The Experts below are selected from a list of 228 Experts worldwide ranked by ideXlab platform
Chung-huang Yang - One of the best experts on this subject based on the ideXlab platform.
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Modular Arithmetic: From Ancient India to Public-Key Cryptography
2015Co-Authors: T. R. N. Rao, Chung-huang YangAbstract:Abstract. We begin with an algorithm from Aryabhatiya, for solving the indeterminate equation a·x + c = b·y of degree one (also known as Diophantine equation) and its extension to solve the system of two residues X mod mi = Xi (for i =1, 2). This contribution known as Aryabhatiya Algorithm (AA) is very profound in the sense that the problem of two congruences was solved with just one modular inverse operation and a modular reduction to a smaller modulus than the compound modulus. We extend AA to any set of t residues and is stated as Aryabhata Remainder Theorem (ART) and an iterative algorithm is given to solve for t moduli mi (i=1, 2,…, t). The ART, which has much in common with Extended Euclidean Algorithm (EEA), Chinese Remainder Theorem (CRT) and Garner’s algorithm (GA), is shown to have a complexity comparable or better than CRT and GA. Key words: Diophantine equation, Aryabhata, systems of congruences, modular arithmetic, residue number system, modular inverse
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Aryabhata Remainder Theorem: Relevance to Public-Key Crypto-Algorithms
Circuits Systems and Signal Processing, 2006Co-Authors: T. R. N. Rao, Chung-huang YangAbstract:Public-key crypto-algorithms are widely employed for authentication, signatures, secret-key generation and access control. The new range of public-key sizes for RSA and DSA has gone up to 1024 bits and beyond. The elliptic-curve key range is from 162 bits to 256 bits. Many varied software and hardware algorithms are being developed for implementation for smart-card crypto-coprocessors and for public-key infrastructure. We begin with an algorithm from Aryabhatiya for solving the indeterminate equation a · x + c = b · y of degree one (also known as the Diophantine equation) and its extension to solve the system of two residues X mod m_i = X_i (for i = 1,2). This contribution known as the Aryabhatiya algorithm (AA) is very profound in the sense that the problem of two congruences was solved with just one modular inverse operation and a modular reduction to a smaller modulus than the compound modulus. We extend AA to any set of t residues, and this is stated as the Aryabhata remainder theorem (ART). An iterative algorithm is also given to solve for t moduli m_i (i = 1, 2,... , t). The ART, which has much in common with the extended Euclidean algorithm (EEA), Chinese remainder theorem (CRT) and Garner's algorithm (GA), is shown to have a complexity comparable to or better than that of the CRT and GA.
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Aryabhata REMAINDER THEOREM: RELEVANCE TO
2004Co-Authors: T. R. N. Rao, Chung-huang YangAbstract:Abstract. Public-key crypto-algorithms are widely employed for authentication, signatures, secret-key generation and access control. The new range of public-key sizes for RSA and DSA has gone up to 1024 bits and beyond. The elliptic-curve key range is from 162 bits to 256 bits. Many varied software and hardware algorithms are being developed for implementation for smart-card crypto-coprocessors and for public-key infrastructure. We begin with an algorithm from Aryabhatiya for solving the indeterminate equation a ·x +c = b · y of degree one (also known as the Diophantine equation) and its extension to solve the system of two residues X mod mi = Xi (for i = 1, 2). This contribution known as the Aryabhatiya algorithm (AA) is very profound in the sense that the problem of two congruences was solved with just one modular inverse operation and a modular reduction to a smaller modulus than the compound modulus. We extend AA to any set of t residues, and this is stated as the Aryabhata remainder theorem (ART). An iterative algorithm is also given to solve for t moduli mi (i = 1, 2,...,t). The ART, which has much in common with the extended Euclidean algorithm (EEA), Chinese remainder theorem (CRT) and Garner’s algorithm (GA), is shown to have a complexity comparable to or better than that of the CRT and GA
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Relevance to public-key crypto-algorithms
2004Co-Authors: T. R. N. Rao, Chung-huang Yang, Crypto LabAbstract:Abstract. Public-key crypto-algorithms are widely employed for authentication, signatures, secret-key generation and access control. The new range of public-key sizes for RSA and DSA has gone up to 1024 bits and beyond. Elliptic-curve key range is from 162 bits to 256 bits. Many varied software and hardware algorithms are being developed for implementation for smart-card crypto-coprocessors and for public-key infrastructure. We begin with an algorithm from Aryabhatiya, for solving the indeterminate equation a·x + c = b·y of degree one (also known as Diophantine equation) and its extension to solve the system of two residues X mod m i = X i (for i =1, 2). This contribution known as Aryabhatiya Algorithm (AA) is very profound in the sense that the problem of two congruences was solved with just one modular inverse operation and a modular reduction to a smaller modulus than the compound modulus. We extend AA to any set of t residues and is stated as Aryabhata Remainder Theorem (ART) and an iterative algorithm is given to solve for t moduli mi (i=1, 2,…, t). The ART, which has much in common with Extended Euclidean Algorithm (EEA), Chinese Remainder Theorem (CRT) and Garner’s algorithm (GA), is shown to have a complexity comparable or better than CRT and GA. 1
