memory – than the existing solutions that achieve it by executing KAS and AE separately. Our first KAS-AE construction is built by using the cryptographic primitive MLE (EUROCRYPT 2013) as a black box; the other two constructions (which are the most efficient ones) have been derived by cleverly tweaking the hash function FP (Indocrypt 2012) and the authenticated encryption scheme APE (FSE 2014). This high efficiency of our constructions is critically achieved by using two techniques: design of a mechanism for reverse decryption used for reduction of time complexity, and a novel key management scheme for optimizing storage requirements when organizational hierarchy forms an arbitrary access graph (instead of a linear graph). We observe that constructing a highly efficient KAS-AE scheme using primitives other than MLE, FP and APE is a non-trivial task. We leave it as an open problem. Finally, we provide a detailed comparison of all the KAS-AE schemes.
2018
TOSC
On the Generalization of Butterfly Structure
Butterfly structure was proposed in CRYPTO 2016 [PUB16], and it can generate permutations over
2018
TOSC
Cryptanalysis of AES-PRF and Its Dual 📺
A dedicated pseudorandom function (PRF) called AES-PRF was proposed by Mennink and Neves at FSE 2018 (ToSC 2017, Issue 3). AES-PRF is obtained from AES by using the output of the 5-th round as the feed-forward to the output state. This paper presents extensive security analysis of AES-PRF and its variants. Specifically, we consider unbalanced variants where the output of the s-th round is used as the feed-forward. We also analyze the security of “dual” constructions of the unbalanced variants, where the input state is used as the feed-forward to the output of the s-th round. We apply an impossible differential attack, zero-correlation linear attack, traditional differential attack, zero correlation linear distinguishing attack and a meet-in-the-middle attack on these PRFs and reduced round versions. We show that AES-PRF is broken whenever s ≤ 2 or s ≥ 6, or reduced to 7 rounds, and Dual-AES-PRF is broken whenever s ≤ 4 or s ≥ 8. Our results on AES-PRF improve the initial security evaluation by the designers in various ways, and our results on Dual-AES-PRF give the first insight to its security.
2018
TOSC
Cryptanalysis of Low-Data Instances of Full LowMCv2 📺
LowMC is a family of block ciphers designed for a low multiplicative complexity. The specification allows a large variety of instantiations, differing in block size, key size, number of S-boxes applied per round and allowed data complexity. The number of rounds deemed secure is determined by evaluating a number of attack vectors and taking the number of rounds still secure against the best of these. In this paper, we demonstrate that the attacks considered by the designers of LowMC in the version 2 of the round-formular were not sufficient to fend off all possible attacks. In the case of instantiations of LowMC with one of the most useful settings, namely with few applied S-boxes per round and only low allowable data complexities, efficient attacks based on difference enumeration techniques can be constructed. We show that it is most effective to consider tuples of differences instead of simple differences, both to increase the range of the distinguishers and to enable key recovery attacks. All applications for LowMC we are aware of, including signature schemes like Picnic and more recent (ring/group) signature schemes have used version 3 of the roundformular for LowMC, which takes our attack already into account.
2018
TOSC
On Efficient Constructions of Lightweight MDS Matrices
The paper investigates the maximum distance separable (MDS) matrix over the matrix polynomial residue ring. Firstly, by analyzing the minimal polynomials of binary matrices with 1 XOR count and element-matrices with few XOR counts, we present an efficient method for constructing MDS matrices with as few XOR counts as possible. Comparing with previous constructions, our corresponding constructions only cost 1 minute 27 seconds to 7 minutes, while previous constructions cost 3 days to 4 weeks. Secondly, we discuss the existence of several types of involutory MDS matrices and propose an efficient necessary-and-sufficient condition for identifying a Hadamard matrix being involutory. According to the condition, each involutory Hadamard matrix over a polynomial residue ring can be accurately and efficiently searched. Furthermore, we devise an efficient algorithm for constructing involutory Hadamard MDS matrices with as few XOR counts as possible. We obtain many new involutory Hadamard MDS matrices with much fewer XOR counts than optimal results reported before.
