Divisibility Properties Research Articles

Integral Cryptanalysis of Lightweight Block Cipher PIPO

PIPO is a lightweight block cipher proposed at ICISC 2020, which has a byte-oriented structure suitable for bit-sliced implementation and allows for efficient higher-order masking implementations. In this study, we use bit-based division property techniques to construct 6-round integral distinguishers, and propose key-recovery attacks on 8 rounds of PIPO-64/128 and 10 rounds of PIPO-64/256. The data complexity of both attacks is 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">63</sup> chosen plaintexts and the time complexities are 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">125</sup> and 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">253.8</sup> respectively. Our results complement the security analysis of PIPO, and show that the PIPO structure is resistant to recently researched cryptanalysis methods. Because only differential and linear attacks were carefully considered to determine the number of rounds of PIPO, our work, based on division property, is important for verifying the security margin.

3073 – HUMAN HAEMATOPOIETIC STEM AND PROGENITOR CELLS FORM KINSHIPS THAT ARE SIMILAR IN FATE AND SYNCHRONOUS IN DIVISION

While the heterogeneity of human haematopoietic stem and progenitor cells has become apparent, linking this heterogeneity to functional properties at the single-cell level remains a challenge. Here, we have developed a combination of flow cytometry, live-cell imaging, and mathematical modelling to track the cell division properties and cell commitment in human haematopoietic stem and progenitor cells. We identified two key properties: synchronicity in division and similarity in fate among progenies of individual stem and progenitor cells. Each human stem and progenitor cell give rise to a cellular kinship that share a strong resemblance in fate and divide synchronously, suggesting an intrinsic cellular memory transmitted from mother to daughter cells. Interestingly, we observe that the time to divide shared by two sibling cells is not correlated with the division time of its mother cell. These behaviours are observed and shared across the entire lifespan, from cord blood to aged bone marrow cells. Moreover, we collected evidence that these behaviours are altered in haematological malignancies such as acute myeloid leukemia. Using a multi-agent mathematical model, we were able to recapitulate the cellular early commitment, where two sister cells descending from the same ancestor share a strong correlation in fate. In conclusion, we propose a model that describe the human haematopoiesis as a heterogeneous system formed by homogeneous cell families. While the heterogeneity of human haematopoietic stem and progenitor cells has become apparent, linking this heterogeneity to functional properties at the single-cell level remains a challenge. Here, we have developed a combination of flow cytometry, live-cell imaging, and mathematical modelling to track the cell division properties and cell commitment in human haematopoietic stem and progenitor cells. We identified two key properties: synchronicity in division and similarity in fate among progenies of individual stem and progenitor cells. Each human stem and progenitor cell give rise to a cellular kinship that share a strong resemblance in fate and divide synchronously, suggesting an intrinsic cellular memory transmitted from mother to daughter cells. Interestingly, we observe that the time to divide shared by two sibling cells is not correlated with the division time of its mother cell. These behaviours are observed and shared across the entire lifespan, from cord blood to aged bone marrow cells. Moreover, we collected evidence that these behaviours are altered in haematological malignancies such as acute myeloid leukemia. Using a multi-agent mathematical model, we were able to recapitulate the cellular early commitment, where two sister cells descending from the same ancestor share a strong correlation in fate. In conclusion, we propose a model that describe the human haematopoiesis as a heterogeneous system formed by homogeneous cell families.

Some Divisibility Properties of Lucas Numbers

The Lucas number sequence is a popular number sequence that has been described as similar to the Fibonacci number sequence. A lot of research has been done on this number sequence. Some of these studies are on the divisibility properties of this number sequence. Carlitz (1964) examined the requirement that a given Lucas number can be divided by another Lucas number. After that, many studies have been done on this subject. In the present article, we obtain some divisibility properties of the Lucas Numbers. First, we examine the case $L_{(2n-1)m}/L_{m}$ and then we obtain $L_{\left( 2n-1\right) m}$ using different forms of Lucas numbers.

Arithmetic properties of Fourier coefficients of meromorphic modular forms

We investigate integrality and divisibility properties of Fourier coefficients of meromorphic modular forms of weight $2k$ associated to positive definite integral binary quadratic forms. For example, we show that if there are no non-trivial cusp forms of weight $2k$, then the $n$-th coefficients of these meromorphic modular forms are divisible by $n^{k-1}$ for every natural number $n$. Moreover, we prove that their coefficients are non-vanishing and have either constant or alternating signs. Finally, we obtain a relation between the Fourier coefficients of meromorphic modular forms, the coefficients of the $j$-function, and the partition function.

