Modular Addition Research Articles

On the Probability and Automatic Search of Rotational-XOR Cryptanalysis on ARX Ciphers

Abstract Rotational-XOR cryptanalysis is a very recent technique for ARX ciphers. In this paper, the probability propagation formula of RX-cryptanalysis in modular addition is extended, and the calculation of RX-difference probability for any rotation parameter ($0<k<n$) can be realized. By proposing a concept of RX-offset and constructing the corresponding distribution table, the propagation of RX-difference in modular addition can be derived from the propagation of XOR-difference. Combined with the improvement of the automatic search tool for XOR-differential characteristics of ARX ciphers, we only need to add one more operation in each round, i.e. traverse the possible value of RX-offset and XOR it with the output XOR-difference of modular addition, thus it can achieve the search for RX-differential characteristics. With this method, the RX-differential distinguisher of ARX-C primitives without or with linear key schedule can be searched. For the applications, we have obtained the third-party RX-cryptanalysis results for Alzette and CHAM for the first time as far as we know.

Ediacara growing pains: Modular addition and development in Dickinsonia costata.

Constraining patterns of growth using directly observable and quantifiable characteristics can reveal a wealth of information regarding the biology of the Ediacara Biota - the oldest macroscopic, complex community forming organisms in the fossil record. However, these rely on individuals captured at an instant in time at various growth stages, and so different interpretations can be derived from the same material. Here we leverage newly discovered and well-preserved Dickinsonia costata Sprigg 1947 from South Australia, combined with hundreds of previously described specimens, to test competing hypotheses for the location of module addition. We find considerable variation in the relationship between the total number of modules and body size that cannot be explained solely by expansion and contraction of individuals. Patterns derived assuming new modules differentiated at the anterior result in numerous examples where the oldest module(s) must decrease in size with overall growth, potentially falsifying this hypothesis. Observed polarity as well as the consistent posterior location of defects and indentations support module formation at this end in D. costata. Regardless, changes in repeated units with growth share similarities with those regulated by morphogen gradients in metazoans today, suggesting that these genetic pathways were operating in Ediacaran animals.

Maximums of the Additive Differential Probability of Exclusive-Or

At FSE 2004, Lipmaa et al. studied the additive differential probability adp⊕(α,β → γ) of exclusive-or where differences α,β,γ ∈ Fn2 are expressed using addition modulo 2n. This probability is used in the analysis of symmetric-key primitives that combine XOR and modular addition, such as the increasingly popular Addition-Rotation-XOR (ARX) constructions. The focus of this paper is on maximal differentials, which are helpful when constructing differential trails. We provide the missing proof for Theorem 3 of the FSE 2004 paper, which states that maxα,βadp⊕(α,β → γ) = adp⊕(0,γ → γ) for all γ. Furthermore, we prove that there always exist either two or eight distinct pairs α,β such that adp⊕( α,β → γ) = adp⊕(0,γ → γ), and we obtain recurrence formulas for calculating adp⊕. To gain insight into the range of possible differential probabilities, we also study other properties such as the minimum value of adp⊕(0,γ → γ), and we find all γ that satisfy this minimum value.

Reversible data hiding in encrypted color images using cross-channel correlations

In recent years, the increasing requirements in cloud storage and cloud computing have made it necessary to encrypt digital images for privacy protection. Meanwhile, many reversible data hiding (RDH) algorithms in the encrypted domain have been proposed. However, most of these algorithms are for gray-level images, and the intrinsic cross-channel correlations of color images cannot be utilized to improve the embedding capacity. In this paper, we propose a novel data hiding method for encrypted color images. In the encryption stage, the homomorphic property of encryption is achieved by basic modular addition. During the data hiding process, the cross-channel correlations between R, G and B channels are generated in encrypted domain, and data hiding is performed by the difference histogram shifting. Analysis and experiments demonstrate that the proposed method is secure and the RDH performance is superior.

Quantum search for scaled hash function preimages

We present the implementation of Grover’s algorithm in a quantum simulator to perform a quantum search for preimages of two scaled hash functions, whose design only uses modular addition, word rotation and bitwise exclusive or. Our implementation provides the means to assess with precision the scaling of the number of gates and depth of a full-fledged quantum circuit designed to find the preimages of a given hash digest. The detailed construction of the quantum oracle shows that the presence of AND gates, OR gates, shifts of bits and the reuse of the initial state along the computation require extra quantum resources as compared with other hash functions based on modular additions, XOR gates and rotations. We also track the entanglement entropy present in the quantum register at every step along the computation, showing that it becomes maximal at the inner core of the first action of the quantum oracle, which implies that no classical simulation based on tensor networks would be of relevance. Finally, we show that strategies that suggest a shortcut based on sampling the quantum register after a few steps of Grover’s algorithm can only provide some marginal practical advantage in terms of error mitigation.