2018
TOSC
Cube-Attack-Like Cryptanalysis of Round-Reduced Keccak Using MILP 📺
Cube-attack-like cryptanalysis on round-reduced Keccak was proposed by Dinur et al. at EUROCRYPT 2015. It recovers the key through two phases: the preprocessing phase for precomputing a look-up table and online phase for querying the output and getting the cube sum with which the right key can be retrieved by looking up the precomputed table. It was shown that such attacks are efficient specifically for Keccak-based constructions with small nonce or message block size. In this paper, we provide a mixed integer linear programming (MILP) model for cubeattack- like cryptanalysis on keyed Keccak, which does not impose any unnecessary constraint on cube variables and finds almost optimal cubes by balancing the two phases of cube-attack-like cryptanalysis. Our model is applied to Ketje Jr, Ketje Sr, a Xoodoo-based authenticated encryption and Keccak-MAC-512, all of which have a relatively small nonce or message block size. As a result, time complexities of 5-round attacks on Ketje Jr and 7-round attacks on Ketje Sr can be improved significantly. Meanwhile, 6-round attacks, one more round than the previous best attack, are possible if the key size of Ketje V1 (V2) is reduced to 72 (80) bits. For Xoodoo-based AE in Ketje style, the attack reaches 6 rounds. Additionally, a 7-round attack of Keccak-MAC-512 is achieved. To verify the correctness of our attacks, a 5-round attack on Ketje V1 is implemented and tested practically. It is noted that this work does not threaten the security of any Keccak-based construction.
2018
TOSC
Sound Hashing Modes of Arbitrary Functions, Permutations, and Block Ciphers 📺
Cryptographic hashing modes come in many flavors, including Merkle-Damgård with various types of strengthening, Merkle trees, and sponge functions. As underlying primitives, these functions use arbitrary functions, permutations, or block ciphers. In this work we provide three simple proofs, one per primitive type, that cover all modes where the input to the primitive consists of message bits, chaining value bits, and bits that only depend on the mode and message length. Our approach generalizes and simplifies over earlier attempts of Dodis et al. (FSE 2009) and Bertoni et al. (Int. J. Inf. Sec. 2014). We prove tight indifferentiability bounds for modes using each of these three primitive types provided that the mode satisfies some easy to verify conditions.
2018
TOSC
ShiftRows Alternatives for AES-like Ciphers and Optimal Cell Permutations for Midori and Skinny 📺
We study possible alternatives for ShiftRows to be used as cell permutations in AES-like ciphers. As observed during the design process of the block cipher Midori, when using a matrix with a non-optimal branch number for the MixColumns operation, the choice of the cell permutation, i.e., an alternative for ShiftRows, can actually improve the security of the primitive. In contrast, when using an MDS matrix it is known that one cannot increase the minimum number of active S-boxes by deviating from the ShiftRows-type permutation. However, finding the optimal choice for the cell permutation for a given, non-optimal, MixColumns operation is a highly non-trivial problem. In this work, we propose techniques to speed up the search for the optimal cell permutations significantly. As case studies, we apply those techniques to Midori and Skinny and provide possible alternatives for their cell permutations. We finally state an easy-to-verify sufficient condition on a cell permutation, to be used as an alternative in Midori, that attains a high number of active S-boxes and thus provides good resistance against differential and linear attacks.
2018
TOSC
Functional Graphs and Their Applications in Generic Attacks on Iterated Hash Constructions
We provide a survey about generic attacks on cryptographic hash constructions including hash-based message authentication codes and hash combiners. We look into attacks involving iteratively evaluating identical mappings many times. The functional graph of a random mapping also involves iteratively evaluating the mapping. These attacks essentially exploit properties of the functional graph. We map the utilization space of those properties from numerous proposed known attacks, draw a comparison among classes of attacks about their advantages and limitations. We provide a systematic exposition of concepts of cycles, deep-iterate images, collisions and their roles in cryptanalysis of iterated hash constructions. We identify the inherent relationship between these concepts, such that case-by-case theories about them can be unified into one knowledge system, that is, theories on the functional graph of random mappings. We show that the properties of the cycle search algorithm, the chain evaluation algorithm and the collision search algorithm can be described based on statistic results on the functional graph. Thereby, we can provide different viewpoints to support previous beliefs on individual knowledge. In that, we invite more sophisticated analysis of the functional graph of random mappings and more future exploitations of its properties in cryptanalysis.
