Chinese remainder theorem algorithm Ø To describe the Chinese remainder theorem and its application. Given set G and a binary operation ∗, if each element in the set obeys the following 4 properties, then the set and its operation (G,∗) is called a group. Host and manage packages Security. The theorem is called "Chinese" because it was first stated by the ancient Chinese mathematician Sun Tzu Suan The Chinese Remainder Theorem Let m and n be integers where gcd(m,n)=1, and let b and c be any integers. The structure of the rank of an RNS number, Chinese Remainder Theorem: GCD ( Greatest Common Divisor ) or HCF ( Highest Common Factor ) If GCD(a,b) = 1, then for any remainder ra modulo a and any remainder rb modulo b All algorithms implemented in C#. technique in the chinese remainder theorem we can obtain GCD’s (greatest common divisor) of any 2 co-prime numbers. Exercises 3. ALGORITHMS FOR THE GENERAL CHINESE REMAINDER THEOREM In this section we will present the most important algorithms for the general variant of the Chinese remainder theorem. In its basic form, the Chinese remainder theorem will determine a number \(p\) that, when divided by some given divisors, leaves given remainders. Al­ gorithmically, find ax. 5] for a gentle introduction to this topic. Since this is true for all i, r is a common multiple of the smaller than the least common multiple . The Chinese Remainder Theorem says that certain systems of simultaneous congruences with different moduli have solutions. If a,b Chinese Remainder Theorem synthesis algorithm. This theorem has this name because it is a theorem about remainders and was first discovered in the 3rd century AD by the Chinese mathematician Sunzi in Sunzi Suanjing. By breaking The Chinese Remainder Theorem Last updated: August 7th, 1995. Now, let us verify the Chinese remainder theorem for a system of congruences. We refer to [9, Chap. In this paper, we study the case when the moduli are fixed and can even be chosen by the user. We strongly recommend to refer below post as a prerequisite for this. This is only possible if . The By the Division Algorithm, there are unique numbers q and r such that Now divides both m and , so divides r. 1 Construct the correspondences between the indicated sets. \nonumber \] At this point, Chinese Remainder Theorem Euclidean Algorithm April 11, 2010 1 Algebra We start by discussing algebraic structures and their properties. Der erweiterte euklidische Algorithmus (Details siehe →hier) findet für gegebene a und b die Werte s und t, die die Gleichung s·a + tb = ggT(a,b) erfüllen. This is because for each congruence equation, we need to compute the modular inverse separately, resulting in quadratic time complexity. That is, given a set of moduli $\{m_i\}_{i=1}^{r}$ and Skip to main content. Plan and track work Code The Genius of the Chinese Remainder Theorem. Automate any workflow Packages. Fast Parallel Garner Algorithm for Chi-nese Remainder Theorem. Write better code with AI Security. In the RSA algorithm calculations are made modulo n, where n is a product of two large prime numbers p and q. As proposed algorithm makes use of CRT to encrypt the image, so firstly getting The Chinese Remainder Theorem is a mathematical principle that allows one to solve systems of equations that involve reducing modulo several numbers. Here, we prove the existence of the solution using the mathematical induction. This algorithm We introduce a new type of timing attack which enables the factorization of an RSA-modulus if the exponentiation with the secret exponent uses the Chinese Remainder Theorem and Montgomery's algorithm Its standard variant assumes that both exponentiations are carried out with a simple square and multiply algorithm However, although its efficiency implementation of Chinese remainder theorem algorithm in python - Mbaqban/Chinese-remainder-theorem. Then there exists a number X mod m 1m 2, such that X modm If \( n_j \) are pairwise coprime, then the Chinese Remainder Theorem asserts that the system has a unique solution modulo \( N = n_1 * n_2 * * n_j \). The Chinese Remainder Theorem (CRT) plays a significant role in cryptography, particularly in enhancing the efficiency and security of widely used algorithms like RSA. Through an appropriate use of the technique of FFT-trading, we will show that this assumption allows for the gain of an asymptotic factor in the complexity of “Chinese remaindering”. Automate any workflow Codespaces. (i) Closure. At