Fast Software Encryption

We provide new bounds for the pseudo-random function security of keyed sponge constructions. For the case \(c\le b/2\) (c the capacity and b the permutation size), our result improves over all previously-known bounds. A remarkable aspect of our bound is that dependence between capacity and message length is removed, partially solving the open problem posed by Gaži et al. at CRYPTO 2015. Our bound is essentially tight, matching the two types of attacks pointed out by Gaži et al. For the case \(c>b/2\), Gaži et al.’s bound remains the best for the case of single-block output, but for keyed sponges with extendable outputs, our result partly (when query complexity is relatively large) provides better security than Mennink et al.’s bound presented at ASIACRYPT 2015.

Typical AE schemes are supposed to be secure when used as specified. However, they can – and often do – fail miserably when used improperly. As a partial remedy, Rogaway and Shrimpton proposed (nonce-)misuse-resistant AE (MRAE) and the first MRAE scheme SIV (“Synthetic Initialization Vector”). This paper proposes RIV (“Robust Initialization Vector”), which extends the generic SIV construction by an additional call to the internal PRF. RIV inherits the full security assurance from SIV, but unlike SIV and other MRAE schemes, RIV is also provably secure when releasing unverified plaintexts. This follows a recent line of research on “Robust Authenticated Encryption”, similar to the CAESAR candidate AEZ.

An AES-based instantiation of RIV runs at less than 1.5 cpb on current x64 processors. Unlike the proposed instantiation of AEZ, which gains speed by relying on reduced-round AES, our instantiation of RIV is provably secure under the single assumption of the AES being secure.

Spritz is a stream cipher proposed by Rivest and Schuldt at the rump session of CRYPTO 2014. It is intended to be a replacement of the popular RC4 stream cipher. In this paper we propose distinguishing attacks on the full Spritz, based on a short-term bias in the first two bytes of a keystream and a long-term bias in the first two bytes of every cycle of N keystream bytes, where N is the size of the internal permutation. Our attacks are able to distinguish a keystream of the full Spritz from a random sequence with samples of first two bytes produced by \(2^{44.8}\) multiple key-IV pairs or \(2^{60.8}\) keystream bytes produced by a single key-IV pair. These biases are also useful in the event of plaintext recovery in a broadcast attack. In the second part of the paper, we look at a state recovery attack on Spritz, in a special situation when the cipher enters a class of weak states. We determine the probability of encountering such a state, and demonstrate a state recovery algorithm that betters the \(2^{1400}\) step algorithm of Ankele et al. at Latincrypt 2015.

Filter generators are vulnerable to several attacks which have led to well-known design criteria on the Boolean filtering function. However, Rønjom and Cid have observed that a change of the primitive root defining the LFSR leads to several equivalent generators. They usually offer different security levels since they involve filtering functions of the form \(F(x^k)\) where \(k\) is coprime to \((2^n-1)\) and \(n\) denotes the LFSR length. It is proved here that this monomial equivalence does not affect the resistance of the generator against algebraic attacks, while it usually impacts the resistance to correlation attacks. Most importantly, a more efficient attack can often be mounted by considering non-bijective monomial mappings. In this setting, a divide-and-conquer strategy applies based on a search within a multiplicative subgroup of \(\mathbb {F}_{2^n}^*\). Moreover, if the LFSR length \(n\) is not a prime, a fast correlation involving a shorter LFSR can be performed.