2018
TOSC
Conditional Linear Cryptanalysis – Cryptanalysis of DES with Less Than 242 Complexity 📺
In this paper we introduce a new extension of linear cryptanalysis that may reduce the complexity of attacks by conditioning linear approximations on other linear approximations. We show that the bias of some linear approximations may increase under such conditions, so that after discarding the known plaintexts that do not satisfy the conditions, the bias of the remaining known plaintexts increases. We show that this extension can lead to improvements of attacks, which may require fewer known plaintexts and time of analysis. We present several types of such conditions, including one that is especially useful for the analysis of Feistel ciphers. We exemplify the usage of such conditions for attacks by a careful application of our extension to Matsui’s attack on the full 16-round DES, which succeeds to reduce the complexity of the best attack on DES to less than 242. We programmed a test implementation of our attack and verified our claimed results with a large number of runs. We also introduce a new type of approximations, to which we call scattered approximations, and discuss its applications.
2018
TOSC
Revisiting and Improving Algorithms for the 3XOR Problem
The 3SUM problem is a well-known problem in computer science and many geometric problems have been reduced to it. We study the 3XOR variant which is more cryptologically relevant. In this problem, the attacker is given black-box access to three random functions F,G and H and she has to find three inputs x, y and z such that F(x) ⊕ G(y) ⊕ H(z) = 0. The 3XOR problem is a difficult case of the more-general k-list birthday problem. Wagner’s celebrated k-list birthday algorithm, and the ones inspired by it, work by querying the functions more than strictly necessary from an information-theoretic point of view. This gives some leeway to target a solution of a specific form, at the expense of processing a huge amount of data. However, to handle such a huge amount of data can be very difficult in practice. This is why we first restricted our attention to solving the 3XOR problem for which the total number of queries to F, G and H is minimal. If they are n-bit random functions, it is possible to solve the problem with roughly
2018
TOSC
Generating Graphs Packed with Paths Estimation of Linear Approximations and Differentials 📺
When designing a new symmetric-key primitive, the designer must show resistance to known attacks. Perhaps most prominent amongst these are linear and differential cryptanalysis. However, it is notoriously difficult to accurately demonstrate e.g. a block cipher’s resistance to these attacks, and thus most designers resort to deriving bounds on the linear correlations and differential probabilities of their design. On the other side of the spectrum, the cryptanalyst is interested in accurately assessing the strength of a linear or differential attack.While several tools have been developed to search for optimal linear and differential trails, e.g. MILP and SAT based methods, only few approaches specifically try to find as many trails of a single approximation or differential as possible. This can result in an overestimate of a cipher’s resistance to linear and differential attacks, as was for example the case for PRESENT.In this work, we present a new algorithm for linear and differential trail search. The algorithm represents the problem of estimating approximations and differentials as the problem of finding many long paths through a multistage graph. We demonstrate that this approach allows us to find a very large number of good trails for each approximation or differential. Moreover, we show how the algorithm can be used to efficiently estimate the key dependent correlation distribution of a linear approximation, facilitating advanced linear attacks. We apply the algorithm to 17 different ciphers, and present new and improved results on several of these.
2018
TOSC
Frequency-smoothing encryption: preventing snapshot attacks on deterministically encrypted data
Statistical analysis of ciphertexts has been recently used to carry out devastating inference attacks on deterministic encryption (Naveed, Kamara, and Wright, CCS 2015), order-preserving/revealing encryption (Grubbs et al., S&P 2017), and searchable encryption (Pouliot and Wright, CCS 2016). At the heart of these inference attacks is classical frequency analysis. In this paper, we propose and evaluate another classical technique, homophonic encoding, as a means to combat these attacks. We introduce and develop the concept of frequency-smoothing encryption (FSE) which provably prevents inference attacks in the snapshot attack model, wherein the adversary obtains a static snapshot of the encrypted data, while preserving the ability to efficiently and privately make point queries. We provide provably secure constructions for FSE schemes, and we empirically assess their security for concrete parameters by evaluating them against real data. We show that frequency analysis attacks (and optimal generalisations of them for the FSE setting) no longer succeed.
