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M.n.s. Swamy - One of the best experts on this subject based on the ideXlab platform.

  • A high secure approach for medical images transmission over wireless channels
    2012 Annual IEEE India Conference (INDICON), 2012
    Co-Authors: K. Praveenkumar, M.n.s. Swamy, K. Deergha Rao
    Abstract:

    In this paper, a high secure approach for transmission of medical images over wireless channels is presented. In this approach, algorithms based on Chaos and Brahmagupta-Bhaskara (BB) equation are proposed for encryption and decryption of the medical images. and no lossy encoding is used for coding the encrypted medical images. Furthermore, turbo channel coding is proposed to correct the transmission errors over impulsive noisy wireless channels. The efficacy of the proposed approach is illustrated with the implementation results on a medical image.

  • cryptographic applications of Brahmagupta bha spl tilde skara equation
    IEEE Transactions on Circuits and Systems I-regular Papers, 2006
    Co-Authors: N.r. Murthy, M.n.s. Swamy
    Abstract:

    The Brahmagupta-Bha/spl tilde/skara (BB) equation is a quadratic Diophantine equation of the form NX/sup 2/+k=Y/sup 2/, where k is an integer (positive or negative) and N is a positive integer such that /spl radic/N is irrational. A particular case of the BB equation with k=1 is also known as Pell equation in literature. This equation in the Galois Field GF(p), where p is an odd prime has some practically useful properties. Application of these properties in two different fields of cryptography, namely, digital encryption and user authentication are discussed in this paper. For those applications, where software computation of the roots of the BB equation is unacceptable for being too slow, a hardware architecture for using the BB equation in GF(p) is given that is useful for implementation in VLSI form.

  • Cryptographic applications of Brahmagupta-Bha/spl tilde/skara equation
    IEEE Transactions on Circuits and Systems I: Regular Papers, 2006
    Co-Authors: N.r. Murthy, M.n.s. Swamy
    Abstract:

    The Brahmagupta-Bha/spl tilde/skara (BB) equation is a quadratic Diophantine equation of the form NX/sup 2/+k=Y/sup 2/, where k is an integer (positive or negative) and N is a positive integer such that /spl radic/N is irrational. A particular case of the BB equation with k=1 is also known as Pell equation in literature. This equation in the Galois Field GF(p), where p is an odd prime has some practically useful properties. Application of these properties in two different fields of cryptography, namely, digital encryption and user authentication are discussed in this paper. For those applications, where software computation of the roots of the BB equation is unacceptable for being too slow, a hardware architecture for using the BB equation in GF(p) is given that is useful for implementation in VLSI form.

N.r. Murthy - One of the best experts on this subject based on the ideXlab platform.

  • cryptographic applications of Brahmagupta bha spl tilde skara equation
    IEEE Transactions on Circuits and Systems I-regular Papers, 2006
    Co-Authors: N.r. Murthy, M.n.s. Swamy
    Abstract:

    The Brahmagupta-Bha/spl tilde/skara (BB) equation is a quadratic Diophantine equation of the form NX/sup 2/+k=Y/sup 2/, where k is an integer (positive or negative) and N is a positive integer such that /spl radic/N is irrational. A particular case of the BB equation with k=1 is also known as Pell equation in literature. This equation in the Galois Field GF(p), where p is an odd prime has some practically useful properties. Application of these properties in two different fields of cryptography, namely, digital encryption and user authentication are discussed in this paper. For those applications, where software computation of the roots of the BB equation is unacceptable for being too slow, a hardware architecture for using the BB equation in GF(p) is given that is useful for implementation in VLSI form.

  • Cryptographic applications of Brahmagupta-Bha/spl tilde/skara equation
    IEEE Transactions on Circuits and Systems I: Regular Papers, 2006
    Co-Authors: N.r. Murthy, M.n.s. Swamy
    Abstract:

    The Brahmagupta-Bha/spl tilde/skara (BB) equation is a quadratic Diophantine equation of the form NX/sup 2/+k=Y/sup 2/, where k is an integer (positive or negative) and N is a positive integer such that /spl radic/N is irrational. A particular case of the BB equation with k=1 is also known as Pell equation in literature. This equation in the Galois Field GF(p), where p is an odd prime has some practically useful properties. Application of these properties in two different fields of cryptography, namely, digital encryption and user authentication are discussed in this paper. For those applications, where software computation of the roots of the BB equation is unacceptable for being too slow, a hardware architecture for using the BB equation in GF(p) is given that is useful for implementation in VLSI form.