In this article, we analyze the circulant structure of generalized circulant matrices to reduce the search space for finding lightweight MDS matrices. We first show that the implementation of circulant matrices can be serialized and can achieve similar area requirement and clock cycle performance as a serial-based implementation. By proving many new properties and equivalence classes for circulant matrices, we greatly reduce the search space for finding lightweight maximum distance separable (MDS) circulant matrices. We also generalize the circulant structure and propose a new class of matrices, called cyclic matrices, which preserve the benefits of circulant matrices and, in addition, have the potential of being self-invertible. In this new class of matrices, we obtain not only the MDS matrices with the least XOR gates requirement for dimensions from \(3 \times 3\) to \(8 \times 8\) in \({\text {GF}}(2^4)\) and \({\text {GF}}(2^8)\), but also involutory MDS matrices which was proven to be non-existence in the class of circulant matrices. To the best of our knowledge, the latter matrices are the first of its kind, which have a similar matrix structure as circulant matrices and are involutory and MDS simultaneously. Compared to the existing best known lightweight matrices, our new candidates either outperform or match them in terms of XOR gates required for a hardware implementation. Notably, our work is generic and independent of the metric for lightweight. Hence, our work is applicable for improving the search for efficient circulant matrices under other metrics besides XOR gates.

In the present paper, we investigate the problem of constructing MDS matrices with as few bit XOR operations as possible. The key contribution of the present paper is constructing MDS matrices with entries in the set of \(m\times m\) non-singular matrices over \(\mathbb {F}_2\) directly, and the linear transformations we used to construct MDS matrices are not assumed pairwise commutative. With this method, it is shown that circulant involutory MDS matrices, which have been proved do not exist over the finite field \(\mathbb {F}_{2^m}\), can be constructed by using non-commutative entries. Some constructions of \(4\times 4\) and \(5\times 5\) circulant involutory MDS matrices are given when \(m=4,8\). To the best of our knowledge, it is the first time that circulant involutory MDS matrices have been constructed. Furthermore, some lower bounds on XORs that required to evaluate one row of circulant and Hadamard MDS matrices of order 4 are given when \(m=4,8\). Some constructions achieving the bound are also given, which have fewer XORs than previous constructions.

We explore the feasibility of applying SAT solvers to optimizing implementations of small functions such as S-boxes for multiple optimization criteria, e.g., the number of nonlinear gates and the number of gates. We provide optimized implementations for the S-boxes used in Ascon, ICEPOLE, Joltik/Piccolo, Keccak/Ketje/Keyak, LAC, Minalpher, PRIMATEs, Prøst, and RECTANGLE, most of which are candidates in the secound round of the CAESAR competition. We then suggest a new method to optimize for circuit depth and we make tooling publicly available to find efficient implementations for several criteria. Furthermore, we illustrate with the 5-bit S-box of PRIMATEs how multiple optimization criteria can be combined.

We provide further evidence that implementing software countermeasures against timing attacks is a non-trivial task and requires domain-specific software development processes: we report an implementation bug in the s2n library, recently released by AWS Labs. This bug (now fixed) allowed bypassing the balancing countermeasures against timing attacks deployed in the implementation of the MAC-then-Encode-then-CBC-Encrypt (MEE-CBC) component, creating a timing side-channel similar to that exploited by Lucky 13.

Although such an attack could only be launched when the MEE-CBC component is used in isolation – Albrecht and Paterson recently confirmed in independent work that s2n’s second line of defence, once reinforced, provides adequate mitigation against current adversary capabilities – its existence serves as further evidence to the fact that conventional software validation processes are not effective in the study and validation of security properties. To solve this problem, we define a methodology for proving security of implementations in the presence of timing attackers: first, prove black-box security of an algorithmic description of a cryptographic construction; then, establish functional correctness of an implementation with respect to the algorithmic description; and finally, prove that the implementation is leakage secure.

We present a proof-of-concept application of our methodology to MEE-CBC, bringing together three different formal verification tools to produce an assembly implementation of this construction that is verifiably secure against adversaries with access to some timing leakage. Our methodology subsumes previous work connecting provable security and side-channel analysis at the implementation level, and supports the verification of a much larger case study. Our case study itself provides the first provable security validation of complex timing countermeasures deployed, for example, in OpenSSL.