2018
TOSC
State-Recovery Attacks on Modified Ketje Jr
In this article we study the security of the authenticated encryption algorithm Ketje against divide-and-conquer attacks. Ketje is a third-round candidate in the ongoing CAESAR competition, which shares most of its design principles with the SHA-3 hash function. Several versions of Ketje have been submitted, with different sizes for its internal state. We describe several state-recovery attacks on the smaller variant, called Ketje Jr. We show that if one increases the amount of keystream output after each round from 16 bits to 40 bits, Ketje Jr becomes vulnerable to divide-and-conquer attacks with time complexities 271.5 for the original version and 282.3 for the current tweaked version, both with a key of 96 bits. We also propose a similar attack when considering rates of 32 bits for the non-tweaked version. Our findings do not threaten the security of Ketje, but should be taken as a warning against potential future modifications that would aim at increasing the performance of the algorithm.
2018
TOSC
On the Boomerang Uniformity of Cryptographic Sboxes 📺
The boomerang attack is a cryptanalysis technique against block ciphers which combines two differentials for the upper part and the lower part of the cipher. The dependency between these two differentials then highly affects the complexity of the attack and all its variants. Recently, Cid et al. introduced at Eurocrypt’18 a new tool, called the Boomerang Connectivity Table (BCT) that permits to simplify this complexity analysis, by storing and unifying the different switching probabilities of the cipher’s Sbox in one table. In this seminal paper a brief analysis of the properties of these tables is provided and some open questions are raised. It is being asked in particular whether Sboxes with optimal BCTs exist for even dimensions, where optimal means that the maximal value in the BCT equals the lowest known differential uniformity. When the dimension is even and differs from 6, such optimal Sboxes correspond to permutations such that the maximal value in their DDT and in their BCT equals 4 (unless APN permutations for such dimensions exist). We provide in this work a more in-depth analysis of boomerang connectivity tables, by studying more closely differentially 4-uniform Sboxes. We first completely characterize the BCT of all differentially 4-uniform permutations of 4 bits and then study these objects for some cryptographically relevant families of Sboxes, as the inverse function and quadratic permutations. These two families provide us with the first examples of differentially 4-uniform Sboxes optimal against boomerang attacks for an even number of variables, answering the above open question.
2018
TOSC
Lightweight and Side-channel Secure 4 × 4 S-Boxes from Cellular Automata Rules 📺
This work focuses on side-channel resilient design strategies for symmetrickey cryptographic primitives targeting lightweight applications. In light of NIST’s lightweight cryptography project, design choices for block ciphers must consider not only security against traditional cryptanalysis, but also side-channel security, while adhering to low area and power requirements. In this paper, we explore design strategies for substitution-permutation network (SPN)-based block ciphers that make them amenable to low-cost threshold implementations (TI) - a provably secure strategy against side-channel attacks. The core building blocks for our strategy are cryptographically optimal 4×4 S-Boxes, implemented via repeated iterations of simple cellular automata (CA) rules. We present highly optimized TI circuits for such S-Boxes, that consume nearly 40% less area and power as compared to popular lightweight S-Boxes such as PRESENT and GIFT. We validate our claims via implementation results on ASIC using 180nm technology. We also present a comparison of TI circuits for two popular lightweight linear diffusion layer choices - bit permutations and MixColumns using almost-maximum-distance-separable (almost-MDS) matrices. We finally illustrate design paradigms that combine the aforementioned TI circuits for S-Boxes and diffusion layers to obtain fully side-channel secure SPN block cipher implementations with low area and power requirements.
2018
TOSC
Revisiting Variable Output Length XOR Pseudorandom Function
Let σ be some positive integer and C ⊆ {(i, j) : 1 ≤ i < j ≤ σ}. The theory behind finding a lower bound on the number of distinct blocks P1, . . . , Pσ ∈ {0, 1}n satisfying a set of linear equations {Pi ⊕Pj = ci,j : (i, j) ∈ C} for some ci,j ∈ {0, 1}n, is called mirror theory. Patarin introduced the mirror theory and provided a proof for this. However, the proof, even for a special class of equations, is complex and contains several non-trivial gaps. As an application of mirror theory, XORP[w] (known as XOR construction) returning (w−1) block output, is a pseudorandom function (PRF) for some parameter w, called width. The XOR construction can be seen as a basic structure of some encryption algorithms, e.g., the CENC encryption and the CHM authenticated encryption, proposed by Iwata in 2006. Due to potential application of XORP[w] and the nontrivial gaps in the proof of mirror theory, an alternative simpler analysis of PRF-security of XORP[w] would be much desired. Recently (in Crypto 2017) Dai et al. introduced a tool, called the χ2 method, for analyzing PRF-security. Using this tool, the authors have provided a proof of PRF-security of XORP[2] without relying on the mirror theory. In this paper, we resolve the general case; we apply the χ2 method to obtain a simpler security proof of XORP[w] for any w ≥ 2. For w = 2, we obtain a tighter bound for a wider range of parameters than that of Dai et al.. Moreover, we consider variable width construction XORP[∗] (in which the widths are chosen by adversaries adaptively), and also provide variable output length pseudorandom function (VOLPRF) security analysis for it. As an application of VOLPRF, we propose an authenticated encryption which is a simple variant of CHM or AES-GCM and provides much higher security than those at the cost of one extra blockcipher call for every message.