the end of this 36 Modifikasi Algoritma Kriptografi RSA Multiprima Menggunakan Chinese Remainder Theorem dan Garner’s Algorithm Fatimah Putri Johari#1, Dewi Murni*2, Hendra Syarifuddin *3 #Student of Mathematics Department Universitas Negeri Padang, Indonesia *Lecturers of Mathematics DepartmentUniversitas Negeri Padang, Indonesia 1fatimahputry27@gmail. txt) or read online for free. 1) The document provides examples and solutions related to the Chinese Remainder Theorem, systems of modular congruences, and quadratic residues and primality testing. " Definition of Chinese remainder theorem, possibly with links to more information and implementations. The Chinese Remainder Theorem; Share. Contribute to TheAlgorithms/C-Sharp development by creating an account on GitHub. 0. 0 = 1 with Euclidean Algorithm, then ax. k, k > 0. jxu commented Jan 14, 2023. We design the """ Chinese Remainder Theorem: GCD ( Greatest Common Divisor ) or HCF ( Highest Common Factor ) If GCD(a,b) = 1, then for any remainder ra modulo a and any remainder rb modulo b there exists integer n, such that n = ra (mod a) and n = ra(mod b). b mod m. gcd(mi, mj) = 1, for = j. A proof can be found here . We observe that (p − 1) and (q − 1) are even and that we cannot directly apply the Chinese Remainder Theorem Fermat’s little theorem; Polynomial version of Fermat's little theorem; Euler’s totient function; Euler’s theorem; Chinese remainder theorem; Bézout's identity; Extended euclidean algorithm; Diophantine equations; Inverse of a modular matrix; Modular determinant; Discrete logarithms; Invertible numbers modulo n; Primitive root modulo n The Chinese Remainder Theorem (CRT) and underlying algorithm allows to work with multiple moduli The general idea is to compute a large integer X knowing only its remainders modulo a small set of integers (called moduli) The principles of this method was established sometime in the 3rd and 5th century in China A Chinese mathematician Sun Tzu or Sunzi is known to be Maximum Likelihood Estimation Based Complex-Valued Robust Chinese Remainder Theorem and Its Fast Algorithm. It is used in cryptography and computer science for The Chinese Remainder Theorem states that for positive integers num [0], num [1], , num [k-1] that are pairwise coprime, and any given sequence of integers rem [0], rem [1], The general form is given by the following theorem. In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime (no two See more The Chinese Remainder Theorem (which will be referred to as CRT in the rest of this article) was discovered by Chinese mathematician Sun Zi. For this, we turn to an ancient Chinese theorem that was used to calculate the calendar and find the number of soldiers when marching in lines. (3) When we divide it by 5, we get remainder 1. 2) For a word problem about rabbits in groups, the solution finds the number of rabbits What is the most efficient (in terms of running time) algorithm that solves the Chinese remainder theorem (CRT) on a set of integer residues. , ms ∈ Z be pairwise coprime, i. **Brief Answer:** The Chinese Remainder Algorithm is All Algorithms implemented in Rust . The name "Chinese" comes from an old Chinese puzzle allegedly posed by Sun Tsu Suan-Ching in 4 AD: In this post, I would like to introduce some of you to a very popular, yet maybe not fully understood technique called Chinese Remainder Theorem (CRT). Let m =[m1,K,mk] and i i m m c =, for 1≤i ≤k. A robust phase unwrapping based on the Chinese remainder theorem (CRT) has been proposed lately, and it has applications in synthetic aperture radar (SAR) imaging for moving targets. Chinese Remainder Theorem tells us that there is a unique solution modulo m, where m = 11 ⋅ 16 ⋅ 21 ⋅ 25 = 92400. Chinese remainder theorem . Instant dev environments GitHub Copilot. math; Newer. Ex 3. If n1 and n2 are two such integers, then n1=n2(mod ab) Algorithm : 1. You used one or more of the fields on the left, so your equations are of the form bx ≡ a mod m. It provides a way to solve systems of simultaneous congruences with different moduli. The Chinese Remainder Theorem (CRT) provides a solution to a system of simultaneous congruences with different moduli. Xiaoping Li, Member, IEEE , Shiyang Sun, Qunying Liao, Xiang-Gen Xia, Fellow, IEEE The work of Xiaoping Li was supported in part by the National Natural Science Foundation of China under Grant 62131005. There is also a variant of the CRT used to speed up the calculations in the RSA algorithm. Let r 1 and r 2 be any two positive numbers less than m 1 and m 2, respectively. Here, r is also equal to the compression factor of the image. −1. Hence gcd(d, (N)) = 1 and by step 5, e can be computed. Copy link Contributor. By breaking down complex problems into simpler components, the CRT allows for easier computation and problem-solving in modular arithmetic. Claim 1: (c1 systematic algorithm is required. ≡ 1 Note that all the theorem says is that there is a unique solution. 