Implementations of white-box cryptography aim to protect a secret key in a white-box environment in which an adversary has full control over the execution process and the entire environment. Its fundamental principle is the map of the cryptographic architecture, including the secret key, to a number of encoded tables that shall resist the inspection and decomposition of an attacker. In a gray-box scenario, however, the property of hiding required implementation details from the attacker could be used as a promising mitigation strategy against side-channel attacks (SCA). In this work, we present a first white-box implementation of AES on reconfigurable hardware for which we evaluate this approach assuming a gray-box attacker. We show that – unfortunately – such an implementation does not provide sufficient protection against an SCA attacker. We continue our evaluations by a thorough analysis of the source of the observed leakage, and present additional results which can be used to build stronger white-box designs.

Masking is a popular countermeasure to thwart side-channel attacks on embedded systems. Many proposed masking schemes, even carrying “security proofs”, are eventually broken because they are flawed by design. The security validation process is nowadays a lengthy, tedious and manual process. In this paper, we report on a method to verify the soundness of a masking scheme before implementing it on a device. We show that by instrumenting a high-level implementation of the masking scheme and by applying leakage detection techniques, a system designer can quickly assess at design time whether the masking scheme is flawed or not, and to what extent. Our method requires not more than working high-level source code and is based on simulation. Thus, our method can be used already in the very early stages of design. We validate our approach by spotting in an automated fashion first-, second- and third-order flaws in recently published state-of-the-art schemes in a matter of seconds with limited computational resources. We also present a new second-order flaw on a table recomputation scheme, and show that the approach is useful when designing a hardware masked implementation.

Side-channel attacks are an important concern for the security of cryptographic algorithms. To counteract it, a recent line of research has investigated the use of software encoding functions such as dual-rail rather than the well known masking countermeasure. The core idea consists in encoding the sensitive data with a fixed Hamming weight value and perform all operations following this fashion. This new set of countermeasures applies to all devices that leak a function of the Hamming weight of the internal variables. However when the leakage model deviates from this idealized model, the claimed security guarantee vanishes. In this work, we introduce a framework that aims at building customized encoding functions according to the precise leakage model based on stochastic profiling. We specifically investigate how to take advantage of adversary’s knowledge of the physical leakage to select the corresponding optimal encoding. Our solution has been evaluated within several security metrics, proving its efficiency against side-channel attacks in realistic scenarios. A concrete experimentation of our proposal to protect the PRESENT Sbox confirms its practicability. In a realistic scenario, our new custom encoding achieves a hundredfold reduction in leakage compared to the dual-rail, although using the same code length.

Key schedules in block ciphers are often highly simplified, which causes weakness that can be exploited in many attacks. At ASIACRYPT 2011, Dunkelman et al. proposed a technique using the weakness in the key schedule of AES, called key-bridging technique, to improve the overall complexity. The advantage of key-bridging technique is that it allows the adversary to deduce some sub-key bits from some other sub-key bits, even though they are separated by many key mixing steps. Although the relations of successive rounds may be easy to see, the relations of two rounds separated by some mixing steps are very hard to find. In this paper, we describe a versatile and powerful algorithm for searching key-bridging technique on word-oriented and bit-oriented block ciphers. To demonstrate the usefulness of our approach, we apply our tool to the impossible differential and multidimensional zero correlation linear attacks on 23-round LBlock, 23-round TWINE-80 and 25-round TWINE-128. To the best of our knowledge, these results are the currently best results on LBlock and TWINE in the single-key setting.