On Two Problems Related to Divisibility Properties of z(n)

The order of appearance (in the Fibonacci sequence) function z:Z≥1→Z≥1 is an arithmetic function defined for a positive integer n as z(n)=min{k≥1:Fk≡0(modn)}. A topic of great interest is to study the Diophantine properties of this function. In 1992, Sun and Sun showed that Fermat’s Last Theorem is related to the solubility of the functional equation z(n)=z(n2), where n is a prime number. In addition, in 2014, Luca and Pomerance proved that z(n)=z(n+1) has infinitely many solutions. In this paper, we provide some results related to these facts. In particular, we prove that limsupn→∞(z(n+1)−z(n))/(logn)2−ϵ=∞, for all ϵ∈(0,2).

On involutive division on monoids

We consider an arbitrary monoid MM, on which an involutive division is introduced, and the set of all its finite subsets SetMM. Division is considered as a mapping d:SetMM{d:SetM \times M}, whose image d(U,m){d(U,m)} is the set of divisors of mm in UU. The properties of division and involutive division are defined axiomatically. Involutive division was introduced in accordance with the definition of involutive monomial division, introduced by V.P. Gerdt and Yu.A. Blinkov. New notation is proposed that provides brief but explicit allowance for the dependence of division on the SetMM element. The theory of involutive completion (closures) of sets is presented for arbitrary monoids, necessary and sufficient conditions for completeness (closedness) - for monoids generated by a finite set XX. The analogy between this theory and the theory of completely continuous operators is emphasized. In the last section, we discuss the possibility of solving the problem of replenishing a given set by successively expanding the original domain and its connection with the axioms used in the definition of division. All results are illustrated with examples of Thomas monomial division.

Moment Infinite Divisibility of Weighted Shifts: Sequence Conditions

We consider weighted shift operators having the property of moment infinite divisibility; that is, for any \(p > 0\), the shift is subnormal when every weight (equivalently, every moment) is raised to the p-th power. By reconsidering sequence conditions for the weights or moments of the shift, we obtain a new characterization for such shifts, and we prove that such shifts are, under mild conditions, robust under a variety of operations and also rigid in certain senses. In particular, a weighted shift whose weight sequence has a limit is moment infinitely divisible if and only if its Aluthge transform is. As a consequence, we prove that the Aluthge transform maps the class of moment infinitely divisible weighted shifts bijectively onto itself. We also consider back-step extensions, subshifts, and completions.

The Proof of a Conjecture on the Density of Sets Related to Divisibility Properties of z(n)

Let (Fn)n be the sequence of Fibonacci numbers. The order of appearance (in the Fibonacci sequence) of a positive integer n is defined as z(n)=min{k≥1:n∣Fk}. Very recently, Trojovská and Venkatachalam proved that, for any k≥1, the number z(n) is divisible by 2k, for almost all integers n≥1 (in the sense of natural density). Moreover, they posed a conjecture that implies that the same is true upon replacing 2k by any integer m≥1. In this paper, in particular, we prove this conjecture.

On the infinite Borwein product raised to a positive real power

In this paper, we study properties of the coefficients appearing in the q-series expansion of prod _{nge 1}[(1-q^n)/(1-q^{pn})]^delta , the infinite Borwein product for an arbitrary prime p, raised to an arbitrary positive real power delta . We use the Hardy–Ramanujan–Rademacher circle method to give an asymptotic formula for the coefficients. For p=3 we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent delta is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product. We conclude with an Appendix presenting several new conjectures on precise sign patterns of infinite products raised to a real power which are similar to the conjecture we made in the p=3 case.

On infinitely divisible multivariate gamma distributions

We provide an explicit construction of variables from a bivariate gamma distribution which is infinitely divisible. These distributions are currently only known through their (complicated) density functions or moment generating functions. The sampling construction uses independent gamma and Poisson random variables and sheds further light on these distributions and their infinitely divisible property. The extension to multivariate gamma infinitely divisible variables is also provided, as well as a negative binomial case.