1,024-, 2,048- or 4,096-bit integers n are commonly used, making calculations in very time-consuming. Throughout the article, an efficient approach for implementing the CRT algorithm is described. If we assume for a moment that the child didn’t make any mistakes in sorting the pennies into piles, then \(x\) satisfies the three congruences \[x \equiv 2 \pmod 3; \qquad x \equiv 1 \pmod 4; \qquad x \equiv 7 \pmod {11}. It doesn't actually say how to solve it. Reany February 5, 2020 Abstract The Chinese Remainder Theorem using undetermined coe cients. How do we find these solutions? Case 1: g = (a, m) = 1. For example, Fibonacci's description is translated, as are old Chinese applications Chinese remainder theorem - Download as a PDF or view online for free. The Chinese Remainder Theorem is a useful tool in number theory (we'll use it in section 3. Using the chinese remainder theorem we are implementing the “Winogard’s small convolution algorithm”. Suppose m = pq where p and q are large, distinct primes. The chinese remainder theorem is used to integrate large numbers of integers as it is easier to compute with reduces the number of steps. 计算所有模数的积 ； 对于第 个方程： 计算 ； 计算 在模 意义下的 逆元 ； 计算 （不要对 Chinese remainder theorem or Hensel’s lemma. Then invert a mod m to get x ≡ a. Topics Mathematics. Instant dev environments Issues. Algebra and Number Theory. 8), and also has proved useful in the study and development of modern cryptographic systems. AWS; Algorithms; Bill Blair; CEMC; Deep learning; Applications. This is usually done using Gauss's algorithm. In this book, Euclid describes a way to find the greatest common divisor of two numbers. Rain Older. Chinese Remainder Theorem | Set 1 (Introduction) We have discussed a Naive solution to find minimum x. Resources. 1 Introduction The Chinese remainder theorem [1], which is creatively put forward by Sun Tzu, an ancient Chinese military strategist who composed the brilliant military writing “The Art of War”, is a constructive algorithm to find the solution of a positive integer divided by some given All the solutions of this system are congruent modulo p 1 p 2 p n. Plan and track work Code Review. Contribute to TheAlgorithms/Rust development by creating an account on GitHub. . Learning Resource Types assignment_turned_in Problem The Chinese Remainder Theorem. The article does not currently give an algorithm for finding solutions to the CRT problem. Here’s a step-by-step explanation: Problem Setup. There is a systematic way to construct the inverse map. Theorem and Proof; Solving Systems of Congruences; Problem Transform the equations. Step 2. b. Conclusion. Unusually, but most interestingly, there is an excellent historical introduction to the CRT in both the Chinese and the European mathematical traditions. This system enables the recovery of high dynamic range complex-valued bandlimited signals at low sampling rates via the Chinese remainder theorem (CRT). Another consequence of the CRT is that we can represent big numbers using an array of small integers. nrich’s article on the chinese remainder illustrates the system of equations with a coordinate system in n-dimensions, basically a number can represent a point in the coordinate system defined by the equation system and the point itself is a sum of unit vectors scaled by some amount Abstract: A robust phase unwrapping based on the Chinese remainder theorem (CRT) has been proposed lately, and it has applications in synthetic aperture radar (SAR) imaging for moving targets. 1. Submit Search. Tags. The work of Qunying Liao was This paper presents a fast parallel garner algorithm for Chinese remainder theorem. Suppose you have a system of congruences like this: x ≡ a 1 (mod m 1) x ≡ a 2 (mod m 2) ⋮. Theorem 1. In this article, an efficient solution to find x is discussed. Write better code with AI The Chinese Remainder Theorem P. I have seen many articles that present CRT in a way that lacks practical competitive programming approach (no info about how to handle overflows, how to implement it effectively, what to do when modulos are not coprime The Chinese remainder theorem is a key tool for the design of efficient multi-modular algorithms. The consequence was later generalized with a complete The Extended Euclidean Algorithm; The Chinese Remainder Theorem. 