In recent years, Mixed Integer Linear Programming (MILP) has been successfully applied in searching for differential characteristics and linear approximations in block ciphers and has produced the significant results for some ciphers such as SIMON (a family of lightweight and hardware-optimized block ciphers designed by NSA) etc. However, in the literature, the MILP-based automatic search algorithm for differential characteristics and linear approximations is still infeasible for block ciphers such as ARX constructions. In this paper, we propose an MILP-based method for automatic search for differential characteristics and linear approximations in ARX ciphers. By researching the properties of differential characteristic and linear approximation of modular addition in ARX ciphers, we present a method to describe the differential characteristic and linear approximation with linear inequalities under the assumptions of independent inputs to the modular addition and independent rounds. We use this representation as an input to the publicly available MILP optimizer Gurobi to search for differential characteristics and linear approximations for ARX ciphers. As an illustration, we apply our method to Speck, a family of lightweight and software-optimized block ciphers designed by NSA, which results in the improved differential characteristics and linear approximations compared with the existing ones. Moreover, we provide the improved differential attacks on Speck48, Speck64, Speck96 and Speck128, which are the best attacks on them in terms of the number of rounds.

We propose the first adaptation of Matsui’s algorithm for finding the best differential and linear trails to the class of ARX ciphers. It is based on a branch-and-bound search strategy, does not use any heuristics and returns optimal results. The practical application of the new algorithm is demonstrated on reduced round variants of block ciphers from the Speck family. More specifically, we report the probabilities of the best differential trails for up to 10, 9, 8, 7, and 7 rounds of Speck32, Speck48, Speck64, Speck96 and Speck128 respectively, together with the exact number of differential trails that have the best probability. The new results are used to compute bounds, under the Markov assumption, on the security of Speck against single-trail differential cryptanalysis. Finally, we propose two new ARX primitives with provable bounds against single-trail differential and linear cryptanalysis – a long standing open problem in the area of ARX design.

In typical applications of homomorphic encryption, the first step consists for Alice to encrypt some plaintext m under Bob’s public key \(\mathsf {pk}\) and to send the ciphertext \(c = \mathsf {HE}_{\mathsf {pk}}(m)\) to some third-party evaluator Charlie. This paper specifically considers that first step, i.e. the problem of transmitting c as efficiently as possible from Alice to Charlie. As previously noted, a form of compression is achieved using hybrid encryption. Given a symmetric encryption scheme \(\mathsf {E}\), Alice picks a random key k and sends a much smaller ciphertext \(c' = (\mathsf {HE}_{\mathsf {pk}}(k), \mathsf {E}_k(m))\) that Charlie decompresses homomorphically into the original c using a decryption circuit \(\mathcal {C}_{{\mathsf {E}^{-1}}}\).

In this paper, we revisit that paradigm in light of its concrete implementation constraints; in particular \(\mathsf {E}\) is chosen to be an additive IV-based stream cipher. We investigate the performances offered in this context by Trivium, which belongs to the eSTREAM portfolio, and we also propose a variant with 128-bit security: Kreyvium. We show that Trivium, whose security has been firmly established for over a decade, and the new variant Kreyvium have an excellent performance.

We show several constructions based on the AES round function that can be used as building blocks for MACs and authenticated encryption schemes. They are found by a search of the space of all secure constructions based on an efficient design strategy that has been shown to be one of the most optimal among all the considered. We implement the constructions on the latest Intel’s processors. Our benchmarks show that on Intel Skylake the smallest construction runs at 0.188 c/B, while the fastest at only 0.125 c/B, i.e. five times faster than AES-128.

Ciphers that do not use S-boxes have been discussed for the demand on lightweight cryptosystems, and their round functions consist of and, rotation, and xor. Especially, the Simon family is one of the most famous ciphers, and there are many cryptanalyses again the Simon family. However, it is very difficult to guarantee the security because we cannot use useful techniques for S-box-based ciphers. Very recently, the division property, which is a new technique to find integral characteristics, was shown in Eurocrypt 2015. The technique is powerful for S-box-based ciphers, and it was used to break, for the first time, the full MISTY1 in CRYPTO 2015. However, it has not been applied to non-S-box-based ciphers like the Simon family effectively, and only the existence of the 10-round integral characteristic on Simon32 was proven. On the other hand, the experimental characteristic, which possibly does not work for all keys, covers 15 rounds, and there is a 5-round gap. To fill the gap, we introduce a bit-based division property, and we apply it to show that the experimental 15-round integral characteristic always works for all keys. Though the bit-based division property finds more accurate integral characteristics, it requires much time and memory complexity. As a result, we cannot apply it to symmetric-key ciphers whose block length is over 32. Therefore, we alternatively propose a method for designers. The method works for ciphers with large block length, and it shows “provable security” against integral cryptanalyses using the division property. We apply this technique to the Simon family and show that Simon48, 64, 96, and 128 probably do not have 17-, 20-, 25-, and 29-round integral characteristics, respectively.