Computationally Efficient Approach to Implementation of the Chinese Remainder Theorem Algorithm in Minimally Redundant Residue Number System

In this paper, we deal with the critical problem of performing non-modular operations in the Residue Number System (RNS). The Chinese Remainder Theorem (CRT) is widely used in many modern computer applications. Throughout the article, an efficient approach for implementing the CRT algorithm is described. The structure of the rank of an RNS number, a principal positional characteristic of the residue code, is investigated. It is shown that the rank of a number can be represented by a sum of an inexact rank and a two-valued correction to it. We propose a new variant of minimally redundant RNS, which provides low computational complexity for the rank calculation, and its effectiveness analyzed concerning conventional non-redundant RNS. Owing to the extension of the residue code, by adding the excess residue modulo 2, the complexity of the rank calculation goes down from Oleft (k^{2}right ) to Oleft (kright ) with respect to required modular addition operations and lookup tables, where k equals the number of non-redundant RNS moduli.

Quantum Modular Adder over GF(2n − 1) without Saving the Final Carry

Addition is the most basic operation of computing based on a bit system. There are various addition algorithms considering multiple number systems and hardware, and studies for a more efficient addition are still ongoing. Quantum computing based on qubits as the information unit asks for the design of a new addition because it is, physically, wholly different from the existing frequency-based computing in which the minimum information unit is a bit. In this paper, we propose an efficient quantum circuit of modular addition, which reduces the number of gates and the depth. The proposed modular addition is for the Galois Field GF(2n−1), which is important as a finite field basis in various domains, such as cryptography. Its design principle was from the ripple carry addition (RCA) algorithm, which is the most widely used in existing computers. However, unlike conventional RCA, the storage of the final carry is not needed due to modifying existing diminished-1 modulo 2n−1 adders. Our proposed adder can produce modulo sum within the range 0,2n−2 by fewer qubits and less depth. For comparison, we analyzed the proposed quantum addition circuit over GF(2n−1) and the previous quantum modular addition circuit for the performance of the number of qubits, the number of gates, and the depth, and simulated it with IBM’s simulator ProjectQ.

Probabilistic properties of modular addition

Изучается возможность применения разностного криптоанализа к операции сложения по модулю $2^n$, используемой в различных криптосистемах. Получена аналитическая формула математического ожидания энтропии $H_n$ таблицы распределений разностей соответствующего отображения. Рассматриваются также моменты $2^{H_n}$. В частности, получены асимптотические неравенства для $\mathbb{E}2^{qH_n}$ (для $q \in \mathbb{N}$) и $\mathbb{D}2^{H_n}$ при $n \to \infty$. Найдена простая аналитическая формула для числа строк в таблице, задающих одно и то же распределение. Она позволяет эффективно вычислять статистические характеристики энтропии.

Size-control in the synthesis of oxo-bridged phosphazane macrocycles via a modular addition approach

Inorganic macrocycles remain largely underdeveloped compared with their organic counterparts due to the challenges involved in their synthesis. Among them, cyclodiphosphazane macrocycles have shown to be promising candidates for supramolecular chemistry applications due to their ability to encapsulate small molecules or ions within their cavities. However, further developments have been handicapped by the lack of synthetic routes to high-order cyclodiphosphazane macrocycles. Moreover, current approaches allow little control over the size of the macrocycles formed. Here we report the synthesis of high-order oxygen-bridged phosphazane macrocycles via a “3 + n cyclisation” (n = 1 and 3). Using this method, an all-PIII high-order hexameric cyclodiphosphazane macrocycle was isolated, displaying a larger macrocyclic cavity than comparable organic crown-ethers. Our approach demonstrates that increasing building block complexity enables precise control over macrocycle size, which will not only generate future developments in both the phosphazane and main group chemistry but also in the fields of supramolecular chemistry.