2018
TOSC
Short Non-Malleable Codes from Related-Key Secure Block Ciphers
A non-malleable code is an unkeyed randomized encoding scheme that offers the strong guarantee that decoding a tampered codeword either results in the original message, or in an unrelated message. We consider the simplest possible construction in the computational split-state model, which simply encodes a message m as k||Ek(m) for a uniformly random key k, where E is a block cipher. This construction is comparable to, but greatly simplifies over, the one of Kiayias et al. (ACM CCS 2016), who eschewed this simple scheme in fear of related-key attacks on E. In this work, we prove this construction to be a strong non-malleable code as long as E is (i) a pseudorandom permutation under leakage and (ii) related-key secure with respect to an arbitrary but fixed key relation. Both properties are believed to hold for “good” block ciphers, such as AES-128, making this non-malleable code very efficient with short codewords of length |m|+2τ (where τ is the security parameter, e.g., 128 bits), without significant security penalty.
2018
TOSC
Double-block Hash-then-Sum: A Paradigm for Constructing BBB Secure PRF 📺
SUM-ECBC (Yasuda, CT-RSA 2010) is the first beyond birthday bound (BBB) secure block cipher based deterministic MAC. After this work, some more BBB secure deterministic MACs have been proposed, namely PMAC_Plus (Yasuda, CRYPTO 2011), 3kf9 (Zhang et al., ASIACRYPT 2012) and LightMAC_Plus (Naito, ASIACRYPT 2017). In this paper, we have abstracted out the inherent design principle of all these BBB secure MACs and present a generic design paradigm to construct a BBB secure pseudo random function, namely Double-block Hash-then- Sum or in short (DbHtS). A DbHtS construction, as the name implies, computes a double block hash on the message and then sum the encrypted output of the two hash blocks. Our result renders that if the underlying hash function meets certain security requirements (namely cover-free and block-wise universal advantage is low), DbHtS construction provides 2n/3-bit security. We demonstrate the applicability of our result by instantiating all the existing beyond birthday secure deterministic MACs (e.g., SUM-ECBC, PMAC_Plus, 3kf9, LightMAC_Plus) as well as a simple two-keyed variant for each of them and some algebraic hash based constructions.
2018
TOSC
Adiantum: length-preserving encryption for entry-level processors 📺
We present HBSH, a simple construction for tweakable length-preserving encryption which supports the fastest options for hashing and stream encryption for processors without AES or other crypto instructions, with a provable quadratic advantage bound. Our composition Adiantum uses NH, Poly1305, XChaCha12, and a single AES invocation. On an ARM Cortex-A7 processor, Adiantum decrypts 4096-byte messages at 10.6 cycles per byte, over five times faster than AES-256-XTS, with a constant-time implementation. We also define HPolyC which is simpler and has excellent key agility at 13.6 cycles per byte.
2018
TOSC
MDS Matrices with Lightweight Circuits 📺
MDS matrices are an important element for the design of block ciphers such as the AES. In recent years, there has been a lot of work on the construction of MDS matrices with a low implementation cost, in the context of lightweight cryptography. Most of the previous efforts focused on local optimization, constructing MDS matrices with coefficients that can be efficiently computed. In particular, this led to a matrix with a direct xor count of only 106, while a direct implementation of the MixColumn matrix of the AES requires 152 bitwise xors. More recently, techniques based on global optimization have been introduced, where the implementation can reuse some intermediate variables. In particular, Kranz et al. used optimization tools to find a good implementation from the description of an MDS matrix. They have lowered the cost of implementing the MixColumn matrix to 97 bitwise xors, and proposed a new matrix with only 72 bitwise xors, the lowest cost known so far. In this work we propose a different approach to global optimization. Instead of looking for an optimized circuit of a given matrix, we run a search through a space of circuits, to find optimal circuits yielding MDS matrices. This results in MDS matrices with an even lower cost, with only 67 bitwise xors.