We introduce the high-degree indicator matrix (HDIM), an object closely related with both the linear approximation table and the algebraic normal form (ANF) of a permutation. We show that the HDIM of a Feistel Network contains very specific patterns depending on the degree of the Feistel functions, the number of rounds and whether the Feistel functions are 1-to-1 or not. We exploit these patterns to distinguish Feistel Networks, even if the Feistel Network is whitened using unknown affine layers. We also present a new type of structural attack exploiting monomials that cannot be present at round \(r-1\) to recover the ANF of the last Feistel function of a r-round Feistel Network. Finally, we discuss the relations between our findings, integral attacks, cube attacks, Todo’s division property and the congruence modulo 4 of the Linear Approximation Table.

Integral attacks form a powerful class of cryptanalytic techniques that have been widely used in the security analysis of block ciphers. The integral distinguishers are based on balanced properties holding with probability one. To obtain a distinguisher covering more rounds, an attacker will normally increase the data complexity by iterating through more plaintexts with a given structure under the strict limitation of the full codebook. On the other hand, an integral property can only be deterministically verified if the plaintexts cover all possible values of a bit selection. These circumstances have somehow restrained the applications of integral cryptanalysis.

While impossible differential cryptanalysis is a well-known and popular cryptanalytic method, errors in the analysis are often discovered and many papers in the literature present flaws. Wishing to solve that, Boura et al. [1] presented at ASIACRYPT’14 a generic vision of impossible differential attacks with the aim of simplifying and helping the construction and verification of this type of cryptanalysis. In particular, they gave generic complexity analysis formulas for mounting such attacks and develop new ideas for optimizing them.

In this paper we carefully study this generic formula and show impossible differential attacks for which the real time complexity is much higher than estimated by it. In particular, we show that the impossible differential attack against 25-round TWINE-128, presented at FSE’15 by Biryukov et al. [2], actually has a complexity higher than the natural bound of exhaustive search.

Simon is a lightweight block cipher family proposed by NSA in 2013. It has drawn many cryptanalysts’ attention and varieties of cryptanalysis results have been published, including differential, linear, impossible differential, integral cryptanalysis and so on. In this paper, we give the improved linear attacks on all reduced versions of Simon with dynamic key-guessing technique, which was proposed to improve the differential attack on Simon recently. By establishing the boolean function of parity bit in the linear hull distinguisher and reducing the function according to the property of AND operation, we can guess different subkeys (or equivalent subkeys) for different situations, which decrease the number of key bits involved in the attack and decrease the time complexity in a further step. As a result, 23-round Simon32/64, 24-round Simon48/72, 25-round Simon48/96, 30-round Simon64/96, 31-round Simon64/128, 37-round Simon96/96, 38-round Simon96/144, 49-round Simon128/128, 51-round Simon128/192 and 53-round Simon128/256 can be attacked. As far as we know, our attacks on most reduced versions of Simon are the best compared with the previous cryptanalysis results. However, this does not shake the security of Simon family with full rounds.

In recent work, Bellare, Hoang, and Keelveedhi (CRYPTO 2013) introduced a new abstraction called Universal Computational Extractors (UCEs), and showed how they can replace random oracles (ROs) across a wide range of cryptosystems. We formulate a new framework, called Interactive Computational Extractors (ICEs), that extends UCEs by viewing them as models of ROs under unpredictable (aka. high-entropy) queries. We overcome a number of limitations of UCEs in the new framework, and in particular prove the adaptive RKA and semi-adaptive KDM securities of a highly efficient symmetric encryption scheme using ICEs under key offsets.