The Proof of a Conjecture Related to Divisibility Properties of z(n)

The order of appearance of n (in the Fibonacci sequence) z(n) is defined as the smallest positive integer k for which n divides the k—the Fibonacci number Fk. Very recently, Trojovský proved that z(n) is an even number for almost all positive integers n (in the natural density sense). Moreover, he conjectured that the same is valid for the set of integers n≥1 for which the integer 4 divides z(n). In this paper, among other things, we prove that for any k≥1, the number z(n) is divisible by 2k for almost all positive integers n (in particular, we confirm Trojovský’s conjecture).

Ideal structure of rings of analytic functions with non-Archimedean metrics

The work of Helmer [Divisibility properties of integral functions, Duke Math. J. 6(2) (1940) 345–356] applied algebraic methods to the field of complex analysis when he proved the ring of entire functions on the complex plane is a Bezout domain (i.e. all finitely generated ideals are principal). This inspired the work of Henriksen [On the ideal structure of the ring of entire functions, Pacific J. Math. 2(2) (1952) 179–184. On the prime ideals of the ring of entire functions, Pacific J. Math. 3(4) (1953) 711–720] who proved a correspondence between the maximal ideals within the ring of entire functions and ultrafilters on sets of zeroes as well as a correspondence between the prime ideals and growth rates on the multiplicities of zeroes. We prove analogous results on rings of analytic functions in the non-Archimedean context: all finitely generated ideals in the ring of analytic functions on an annulus of a characteristic zero non-Archimedean field are two-generated but not guaranteed to be principal. We also prove the maximal and prime ideal structure in the non-Archimedean context is similar to that of the ordinary complex numbers; however, the methodology has to be significantly altered to account for the failure of Weierstrass factorization on balls of finite radius in fields which are not spherically complete, which was proven by Lazard [Les zeros d’une function analytique d’une variable sur un corps value complet, Publ. Math. l’IHES 14(1) (1942) 47–75].

Integral Distinguishers of the Full-Round Lightweight Block Cipher SAT_Jo

Integral cryptanalysis based on division property is a powerful cryptanalytic method whose range of successful applications was recently extended through the use of Mixed-Integer Linear Programming (MILP). Although this technique was demonstrated to be efficient in specifying distinguishers of reduced round versions of several families of lightweight block ciphers (such as SIMON, PRESENT, and few others), we show that this method provides distinguishers for a full-round block cipher SAT_Jo. SAT_Jo cipher is very similar to the well-known PRESENT block cipher, which has successfully withstood the known cryptanalytic methods. The main difference compared to PRESENT, which turns out to induce severe weaknesses of SAT_Jo algorithm, is its different choice of substitution boxes (S-boxes) and the bit-permutation layer for the reasons of making the cipher highly resource-efficient. Even though the designers provided a security analysis of this scheme against some major generic cryptanalytic methods, an application of the bit-division property in combination with MILP was not considered. By specifying integral distinguishers for the full-round SAT_Jo algorithm using this method, we essentially disapprove its use in intended applications. Using a 30-round distinguisher, we also describe a subkey recovery attack on the SAT_Jo algorithm whose time complexity is about 2 66 encryptions (noting that SAT_Jo is designed to provide 80 bits of security). Moreover, it seems that the choice of bit-permutation induces weak division properties since replacing the original bit-permutation of SAT_Jo by the one used in PRESENT immediately renders integral distinguishers inefficient.

A note on ratios of Fibonacci hybrid and Lucas hybrid numbers

Irmak recently asked an open question related to divisibility properties of Fibonacci and Lucas quaternions [4, p. 374]. In this paper, we give an answer to Fibonacci and Lucas hybrid number version of this question.

Nonlinear Conditions for Ultradifferentiability

A remarkable theorem of Joris states that a function f is C^infty if two relatively prime powers of f are C^infty . Recently, Thilliez showed that an analogous theorem holds in Denjoy–Carleman classes of Roumieu type. We prove that a division property, equivalent to Joris’s result, is valid in a wide variety of ultradifferentiable classes. Generally speaking, it holds in all dimensions for non-quasianalytic classes. In the quasianalytic case we have general validity in dimension one, but we also get validity in all dimensions for certain quasianalytic classes.