2018
TOSC
Key-Recovery Attacks on Full Kravatte
Best Paper FSE 2018
This paper presents a cryptanalysis of full Kravatte, an instantiation of the Farfalle construction of a pseudorandom function (PRF) with variable input and output length. This new construction, proposed by Bertoni et al., introduces an efficiently parallelizable and extremely versatile building block for the design of symmetric mechanisms, e.g. message authentication codes or stream ciphers. It relies on a set of permutations and on so-called rolling functions: it can be split into a compression layer followed by a two-step expansion layer. The key is expanded and used to mask the inputs and outputs of the construction. Kravatte instantiates Farfalle using linear rolling functions and permutations obtained by iterating the Keccak round function.We develop in this paper several attacks against this PRF, based on three different attack strategies that bypass part of the construction and target a reduced number of permutation rounds. A higher order differential distinguisher exploits the possibility to build an affine space of values in the cipher state after the compression layer. An algebraic meet-in-the-middle attack can be mounted on the second step of the expansion layer. Finally, due to the linearity of the rolling function and the low algebraic degree of the Keccak round function, a linear recurrence distinguisher can be found on intermediate states of the second step of the expansion layer. All the attacks rely on the ability to invert a small number of the final rounds of the construction. In particular, the last two rounds of the construction together with the final masking by the key can be algebraically inverted, which allows to recover the key.The complexities of the devised attacks, applied to the Kravatte specifications published on the IACR ePrint in July 2017, or the strengthened version of Kravatte recently presented at ECC 2017, are far below the security claimed.
2018
TOSC
Distinguishing Attack on NORX Permutation
NORX is a permutation-based authentication scheme which is currently a third-round candidate of the ongoing CAESAR competition. The security bound of NORX is derived from the sponge construction applied to an ideal underlying permutation. In this paper, we show that the NORX core permutation is non-ideal with a new distinguishing attack. More specifically, we can distinguish NORX64 permutation with 248.5 queries and distinguish NORX32 permutation with 264.7 queries using carefully crafted differential-linear attacks. We have experimentally verified the distinguishing attack on NORX64 permutation. Although the distinguishing attacks reveal the weakness of the NORX permutation, it does not directly threat the security of the NORX authenticated encryption scheme.
2018
TOSC
Generalized Nonlinear Invariant Attack and a New Design Criterion for Round Constants 📺
The nonlinear invariant attack was introduced at ASIACRYPT 2016 by Todo et al.. The attack has received extensive attention of cryptographic community due to its practical application on the full-round block ciphers SCREAM, iSCREAM, and Midori64. However, the attack heavily relies on the choice of round constants and it becomes inefficient in the case these constants nonlinearly affect the so-called nonlinear invariants. In this article, to eliminate the impact from the round constants, a generalized nonlinear invariant attack which uses a pair of constants in the input of nonlinear invariants is proposed. The efficiency of this extended framework is practically confirmed by mounting a distinguishing attack on a variant of full-round iSCREAM cipher under a class of 280 weak keys. The considered variant of iSCREAM is however resistant against nonlinear invariant attack of Todo et al.. Furthermore, we investigate the resistance of block ciphers against generalized nonlinear invariant attacks with respect to the choice of round constants in an extended framework. We introduce a useful concept of closed-loop invariants of the substitution box (S-box) and show that the choice of robust round constants is closely related to the existence of linear structure of the closed-loop invariants of the substitution layer. In particular, we demonstrate that the design criteria for the round constants in Beierle et al.’s work at CRYPTO 2017 is not an optimal strategy. The round constants selected using this method may induce certain weaknesses that can be exploited in our generalized nonlinear invariant attack model. This scenario is efficiently demonstrated in the case of a slightly modified variant of the Midori64 block cipher.