An image encryption scheme based on double chaotic cyclic shift and Josephus problem

In recent years, image encryption has become a very popular and effective method to protect digital images. Among many image encryption methods, the image encryption algorithm based on chaotic map has extensive application and profound theoretical basis. In this paper, a novel image encryption algorithm is proposed by combining chaotic map with Josephus problem. The whole encryption process adopts the classical permutation–diffusion structure. In the permutation process, the double chaotic cycle shift algorithm has been proposed by improving the chaotic shift transform method. In the diffusion stage, we extend the definition of Josephus problem by using chaotic map to preserve the Josephus sequence diversity. We then divide the image into sub-blocks and pick one randomly according to Josephus sequence. Furthermore, we change the pixel values of the image through performing modular addition and XOR operations on the pixels between blocks. Through the above process, we get the image encryption algorithm based on double chaotic cycle shift and Josephus problem (JP-DCCS). In the simulation experiment and security analysis, we compare JP-DCCS with the other advanced image encryption algorithms. The results show that JP-DCCS has better performance and higher security.

Taming the Merge Operator

Abstract Calculi with disjoint intersection types support a symmetric merge operator with subtyping. The merge operator generalizes record concatenation to any type, enabling expressive forms of object composition, and simple solutions to hard modularity problems. Unfortunately, recent calculi with disjoint intersection types and the merge operator lack a (direct) operational semantics with expected properties such as determinism and subject reduction, and only account for terminating programs. This paper proposes a type-directed operational semantics (TDOS) for calculi with intersection types and a merge operator. We study two variants of calculi in the literature. The first calculus, called λ i , is a variant of a calculus presented by Oliveira et al. (2016) and closely related to another calculus by Dunfield (2014). Although Dunfield proposes a direct small-step semantics for her calculus, her semantics lacks both determinism and subject reduction. Using our TDOS, we obtain a direct semantics for λ i that has both properties. The second calculus, called λ i + , employs the well-known subtyping relation of Barendregt, Coppo and Dezani-Ciancaglini (BCD). Therefore, λ i + extends the more basic subtyping relation of λ i , and also adds support for record types and nested composition (which enables recursive composition of merged components). To fully obtain determinism, both λ i and λ i + employ a disjointness restriction proposed in the original λ i calculus. As an added benefit the TDOS approach deals with recursion in a straightforward way, unlike previous calculi with disjoint intersection types where recursion is problematic. We relate the static and dynamic semantics of λ i to the original version of the calculus and the calculus by Dunfield. Furthermore, for λ i + , we show a novel formulation of BCD subtyping, which is algorithmic, has a very simple proof of transitivity and allows for the modular addition of distributivity rules (i.e. without affecting other rules of subtyping). All results have been fully formalized in the Coq theorem prover.

Method for performing the operation of adding the remainder of numbers modulo

One of the components of a computer system (CS) in a positional binary number system (PNS) is an adder of two numbers. In particular, adders modulo mi of two numbers are also components of the CS. This type of modulo adders is widely used both in the PNS and in the non-positional number system in the residual classes (RNS). An important and urgent scientific and applied problem is the problem of constructing the adders, which operate by modulus mi, that is an arbitrary RNS modulo. If the remainders ai and bi of both numbers A and B in RNS are represented in a binary PNS, then the adder of two residuals ai and bi by modulus mi is a sequential set of n binary one-bit adders (BOBA). The purpose of the article is to develop a method for performing the operation of modular addition (ai + bi)mod mi of two remainders of numbers by an arbitrary modulo mi based on the use of a positional binary adder modulo M = 2n - 1. The proposed method is based on the use of the well-known structure of positional binary adders modulo M = 2n - 1. Technically, the problem of creating the structure of the modular adder is formulated as follows. It is necessary to provide conditions under which the initial adder in PNS modulo M performs the addition operation modulo mi. This procedure is carried out by introducing additional connections as X¯i­j in the positional adder modulo M = 2n - 1, where the expression X¯i­j denotes one-way connection between the output of the j-th BOBA and the input of the i-th BOBA.

Algorithm for constructing the adder of residues of two numbers modulus

An urgent scientific and applied problem is the problem of constructing the adder structure, which is performed on logical elements with two stable states and operates according to an arbitrary modulo mi. This type of adder is used both in the positional binary number system (PNS) and in the non-positional number system in residual classes (RNS). If the residuals ai and bi of numbers А = (a1||a2||…||ai||…||ak ) and B = (b1||b2||…||bi||…||bk ), represented in the RNS are given in a binary PNS, then the adder of two residuals ai and bi modulo mi is a set of n = [log2 (mi - 1)+1] binary one-bit adders (BOBA). Simultaneously, all BOBA are connected as positional binary adders. The purpose of the article is to develop an algorithm for constructing the adder structure of two residuals ai and bi of numbers A and B for an arbitrary modular value mi of RNS. This process is realized by organizing new inter-bit connections of BOBA, using a positional adder modulo M = 2n - 1. It is noted, that there are special sets of modules that are used when processing data in RNS. So, when performing the operation of modular addition of the remainders of numbers, one of three mutually pairwise primes (of the form M = 2n - 1, M = 2n or M = 2n + 1) can be used. It is shown that in order to synthesize an adder modulo mi RNS, in the adder structure modulo M, it is necessary to appropriately form the additional connections.