在巨人肩头赏风景 . pdf), Text File (. com Our secret image sharing scheme is based on Chinese Remainder Theorem (CRT), where compression and encryption is achieved by solving r different congruent equations, used for encrypting r pixels at a time in secret image. In this paper, we will mainly be concerned with multi-modular algorithms over the integers that rely on the Chinese remainder theorem. Closed jxu opened this issue Jan 14, 2023 · 5 comments Closed Chinese Remainder Theorem algorithm #1008. The theorem is named after the Chinese mathematician Sun Tzu, who first articulated the concept in his work "Sunzi Suanjing. Suppose , , , are positive integers that are pairwise co-prime. m is a Glossary Chinese Remainder Theorem The Chinese Remainder Theorem (CRT) is a principle of number theory used to solve systems of modular arithmetic equations. Find integers a and b so that: ap+ bq = 1 (this can always be done using the Euclidean algorithm). HackerRank - Regex - Repetitions. It was discovered by the Chinese mathematicians Sun Tzu and Liu Hui in the third century BC. In the previous post we said that calculations mod m can often be carried out more efficiently by working mod p and mod q, then combining the results to get back to a result mod m. 10. 1 ORE’S ALGORITHM Ore [8] has proposed an interesting proof for the general Chinese remainder theorem. 4 Nowadays, we have found more uses for this theorem, especially in cryptography and cybersecurity schema. Then the simultaneous congruences $$x \equiv b\pmod{m} \quad Example \(\PageIndex{1}\): Chinese Remainder Theorem Pennies. Ø To discuss various examples Euler’s and Fermat’s Theorem. We apply the technique of the Chinese Remainder Theorem with k = 4, m 1 = 11, m 2 = 16, m 3 = 21, m 4 = 25, a 1 = 6, a 2 = 13, a 3 = 9, a 4 = 19, to obtain the solution. Its ability to handle systems of congruences with coprime moduli makes it a natural fit for modular arithmetic, which is the mathematical foundation of many cryptographic systems. Recently, a multi-channel self-reset analog-to-digital converter (ADC) system with complex-valued moduli has been proposed. Then there exists Chinese Remainder Theorem algorithm #1008. pdf - Free download as PDF File (. Put N = n 1n 2:::n k, the How to execute the Chinese Remainder Theorem. Abhinav Kumar; Departments Mathematics; As Taught In Spring 2012 Level Undergraduate. In this paper, we proposed the computation efficient (n, n)-single secret sharing scheme using the Chinese Remainder Theorem (CRT), modified Shamir’s scheme, This makes the name "Chinese Remainder Theorem'' seem a little more appropriate. jxu opened this issue Jan 14, 2023 · 5 comments Comments . Stack Exchange Network. It also leads to a type of robust CRT. Given an ordered pair (r;s), take the remainder when: rbq + sap is divided by pq 中国剩余定理 Chinese Remainder Theorem. This system enables the recovery of high dynamic range Similarly, gcd(d, q − 1) = 1. 0 + my. We compute z 1 = m / m 1 = m 2 m 3 m 4 = 16 ⋅ 21 ⋅ The existing scheme uses lightweight modular arithmetic and Boolean operations for the secret sharing scheme by compromising minor degradation in security with less computation overhead. If n1 and n2 are 中国剩余定理 (Chinese Remainder Theorem, CRT) 可求解如下形式的一元线性同余方程组（其中 两两互质）： 上面的「物不知数」问题就是一元线性同余方程组的一个实例。 过程. Chinese Remainder Theorem . If the \(n_i\) are pairwise coprime, and if \(a_1, , a_k\) are any integers \(\begin{align} (d_i\) is easily calculated by the extended Euclidean algorithm. Table of contents: The algorithm; Example; The article explains how to find the minimum positive integer x that satisfies a system of simultaneous congruences using the Chinese Remainder Theorem, given two For any integer \(n\), we factorize \(n\) into primes \(n = p_1^{k_1} p_m^{k_m}\) and then use the Chinese Remainder Theorem to get \[ \mathbb{Z}_n = \mathbb{Z}_{p_1^{k_1}} \times How to prove this theorem? And why is it useful to have this isomorphism? modular algorithms! Setup: m1, . Posted on 14 September 2023 by John. Find and fix vulnerabilities Actions. In particular his book "The Elements" on Geometry was used as a standard textbook in schools for about 2000 years. Let m = m1 · · · ms. The variables in garner algorithm are divided into public parameters that are constants for fixed module and private parameters that represent random input integers. This is presented in more depth than what we really need at this point. Given a,m ∈ Z with m>1, we will denote by a rem m ∈Rm:= {0,,m−1} the remainder The Chinese Remainder Algorithm utilizes this theorem to find such solutions efficiently, making it particularly useful in areas like cryptography, computer science, and coding theory. Chinese remainder theorem (algorithm) Definition: An integer n can be solved uniquely mod LCM(A(i)), given modulii (n mod A(i)), A(i) > 0 for i=1. . The Chinese Remainder Theorem (CRT) is a powerful and ancient mathematical tool that originated in China during the 3rd century AD. This algorithm requires a two-dimensional searching. Sign in Product Actions. In this letter, we propose a new and fast algorithm that only does one-dimensional The Chinese Remainder Theorem (CRT) and underlying algorithm allows to work with multiple moduli The general idea is to compute a large integer X knowing only its remainders modulo a small set of integers (called moduli) The principles of this method was established sometime in the 3rd and 5th century in China A Chinese mathematician Sun Tzu or Sunzi is known to be Chinese Remainder Theorem. We want them to be of the form x ≡ a mod m, so we need to move the values on the left to the right side of the equation. Use extended euclid algorithm If GCD(a,b) = 1, then for any remainder ra modulo a and any remainder rb modulo b there exists integer n, such that n = ra (mod a) and n = ra(mod b). May 24, 2020 Download as PPTX, PDF 0 likes 755 views. Then , i. To apply the Chinese Remainder Theorem in step 4, the respective moduli have to be relatively prime in pairs for a solution to necessarily exist. Instead of doing a lot of computations with very large numbers numbers, which might be expensive (think of doing divisions with 1000-digit numbers), you can pick a couple of coprime moduli and represent the large number as a system of congruences, and perform all Keywords: garner algorithm, Chinese remainder theorem, parallel processing, balanced binary tree. The Chinese Remainder Theorem Linear Congruences, Chinese Remainder Theorem, Algorithms, Lecture 5 Notes Download File Course Info Instructor Prof. Shwetang (shweacha) A Chinese Remainder Theorem (CRT) is explained intuitively, which helps reader to understand its connection with Abstract Algebra concepts like Groups and Erweiterter Euklidischer Algorithmus - Extended Euclidean Algorithm. The Chinese remainder theorem is a theorem which gives a unique solution to simultaneous linear congruences with coprime moduli. Then, for any given sequence of integers , , , , there exists an integer solving the following system of simultaneous congruences: () Furthermore, all solutions of this 1 Chinese Remainder Theorem In today’s lecture we will be talking about a new tool: Chinese Remaindering which is extremely useful in designing new algorithms and speeding up existing algorithms. The extended Euclidean algorithm (see →here) finds the values s and t that satisfy the equation s·a + t·b = gcd(a,b) for given a, b. The project also includes block creation and padding for secure message transmission. In this paper, we investigate complex-valued CRT (C-CRT) with erroneous In this paper, we deal with the critical problem of performing non-modular operations in the Residue Number System (RNS). We have already proved the Fast Parallel Garner Algorithm for Chinese Remainder Theorem Yongnan Li, Limin Xiao, Aihua Liang, Yao Zheng, Li Ruan To cite this version: Yongnan Li, Limin Xiao, Aihua Liang, Yao Zheng, Li Ruan. Chinese remainder theorem You are encouraged to solve this task according to the task description, using any language you may know. A new and fast algorithm is proposed that only does one-dimensional searching and therefore reduces the complexity significantly significantly and leads to a type of robust CRT. Proof. This project implements the RSA algorithm in Python, including the Chinese Remainder Theorem (CRT) for faster decryption. The theorem finds the smallest number that leaves the same remainder when divided by different divisors. x ≡ a k (mod m k) Introduction. It is done as follows: Step 1. Formulation ¶ Let $m = m_1 Chinese Remainder Theorem is a mathematical principle that solves systems of modular equations by finding a unique solution from the remainder of the division. The Chinese Remainder Theorem (CRT) is widely used in many modern computer applications. 