2018
TOSC
Searching for Subspace Trails and Truncated Differentials
Grassi et al. [Gra+16] introduced subspace trail cryptanalysis as a generalization of invariant subspaces and used it to give the first five round distinguisher for Aes. While it is a generic method, up to now it was only applied to the Aes and Prince. One problem for a broad adoption of the attack is a missing generic analysis algorithm. In this work we provide efficient and generic algorithms that allow to compute the provably best subspace trails for any substitution permutation cipher.
2018
TOSC
Separable Statistics and Multidimensional Linear Cryptanalysis 📺
Multidimensional linear cryptanalysis of block ciphers is improved in this work by introducing a number of new ideas. Firstly, formulae is given to compute approximate multidimensional distributions of the encryption algorithm internal bits. Conventional statistics like LLR (Logarithmic Likelihood Ratio) do not fit to work in Matsui’s Algorithm 2 for large dimension data, as the observation may depend on too many cipher key bits. So, secondly, a new statistic which reflects the structure of the cipher round is constructed instead. Thirdly, computing the statistic values that will fall into a critical region is presented as an optimisation problem for which an efficient algorithm is suggested. The algorithm works much faster than brute forcing all relevant key bits to compute the statistic. An attack for 16-round DES was implemented. We got an improvement over Matsui’s attack on DES in data and time complexity keeping success probability the same. With 241.81 plaintext blocks and success rate 0.83 (computed theoretically) we found 241.46 (which is close to the theoretically predicted number 241.81) key-candidates to 56-bit DES key. Search tree to compute the statistic values which fall into the critical region incorporated 245.45 nodes in the experiment and that is at least theoretically inferior in comparison with the final brute force. To get success probability 0.85, which is a fairer comparison to Matsui’s results, we would need 241.85 data and to brute force 241.85 key-candidates. That compares favourably with 243 achieved by Matsui.
2018
TOSC
Nonlinear Approximations in Cryptanalysis Revisited 📺
This work studies deterministic and non-deterministic nonlinear approximations for cryptanalysis of block ciphers and cryptographic permutations and embeds it into the well-understood framework of linear cryptanalysis. For a deterministic (i.e., with correlation ±1) nonlinear approximation we show that in many cases, such a nonlinear approximation implies the existence of a highly-biased linear approximation. For non-deterministic nonlinear approximations, by transforming the cipher under consideration by conjugating each keyed instance with a fixed permutation, we are able to transfer many methods from linear cryptanalysis to the nonlinear case. Using this framework we in particular show that there exist ciphers for which some transformed versions are significantly weaker with regard to linear cryptanalysis than their original counterparts.
2018
TOSC
More Accurate Differential Properties of LED64 and Midori64 📺
In differential cryptanalysis, a differential is more valuable than the single trail belonging to it in general. The traditional way to compute the probability of the differential is to sum the probabilities of all trails within it. The automatic tool for the search of differentials based on Mixed Integer Linear Programming (MILP) has been proposed and realises the task of finding multiple trails of a given differential. The problem is whether it is reliable to evaluate the probability of the differential traditionally. In this paper, we focus on two lightweight block ciphers – LED64 and Midori64 and show the more accurate estimation of differential probability considering the key schedule. Firstly, an automated tool based on Boolean Satisfiability Problem (SAT) is put forward to accomplish the automatic search of differentials for ciphers with S-boxes and is applied to LED64 and Midori64. Secondly, we provide an automatic approach to detect the right pairs following a given differential, which can be exploited to calculate the differential property. Applying this technique to the STEP function of LED64, we discover some differentials with enhanced probability. As a result, the previous attacks relying upon high probability differentials can be improved definitely. Thirdly, we present a method to compute an upper-bound of the weak-key ratio for a given differential, which is utilised to analyse 4-round differentials of Midori64. We detect two differentials whose weak-key ratios are much lower than the expected 50%. More than 78% of the keys will make these two differentials being impossible differentials. The idea of the estimation for an upper-bound of the weak-key ratio can be employed for other ciphers and allows us to launch differential attacks more reliably. Finally, we introduce how to compute the enhanced differential probability and evaluate the size of keys achieving the improved probability. Such a property may incur an efficient weak-key attack. For a 4-round differential of Midori64, we obtain an improved differential property for a portion of keys.