Comparative analysis of ARX encryption schemes

ARX encryption algorithms are analyzed, that is, those that use only three operations: modular addition, XOR addition and cyclic shift. 16-bit reduced models of the most famous algorithms of this class are being developed. Among these algorithms are Salsa, Chacha, Cypress, Speckey, Simon, Chaskey. Some of them operate with 4-bit words, others with 8-bit words. By an exhaustive search for models of these algorithms some cryptographic parameters are determined. These parameters are the maximum probability of passing the difference (determines the resistance of the cipher to attacks of differential cryptanalysis); maximum probability of linear approximation (determines the resistance of the cipher to attacks of linear cryptanalysis); non-linear order (determines the resistance of the cipher to interpolation attacks, algebraic cryptanalysis). It is demonstrated that most models with an increase in the number of rounds come to the parameters of random permutations. It is determined that the Simon algorithm model does not possess this property. Several modifications of this algorithm are proposed. Comparing the number of necessary operations to achieve random substitution performance, the most successful ARX schemes were determined. The most efficient 4-bit scheme is the reduced Chaskey model, and the most effective 8-bit one is the modification of the Simon scheme which was proposed in this work. It is shown that, potentially, ARX schemes with a large format of operations are more flexible and efficient, since they require approximately half the number of operations to provide cryptographic parameters of random substitution.

MILP‐based automatic differential search for LEA and HIGHT block ciphers

The authors use the mixed-integer linear programming (MILP) technique for the automatic search for differential characteristics of LEA and HIGHT ciphers. They show that the MILP model of the differential property of modular addition with one constant input can be represented with a much lesser number of linear inequalities compared to the general case. Benefiting from this model for HIGHT block cipher, they can achieve a reduction of 112r out of 480r in the total number of linear constraints for the MILP model of r-round of HIGHT. This saving accelerates the searching process of HIGHT about twice as fast. They enjoy the MILP model to investigate the differential effect of these ciphers and provide a more accurate estimation for the differential probability. Their observations show that despite HIGHT, LEA exhibits a strong differential effect. The results gained by this method improve/extend the previous results as follows. For LEA block cipher, they found more efficient 12- and 13-round differentials whose probabilities are better than the best previous 12- and 13-round differentials for a factor of about 26 and 27, respectively. In the case of HIGHT block cipher, they found new 12- and 13-round differentials, though with the same best-reported probabilities.

High throughput unified architecture of LEA algorithm for image encryption

Among instances when the data and image security in resource constrained environment (smart devices and the like) are to be taken into consideration, lightweight encryption algorithms accede in popularity and are deemed of to be of great merit. In devices as those of the sort mentioned, reducing both power consumption and gate count is always a prime concern. LEA, therefore, triumphs on merit, owing to its low consumption levels of hardware resources, which itself is a result of the adaption of simple modular addition, bitwise rotation and bitwise XOR (ARX) operation instead of the relatively more complex S-box technique. The algorithm in question supports keys of varying sizes three, to be precise: 128, 192 and 256-bit for the intent of providing three levels of image and data security. The hardware implementation of the LEA algorithm for the varying key sizes as mentioned above share significant kinship and hardware resources. This paper puts forward the proposition regarding a high speed low area unified architecture for LEA algorithm for three different sizes of key. The individual hardware implementations of the three key sizes tend to consume more hardware resources than those involved in the proposed unified architecture. The pipelined implementation of the proposed design improves operating frequency to a significant degree, with a little increase in the hardware costs. The design proposed in this paper notably supports three varying tile sizes, i.e. 256×256,128×128,64×64, of an input image. As resolution of an input image may not necessarily be an integer multiple of the size of tile, smaller tile sizes at the image boundary can also be handled by the same architecture. The design hereby proposed has been implemented on both UMC ASIC 0.09-μm and XC5VLX330T FPGA platforms and achieved maximum operating frequencies of 740 MHz and 292 MHz respectively. As its increasingly evident on account of the FPGA implementation outcomes, the proposed unified architecture results in vastly improved levels of operating frequency (34%, 33% and 131%) as opposed to the individually available LEA architectures (corresponding to three different key sizes: 128, 192 and 256-bit, respectively). Proposed architecture shows 28%, 35% and 45% increment in the hardware resources with respect to available LEA architectures with single key. The incorporation of the unified key generation technique and pipeline implementation of the algorithm are the main reasons for the more hardware utilization.