We show both negative and positive feasibility results for ICEs. On the negative side, we demonstrate ICE attacks on the \(\mathsf {HMAC}\) and \(\mathsf {NMAC}\) constructions. On the positive side we show that: (1) ROs are indeed ICE secure, thereby confirming the structural soundness of our definition and enabling a finer layered approach to protocol design in the RO model; and (2) a modified version of Liskov’s Zipper Hash is ICE secure with respect to an underlying fixed-input-length RO, for appropriately restricted classes of adversaries. This brings the first result closer to practice by moving away from variable-input-length ROs. Our security proofs employ techniques from indifferentiability in multi-stage settings.

In an order-preserving encryption scheme, the encryption algorithm produces ciphertexts that preserve the order of their plaintexts. Order-preserving encryption schemes have been studied intensely in the last decade, and yet not much is known about the security of these schemes. Very recently, Boneh et al. (Eurocrypt 2015) introduced a generalization of order-preserving encryption, called order-revealing encryption, and presented a construction which achieves this notion with best-possible security. Because their construction relies on multilinear maps, it is too impractical for most applications and therefore remains a theoretical result.

In this work, we build efficiently implementable order-revealing encryption from pseudorandom functions. We present the first efficient order-revealing encryption scheme which achieves a simulation-based security notion with respect to a leakage function that precisely quantifies what is leaked by the scheme. In fact, ciphertexts in our scheme are only about 1.6 times longer than their plaintexts. Moreover, we show how composing our construction with existing order-preserving encryption schemes results in order-revealing encryption that is strictly more secure than all preceding order-preserving encryption schemes.

We reconsider the formalization of known-key attacks against ideal primitive-based block ciphers. This was previously tackled by Andreeva, Bogdanov, and Mennink (FSE 2013), who introduced the notion of known-key indifferentiability. Our starting point is the observation, previously made by Cogliati and Seurin (EUROCRYPT 2015), that this notion, which considers only a single known key available to the attacker, is too weak in some settings to fully capture what one might expect from a block cipher informally deemed resistant to known-key attacks. Hence, we introduce a stronger variant of known-key indifferentiability, where the adversary is given multiple known keys to “play” with, the informal goal being that the block cipher construction must behave as an independent random permutation for each of these known keys. Our main result is that the 9-round iterated Even-Mansour construction (with the trivial key-schedule, i.e., the same round key xored between permutations) achieves our new “multiple” known-keys indifferentiability notion, which contrasts with the previous result of Andreeva et al. that one single round is sufficient when only a single known key is considered. We also show that the 3-round iterated Even-Mansour construction achieves the weaker notion of multiple known-keys sequential indifferentiability, which implies in particular that it is correlation intractable with respect to relations involving any (polynomial) number of known keys.

Universal hash functions (UHFs) have been extensively used in the design of cryptographic schemes. If we consider the related-key attack (RKA) against these UHF-based schemes, some of them may not be secure, especially those using the key of UHF as a part of the whole key of scheme, due to the weakness of UHF in the RKA setting. In order to solve this issue, we propose a new concept of related-key almost universal hash function, which is a natural extension to almost universal hash function in the RKA setting. We define related-key almost universal (RKA-AU) hash function and related-key almost XOR universal (RKA-AXU) hash function. However almost all the existing UHFs do not satisfy the new definitions. We construct one fixed-input-length universal hash function named RH1 and two variable-input-length universal hash functions named RH2 and RH3. We show that RH1 and RH2 are both RKA-AXU, and RH3 is RKA-AU for the RKD set \(\varPhi ^\oplus \). Furthermore, RH1, RH2 and RH3 are nearly as efficient as previously similar constructions. RKA-AU (RKA-AXU) hash functions can be used as components in the related-key secure cryptographic schemes. If we replace the universal hash functions in the schemes with our corresponding constructions, the problems about related-key attack can be solved for some RKD sets. More specifically, we give four concrete applications of RKA-AU and RKA-AXU in related-key secure message authentication codes and tweakable block ciphers.