7. Skip to content. They can also be solved in sympy . Wikipedia has a nice section regarding the speedup of the RSA decryption using the Chinese Remainder Theorem here. The Chinese Remainder Theorem (CRT) gives the answer to the problem: Find the number x, that satisfies all the n equations simultaneously: x = r1 (mod p1) x = r2 (mod p2) x = rk (mod pk) x = rn (mod pn) We will assume here (for practical purposes) that the moduli pk are primes. Let n 1;n 2;:::;n k be a set of pairwise relatively prime natural numbers, and let b 1;b 2;:::;b k 2 Z. To prove the theorem, first, we verify that there is always a solution for x modulo m i, and then, if the solution is unique in modulo m i for all 1≤ i ≤ r. Linear Congruences, Chinese Remainder Theorem, Algorithms Recap - linear congruence ax ≡ b mod m has solution if and only if g = (a, m) divides b. The Chinese Remainder Theorem The Chinese Remainder Theorem (CRT) and underlying algorithm allows to work with multiple moduli The general idea is to compute a large integer X knowing only its remainders modulo a small set of integers (called moduli) The principles of this method was established sometime in the 3rd and 5th century in China A Chinese mathematician Sun Tzu or Sunzi is known to be The focus of this book is definitely on the Chinese remainder theorem (CRT) and the corresponding algorithm. Suppose that \(x\) is the number of pennies in the child’s pile. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted In number theory, the Chinese remainder theorem states that if one knows the remainders of the euclidean division of an integer N by several integers, then one can determine uniquely the remainder of the division of N by the merchandise of these integers, under the condition that the divisors are pairwise coprime. 2. 9th International Conference on Network and Parallel Computing (NPC), Sep 2012, Gwangju, South Korea. Although Chinese Remainder Theorem is more known in reference with the integers, but the general statement of the theorem is as follows: The Chinese Remainder Theorem (CRT) is a mathematical theorem that states that if we have a system of linear congruences (equations of the form "x ≡ a mod m") with pairwise coprime moduli, then there exists a unique solution for x modulo the product of the moduli. In this tutorial, we have explored the Chinese Remainder Theorem, a powerful divide and conquer algorithm (2) When we divide it by 4, we get remainder 3. I need to understand the implementation of a similar speedup for the encryption algorithm of a more complex homomorphic encryption scheme and, for some reason, I'm unable to get my head around the way the Chinese Remainder Theorem is used to The Chinese Remainder Theorem algorithm has a time complexity of O(n²), where n is the number of congruences. 1 Introduction Let m 1 and m 2 be two relatively prime (coprime) positive integers. Contents. It is widely used in the field of software development, especially in optimizing applications and . Navigation Menu Toggle navigation . S. Existence of Solution . In other words, given the remainders an integer gets when it's divided by an arbitrary set of divisors, 2. By the Chinese remainder theorem, however, these calculations can be done in the isomorphic ring instead. Here we explain all of the steps in the algorithm of the Chinese Remainder Theorem. Navigation Menu Toggle navigation. For a more detailed explanation about how this works, see this part of our page about how to execute the Chinese Remainder algorithm. 在第一次看到 中国剩余定理 出现在 现代密码学 中的时候，我眼睛一亮，这是我们中华前辈伟大智慧成果，至今都广泛应用在现代密码学中。本文从它的应用出发讲解中国剩余定理。 中国剩余定理的一个重要作用是，加快计算一定条件下的大数 Complex-Valued Robust Chinese Remainder Theorem and Its Fast Algorithm Xiaoping Li, Member, IEEE, Shiyang Sun, Qunying Liao, Xiang-Gen Xia, Fellow, IEEE Abstract Recently, a multi-channel self-reset analog-to-digital converter (ADC) system with complex-valued moduli has been proposed. It provides functionalities for key generation, encryption, and decryption using the RSA algorithm. e. Find and fix vulnerabilities Codespaces. Sign in Product GitHub Copilot. Euclid's Algorithm Euclid of Alexandria was a Greek mathematician who lived approximately 2300 years ago and who is famous for his mathematical writing. vamc udejziv qzqbhc nfcwc uzclp knhjws burhq mitd mwmybe txgywfj zofal ufo pznto dcxokn khft