T. R. N. Rao - One of the best experts on this subject based on the ideXlab platform.
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Modular Arithmetic: From Ancient India to Public-Key Cryptography
2015Co-Authors: T. R. N. Rao, Chung-huang YangAbstract:Abstract. We begin with an algorithm from Aryabhatiya, for solving the indeterminate equation a·x + c = b·y of degree one (also known as Diophantine equation) and its extension to solve the system of two residues X mod mi = Xi (for i =1, 2). This contribution known as Aryabhatiya Algorithm (AA) is very profound in the sense that the problem of two congruences was solved with just one modular inverse operation and a modular reduction to a smaller modulus than the compound modulus. We extend AA to any set of t residues and is stated as Aryabhata Remainder Theorem (ART) and an iterative algorithm is given to solve for t moduli mi (i=1, 2,…, t). The ART, which has much in common with Extended Euclidean Algorithm (EEA), Chinese Remainder Theorem (CRT) and Garner’s algorithm (GA), is shown to have a complexity comparable or better than CRT and GA. Key words: Diophantine equation, Aryabhata, systems of congruences, modular arithmetic, residue number system, modular inverse
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Aryabhata Remainder Theorem: Relevance to Public-Key Crypto-Algorithms
Circuits Systems and Signal Processing, 2006Co-Authors: T. R. N. Rao, Chung-huang YangAbstract:Public-key crypto-algorithms are widely employed for authentication, signatures, secret-key generation and access control. The new range of public-key sizes for RSA and DSA has gone up to 1024 bits and beyond. The elliptic-curve key range is from 162 bits to 256 bits. Many varied software and hardware algorithms are being developed for implementation for smart-card crypto-coprocessors and for public-key infrastructure. We begin with an algorithm from Aryabhatiya for solving the indeterminate equation a · x + c = b · y of degree one (also known as the Diophantine equation) and its extension to solve the system of two residues X mod m_i = X_i (for i = 1,2). This contribution known as the Aryabhatiya algorithm (AA) is very profound in the sense that the problem of two congruences was solved with just one modular inverse operation and a modular reduction to a smaller modulus than the compound modulus. We extend AA to any set of t residues, and this is stated as the Aryabhata remainder theorem (ART). An iterative algorithm is also given to solve for t moduli m_i (i = 1, 2,... , t). The ART, which has much in common with the extended Euclidean algorithm (EEA), Chinese remainder theorem (CRT) and Garner's algorithm (GA), is shown to have a complexity comparable to or better than that of the CRT and GA.
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Aryabhata REMAINDER THEOREM: RELEVANCE TO
2004Co-Authors: T. R. N. Rao, Chung-huang YangAbstract:Abstract. Public-key crypto-algorithms are widely employed for authentication, signatures, secret-key generation and access control. The new range of public-key sizes for RSA and DSA has gone up to 1024 bits and beyond. The elliptic-curve key range is from 162 bits to 256 bits. Many varied software and hardware algorithms are being developed for implementation for smart-card crypto-coprocessors and for public-key infrastructure. We begin with an algorithm from Aryabhatiya for solving the indeterminate equation a ·x +c = b · y of degree one (also known as the Diophantine equation) and its extension to solve the system of two residues X mod mi = Xi (for i = 1, 2). This contribution known as the Aryabhatiya algorithm (AA) is very profound in the sense that the problem of two congruences was solved with just one modular inverse operation and a modular reduction to a smaller modulus than the compound modulus. We extend AA to any set of t residues, and this is stated as the Aryabhata remainder theorem (ART). An iterative algorithm is also given to solve for t moduli mi (i = 1, 2,...,t). The ART, which has much in common with the extended Euclidean algorithm (EEA), Chinese remainder theorem (CRT) and Garner’s algorithm (GA), is shown to have a complexity comparable to or better than that of the CRT and GA
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Relevance to public-key crypto-algorithms
2004Co-Authors: T. R. N. Rao, Chung-huang Yang, Crypto LabAbstract:Abstract. Public-key crypto-algorithms are widely employed for authentication, signatures, secret-key generation and access control. The new range of public-key sizes for RSA and DSA has gone up to 1024 bits and beyond. Elliptic-curve key range is from 162 bits to 256 bits. Many varied software and hardware algorithms are being developed for implementation for smart-card crypto-coprocessors and for public-key infrastructure. We begin with an algorithm from Aryabhatiya, for solving the indeterminate equation a·x + c = b·y of degree one (also known as Diophantine equation) and its extension to solve the system of two residues X mod m i = X i (for i =1, 2). This contribution known as Aryabhatiya Algorithm (AA) is very profound in the sense that the problem of two congruences was solved with just one modular inverse operation and a modular reduction to a smaller modulus than the compound modulus. We extend AA to any set of t residues and is stated as Aryabhata Remainder Theorem (ART) and an iterative algorithm is given to solve for t moduli mi (i=1, 2,…, t). The ART, which has much in common with Extended Euclidean Algorithm (EEA), Chinese Remainder Theorem (CRT) and Garner’s algorithm (GA), is shown to have a complexity comparable or better than CRT and GA. 1
Shih-chang Chang - One of the best experts on this subject based on the ideXlab platform.