In this paper, we propose a guess and determine attack against some variants of the \(\pi \)-Cipher family of authenticated ciphers. This family of ciphers is a second-round candidate of the CAESAR competition. More precisely, we show a key recovery attack with time complexity little higher than \(2^{4\omega }\), and low data complexity, against variants of the cipher with \(\omega \)-bit words, when the internal permutation is reduced to 2.5 rounds.

In particular, this gives an attack with time complexity \(2^{72}\) against the variant \(\pi \)16-Cipher096 (using 16-bit words) reduced to 2.5 rounds, while the authors claim 96 bits of security with 3 rounds in their second-round submission. Therefore, the security margin for this variant of \(\pi \)-Cipher is very limited.

The attack can also be applied to lightweight variants that are not included in the CAESAR proposal, and use only two rounds. The lightweight variants \(\pi \)16-Cipher096 and \(\pi \)16-Cipher128 claim 96 bits and 128 bits of security respectively, but our attack can break the full 2 rounds with complexity \(2^{72}\).

Finally, the attack can be applied to reduced versions of two more variants of \(\pi \)-Cipher that were proposed in the first-round submission with 4 rounds: \(\pi \)16-Cipher128 (using 16-bit words) and \(\pi \)32-Cipher256 (using 32-bit words). The attack on 2.5 rounds has complexity \(2^{72}\) and \(2^{137}\) respectively, while the security claim for 4 rounds are 128 bits and 256 bits of security.

NORX is a second round candidate of the ongoing CAESAR competition for authenticated encryption. It is a nonce based authenticated encryption scheme based on the sponge construction. Its two variants denoted by NORX32 and NORX64 provide a security level of 128 and 256 bits, respectively. In this paper, we present a state/key recovery attack for both variants with the number of rounds of the core permutation reduced to 2 (out of 4) rounds. The time and data complexities of the attack for NORX32 are \(2^{119}\) and \( 2^{66} \) respectively, and for NORX64 are \( 2^{234} \) and \( 2^{132} \) respectively, while the memory complexity is negligible. Furthermore, we show a state recovery attack against NORX in the parallel mode using an internal differential attack for 2 rounds of the permutation. The data, time and memory complexities of the attack for NORX32 are \(2^{7.3}\), \(2^{124.3}\) and \(2^{115}\) respectively and for NORX64 are \(2^{6.2}\), \(2^{232.8}\) and \(2^{225}\) respectively. Finally, we present a practical distinguisher for the keystream of NORX64 based on two rounds of the permutation in the parallel mode using an internal differential-linear attack. To the best of our knowledge, our results are the best known results for NORX in nonce respecting manner.

The hash function Kupyna was recently published as the Ukrainian standard DSTU 7564:2014. It is structurally very similar to the SHA-3 finalist Grøstl, but differs in details of the round transformations. Most notably, some of the round constants are added with a modular addition, rather than bitwise xor. This change prevents a straightforward application of some recent attacks, in particular of the rebound attacks on the compression function of similar AES-like hash constructions. However, we show that it is actually possible to mount rebound attacks, despite the presence of modular constant additions. More specifically, we describe collision attacks on the compression function for 6 (out of 10) rounds of Kupyna-256 with an attack complexity of \(2^{70}\), and for 7 rounds with complexity \(2^{125.8}\). In addition, we can use the rebound attack for creating collisions for the round-reduced hash function itself. This is possible for 4 rounds of Kupyna-256 with complexity \(2^{67}\) and for 5 rounds with complexity \(2^{120}\).