K. Ramasubramanian - One of the best experts on this subject based on the ideXlab platform.

  • Munīśvara’s Modification of Brahmagupta’s Rule for Second-Order Interpolation
    Gaṇitānanda, 2019
    Co-Authors: K. Ramasubramanian
    Abstract:

    When the values of a function are tabulated for some discrete values of the argument, the functional values corresponding to intermediary argumental values are obtained ordinarily by linear interpolation. For greater accuracy, higher order technique is necessary. It is known that the famous Indian mathematician Brahmagupta (seventh century ad) gave a rule for second-order interpolation.

  • M. Rangacharya and His Century Old Translation of the Gaṇita-sāra-saṅgraha
    Gaṇitānanda, 2019
    Co-Authors: K. Ramasubramanian
    Abstract:

    Lots of Achievements in ancient Indian Mathematics as reflected in the works of Āryabhaṭa I (born 476 ad), Brahmagupta (seventh century ad) and Bhāskara II (twelfth century) were freshly made known in modern form to the Western world during the nineteenth century. Leading role in this regard was displayed by Western scholars such as R. Barrow, H. T. Colebrooke, S. Davies, C. Hutton, H. Kern, L. Rodet, E. Strachey and John Taylor.

  • Review of Rājamṛgāṅka of Bhojarāja
    Sources and Studies in the History of Mathematics and Physical Sciences, 2019
    Co-Authors: Aditya Kolachana, K. Mahesh, K. Ramasubramanian
    Abstract:

    The Rājamṛgāṅka ascribed to the Paramāra king Bhoja of Dhar is the earliest karaṇa (“hand-book of astronomy”) based on the teachings of the Brāhmasphuṭasiddhānta of Brahmagupta. It also incorporates at places teachings of the Sūryasiddhānta, the Romakasiddhānta, the Khaṇḍakhādyaka of Brahmagupta, the Śiṣyadhīvṛddhida of Lalla, the Laghumānasa of Manjula, and other earlier works.

  • Glimpses from the Āryabhaṭasiddhānta
    Sources and Studies in the History of Mathematics and Physical Sciences, 2019
    Co-Authors: Aditya Kolachana, K. Mahesh, K. Ramasubramanian
    Abstract:

    In a paper entitled “Āryabhaṭa I’s astronomy with midnight day-reckoning” published by me nine years ago in the Gaṇita (Vol. 18, No. I, 1967), I had adduced concrete and conclusive evidence to show that Āryabhaṭa I, the celebrated author of the Āryabhaṭīya, wrote one more work on astronomy which was known as Āryabhaṭasiddhānta. Whereas in the Āryabhaṭīya the day was reckoned from one sunrise to the next, in the Āryabhaṭasiddhānta the day was reckoned from one midnight to the next. This latter work of Āryabhaṭa which adopted midnight day-reckoning was first mentioned by Brahmagupta (628 AD) of Bhinmal in Rajasthan, who was so much impressed by its wide popularity that he epitomised the teachings of this work in his calendrical work bearing the title “Food prepared with sugar candy” (Khaṇḍakhādyaka).

  • Mensuration of quadrilaterals in the Līlāvatī
    Sources and Studies in the History of Mathematics and Physical Sciences, 2019
    Co-Authors: K. Ramasubramanian, Takao Hayashi, Clemency Montelle, S. G. Dani
    Abstract:

    Mensuration with quadrilaterals had received attention in the siddhānta tradition at least since Brahmagupta. However, in Bhāskarācārya’s Līlāvatī we come across some distinctively new features. In this paper an attempt will be made to put the development in historical perspective.

Satyanad Kichenassamy - One of the best experts on this subject based on the ideXlab platform.

  • Brahmagupta's apodictic discourse
    Gaṇita Bhāratī, 2021
    Co-Authors: Satyanad Kichenassamy
    Abstract:

    We continue our analysis of Brahmagupta’s Brāhmasphuṭasiddhānta (India, 628), that had shown that each of his sequences of propositions should be read as an apodictic discourse: a connected discourse that develops the natural consequences of explicitly stated assumptions, within a particular conceptual framework. As a consequence, we established that Brahmagupta did provide a derivation of his results on the cyclic quadrilateral. We analyze here, on the basis of the same principles, further problematic passages in Brahmagupta’s magnum opus, regarding number theory and algebra. They make no sense as sets of rules. They become clear as soon as one reads them as an apodictic discourse, so carefully composed that they leave little room for interpretation. In particular, we show that (i) Brahmagupta indicated the principle of the derivation of the solution of linear congruences (the kuṭṭaka) at the end of chapter 12 and (ii) his algebra in several variables is the result of the extension of operations on numbers to new types of quantities – negative numbers, surds and “non-manifest” variables.