Rotational-XOR Rectangle Cryptanalysis on Round-Reduced Simon

Recently, Ashur and Liu introduced the Rotational-XOR-difference approach which is a modification of rotational cryptanalysis, for an ARX cipher Speck (Ashur and Liu, 2016). In this paper, we apply the Rotational-XOR-difference (RXD) approach to a non-ARX cipher Simon and evaluate its security. First, we studied how to calculate the probability of an RXD for bitwise AND operation that the round function of Simon is based on unlike Speck is on modular addition. Next, we prove that two RXD trails can be connected such that it becomes possible to construct a boomerang/rectangle distinguisher similar to the case using differential characteristics. Finally, we construct related-key rectangle distinguishers for round-reduced versions of Simon with block lengths of 32, 48, and 64, and we suggest a five- or six-round key recovery attack. To our knowledge, it is the first attempt to apply the notion of rotational cryptanalysis for a non-ARX cipher. Although our attack does not show the best results for Simon thus far, the attempt here to define and apply a new cryptanalytic characteristic is meaningful, and we expect further improvements and applications to other ciphers to be made in subsequent studies.

Cryptographic properties of small bijective S-boxes with respect to modular addition

We define affine equivalence of S-boxes with respect to modular addition, and explore its use in cryptanalysis. We have identified classes of small bijective S-boxes with respect to this new equivalence, and experimentally computed their properties.

Organotypic breast tumor model elucidates dynamic remodeling of tumor microenvironment

Fibroblasts are a critical component of tumor microenvironments and associate with cancer cells physically and biochemically during different stages of the disease. Existing cell culture models to study interactions between fibroblasts and cancer cells lack native tumor architecture or scalability. We developed a scalable organotypic model by robotically encapsulating a triple negative breast cancer (TNBC) cell spheroid within a natural extracellular matrix containing dispersed fibroblasts. We utilized an established CXCL12 – CXCR4 chemokine-receptor signaling in breast tumors to validate our model. Using imaging techniques and molecular analyses, we demonstrated that CXCL12-secreting fibroblasts have elevated activity of RhoA/ROCK/myosin light chain-2 pathway and rapidly and significantly contract collagen matrices. Signaling between TNBC cells and CXCL12-producing fibroblasts promoted matrix invasion of cancer cells by activating oncogenic mitogen-activated protein kinase signaling, whereas normal fibroblasts significantly diminished TNBC cell invasiveness. We demonstrated that disrupting CXCL12 – CXCR4 signaling using a molecular inhibitor significantly inhibited invasiveness of cancer cells, suggesting blocking of tumor-stromal interactions as a therapeutic strategy especially for cancers such as TNBC that lack targeted therapies. Our organotypic tumor model mimics native solid tumors, enables modular addition of different stromal cells and extracellular matrix proteins, and allows high throughput compound screening against tumor-stromal interactions to identify novel therapeutics.

Automatic Search for the Linear (Hull) Characteristics of ARX Ciphers: Applied to SPECK, SPARX, Chaskey, and CHAM-64

Linear cryptanalysis is an important evaluation method for cryptographic primitives against key recovery attack. In this paper, we revisit the Walsh transformation for linear correlation calculation of modular addition, and an efficient algorithm is proposed to construct the input-output mask space of specified correlation weight. By filtering out the impossible large correlation weights in the first round, the search space of the first round can be substantially reduced. We introduce a concept of combinational linear approximation table (cLAT) for modular addition with two inputs. When one input mask is fixed, another input mask and the output mask can be obtained by the Splitting-Lookup-Recombination approach. We first split the n-bit fixed input mask into several subvectors and then find the corresponding bits of other masks, and in the recombination phase, pruning conditions can be used. By this approach, a large number of search branches in the middle rounds can be pruned. With the combination of the optimization strategies and the branch-and-bound search algorithm, we can improve the search efficiency for linear characteristics on ARX ciphers. The linear hulls for SPECK32/48/64 with a higher average linear potential (ALP) than existing results have been obtained. For SPARX variants, an 11-round linear trail and a 10-round linear hull have been found for SPARX-64 and a 10-round linear trail and a 9-round linear hull are obtained for SPARX-128. For Chaskey, a 5-round linear trail with a correlation of 2−61 has been obtained. For CHAM-64, 34/35-round optimal linear characteristics with a correlation of 2−31/2−33 are found.