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An Access Control Mechanism Based on the Generalized Aryabhata Remainder Theorem
2013Co-Authors: Yanjun Liu, Shih-chang ChangAbstract:An access control mechanism is a technology to protect the confidential files stored in a database by restricting the access rights of different approved users of these files. In this paper, we propose a novel access control mechanism using the single-key-lock system and the generalized Aryabhata remainder theorem (GART), in which each user is associated with a key and each digital file with a lock. Our mechanism possesses three unique features, 1) a high efficiency of constructing the keys for the users and the locks for the files; 2) a simple operation on the user’s key and on the file’s lock allows the user access to the file; and 3) the keys for adding or deleting a user can be updated easily without affecting the existing keys for other users
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An Efficient Oblivious Transfer Protocol Using Residue Number System
2012Co-Authors: Yanjun Liu, Chin-chen Chang, Shih-chang ChangAbstract:Because the t-out-of-n oblivious transfer (OT) protocol can guarantee the privacy of both participants, i.e., the sender and the receiver, it has been used extensively in the study of cryptography. Recently, Chang and Lee presented a robust t-out-of-n OT protocol based on the Chinese remainder theorem (CRT). In this paper, we use the Aryabhata remainder theorem (ART) to achieve the functionality of a t-out-of-n OT protocol, which is more efficient than Chang and Lee’s mechanism. Analysis showed that our proposed protocol meets the fundamental requirements of a general t-out-of-n OT protocol. We also utilized BAN logic to prove that our proposed protocol maintains the security when messages are transmitted between the sender and the receiver
Jérôme Petit - One of the best experts on this subject based on the ideXlab platform.
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Eugène Jacquet and his Pioneering Study of Indian Numerical Notations
2009Co-Authors: Jérôme PetitAbstract:Eugene Jacquet was one of the earliest European scholars who made a comprehensive study of all the systems of numerical notation which were prevalent in India. In a pioneering paper published in 1830 he put together the scattered pieces of information then available in Europe and built up a large corpus of symbolic words employed in Sanskrit astronomical texts to represent numbers. He was one of the first scholars to realise that this system of word numerals used in Sanskrit texts spread outside India and was to be found in Tibet and Java. Jacquet also discussed the two variants of alphabetic notation employed in astronomical texts, viz. the system designed and employed by Aryabhata in his Aryabhatia and the more widely used Katapayadi alphabetic notation. The present article, after a brief outline of Jacquet’s life, introduces Jacquet’s paper on the numerical notations in India.
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Eugène Jacquet and his Pioneering Study of Indian Numerical Notations
Ganita Bharati (Indian Mathematics): Journal of the Indian Society for History of Mathematics, 2009Co-Authors: Jérôme PetitAbstract:Eugène Jacquet was one of the earliest European scholars who made a comprehensive study of all the systems of numerical notation which were prevalent in India. In a pioneering paper published in 1830 he put together the scattered pieces of information then available in Europe and built up a large corpus of symbolic words employed in Sanskrit astronomical texts to represent numbers. He was one of the first scholars to realise that this system of word numerals used in Sanskrit texts spread outside India and was to be found in Tibet and Java. Jacquet also discussed the two variants of alphabetic notation employed in astronomical texts, viz. the system designed and employed by Aryabhata in his Aryabhatia and the more widely used Katapayadi alphabetic notation. The present article, after a brief outline of Jacquet’s life, introduces Jacquet’s paper on the numerical notations in India.
Abhishek Parakh - One of the best experts on this subject based on the ideXlab platform.
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Aryabhata's Root Extraction Methods
arXiv: History and Overview, 2006Co-Authors: Abhishek ParakhAbstract:We investigate the mathematics behind 1500 year old root extraction methods presented by Aryabhata in his famous mathematical treatise "Aryabhatiya". Also, we look at their computational complexity.