New density results and congruences for Andrews' singular overpartitions

Andrews introduced the singular overpartition function C‾k,i(n) which counts the number of overpartitions of n in which no part is divisible by k and only parts ≡±i(modk) may be overlined. In this article, we study the divisibility properties of C‾4k,k(n) and C‾6k,k(n) by arbitrary powers of 2 and 3 for infinite families of k. For an infinite family of k, we prove that C‾4k,k(n) is almost always divisible by arbitrary powers of 2. We also prove that C‾6k,k(n) is almost always divisible by arbitrary powers of 3 for an infinite family of k. Using a result of Ono and Taguchi on nilpotency of Hecke operators, we find infinite families of congruences modulo arbitrary powers of 2 satisfied by C‾4⋅2α,2α(n) and C‾4⋅3⋅2α,3⋅2α(n).

On the bit-based division property of S-boxes

On the bit-based division property of S-boxes

Modeling for Three-Subset Division Property without Unknown Subset

A division property is a generic tool to search for integral distinguishers, and automatic tools such as MILP or SAT/SMT allow us to evaluate the propagation efficiently. In the application to stream ciphers, it enables us to estimate the security of cube attacks theoretically, and it leads to the best key-recovery attacks against well-known stream ciphers. However, it was reported that some of the key-recovery attacks based on the division property degenerate to distinguishing attacks due to the inaccuracy of the division property. Three-subset division property (without unknown subset) is a promising method to solve this inaccuracy problem, and a new algorithm using automatic tools for the three-subset division property was recently proposed at Asiacrypt2019. In this paper, we first show that this state-of-the-art algorithm is not always efficient and we cannot improve the existing key-recovery attacks. Then, we focus on the three-subset division property without unknown subset and propose another new efficient algorithm using automatic tools. Our algorithm is more efficient than existing algorithms, and it can improve existing key-recovery attacks. In the application to Trivium, we show a 842-round key-recovery attack. We also show that a 855-round key-recovery attack, which was proposed at CRYPTO2018, has a critical flaw and does not work. As a result, our 842-round attack becomes the best key-recovery attack. In the application to Grain-128AEAD, we show that the known 184-round key-recovery attack degenerates to a distinguishing attack. Then, the distinguishing attacks are improved up to 189 rounds, and we also show the best key-recovery attack against 190 rounds. In the application to ACORN, we prove that the 772-round key-recovery attack at ISC2019 is in fact a constant-sum distinguisher. We then give new key-recovery attacks mounting to 773-, 774- and 775-round ACORN. We verify the current best key-recovery attack on 892-round Kreyvium and recover the exact superpoly. We further propose a new attack mounting to 893 rounds.

Conditional positive definiteness as a bridge between k–hyponormality and n–contractivity

For sequences α≡{αn}n=0∞ of positive real numbers, called weights, we study the weighted shift operators Wα having the property of moment infinite divisibility (MID); that is, for any p>0, the Schur power Wαp is subnormal. We first prove that Wα is MID if and only if certain infinite matrices log⁡Mγ(0) and log⁡Mγ(1) are conditionally positive definite (CPD). Here γ is the sequence of moments associated with α, Mγ(0),Mγ(1) are the canonical Hankel matrices whose positive semi-definiteness determines the subnormality of Wα, and log is calculated entry-wise (i.e., in the sense of Schur or Hadamard). Next, we use conditional positive definiteness to establish a new bridge between k–hyponormality and n–contractivity, which sheds significant new light on how the two well known staircases from hyponormality to subnormality interact. As a consequence, we prove that a contractive weighted shift Wα is MID if and only if for all p>0, Mγp(0) and Mγp(1) are CPD.

Quantum Algorithm to Identify Division Property of a Multiset

Division property-based integral attack is the generalization of integral property developed by blending saturation attack and higher-order differential attack. This attack is considered as a chosen-plaintext attack because the cryptanalyst generates a multiset of plaintext which possesses a certain division property. However, in real-world applications, it is required to find the division property of a given multiset which turns the attack into a known-plaintext attack. The problem, finding the division property of a given multiset $$\mathbb {X}$$ of size $$|\mathbb {X}|$$ with each element of n-bit, when solved on a classical computer has the time complexity of $$O(n2^n|\mathbb {X}|)$$ (fixed in both average and worst cases). In this paper, a better and comparable algorithm using quantum computing is presented along with its quantum oracle designs that can find the correct division property of a multiset in the average case time complexity of $$O \left( \log (n)2^n\sqrt{|\mathbb {X}|} \right) $$ and worst case time complexity of $$O\left( \log (n) 2^n|\mathbb {X}| \right) $$ using $$\left( n + \lceil \log |\mathbb {X}|\rceil + p \right) $$ -qubits, where p are the precision qubits required by the quantum counting subroutine.