  • Apodictic discourse in ancient and modern Mathematics
    2020
    Co-Authors: Satyanad Kichenassamy
    Abstract:

    Many important texts in Mathematics have often been felt to be mathematically incomplete. We report on a series of papers that shows that well-known seemingly faulty texts by Brahmagupta, Baudhayana and Tartaglia actually contain the missing information encoded within their discursive structure : they are what we have called apodictic discourses. After reviewing these results, we suggest that (i) Ramanujan may have similarly embedded seemingly missing arguments in the structure of his exposition; (ii) the structure of teaching in the U.K. -- that differed from what is observed in France for example -- could account for the dogmatic style S. Ramanujan chose; (iii) a discourse analysis helps remove blinkers in modern Science as well.

  • Le ``triquadrilatère'' de Brahmagupta
    2015
    Co-Authors: Satyanad Kichenassamy
    Abstract:

    On considère généralement que l'œuvre de Brahmagupta est, avec celle d’Āryabhaṭa I, l’une des sources principales des résultats et concepts d’origine indienne dans les mathématiques modernes. L'auteur s’intéresse ici aux propositions XII.21-32 du Brāhma-sphuṭa-siddhānta ou « Système précisé de Brahmā » (628 ap. J.-C., en vers sanskrits), consacrées aux propriétés d’une figure que Brahmagupta appelle « triquadrilatère », et qu’il ne définit pas. Il se propose de montrer, par l’analyse interne du texte, que la cohérence interne du texte permet de déterminer son objet, et indique en même temps les éléments d’une dérivation possible des résultats énoncés. A ce jour, c'est la seule dérivation connue qui soit compatible avec le texte.

  • le triquadrilatere de Brahmagupta
    2015
    Co-Authors: Satyanad Kichenassamy
    Abstract:

    On considere generalement que l'œuvre de Brahmagupta est, avec celle d’Āryabhaṭa I, l’une des sources principales des resultats et concepts d’origine indienne dans les mathematiques modernes. L'auteur s’interesse ici aux propositions XII.21-32 du Brāhma-sphuṭa-siddhānta ou « Systeme precise de Brahmā » (628 ap. J.-C., en vers sanskrits), consacrees aux proprietes d’une figure que Brahmagupta appelle « triquadrilatere », et qu’il ne definit pas. Il se propose de montrer, par l’analyse interne du texte, que la coherence interne du texte permet de determiner son objet, et indique en meme temps les elements d’une derivation possible des resultats enonces. A ce jour, c'est la seule derivation connue qui soit compatible avec le texte.

  • Brahmagupta’s propositions on the perpendiculars of cyclic quadrilaterals
    Historia Mathematica, 2012
    Co-Authors: Satyanad Kichenassamy
    Abstract:

    We continue a recent analysis of Propositions XII.21-28 of Brahmagupta's Brāhma-sphuṭa-siddhānta (India, 628 A.D.), on the area and diagonals of the cyclic quadrilateral, by examining Propositions XII.29-32, that explain how to determine the perpendiculars as well as all the portions of diagonals and perpendiculars. These results include the result nowadays referred to as "Brahmagupta's theorem" (XII.30-31). Brahmagupta describes both the geometric situation and the key elements of the derivation of his results. We analyze the expression of hypotheses and derivations, using only Brahmagupta's conceptual framework, that does not include the notion of angle, and uses proportion only in a standard form (XII.25).

Kichenassamy Satyanad - One of the best experts on this subject based on the ideXlab platform.

  • Le ``triquadrilatère'' de Brahmagupta: Analyse d'un texte mathématique
    IREM de Reims, 2015
    Co-Authors: Kichenassamy Satyanad
    Abstract:

    International audienceOn considère généralement que l'œuvre de Brahmagupta est, avec celle d’Āryabhaṭa I, l’une des sources principales des résultats et concepts d’origine indienne dans les mathématiques modernes. L'auteur s’intéresse ici aux propositions XII.21-32 du Brāhma-sphuṭa-siddhānta ou « Système précisé de Brahmā » (628 ap. J.-C., en vers sanskrits), consacrées aux propriétés d’une figure que Brahmagupta appelle « triquadrilatère », et qu’il ne définit pas. Il se propose de montrer, par l’analyse interne du texte, que la cohérence interne du texte permet de déterminer son objet, et indique en même temps les éléments d’une dérivation possible des résultats énoncés. A ce jour, c'est la seule dérivation connue qui soit compatible avec le texte

  • L’analyse littéraire au service de l’histoire des mathématiques : critique interne de la géométrie de Brahmagupta
    'PERSEE Program', 2012
    Co-Authors: Kichenassamy Satyanad
    Abstract:

    Kichenassamy Satyanad. L’analyse littéraire au service de l’histoire des mathématiques : critique interne de la géométrie de Brahmagupta . In: Comptes rendus des séances de l'Académie des Inscriptions et Belles-Lettres, 156e année, N. 2, 2012. pp. 781-796

  • Brahmagupta's propositions on the perpendiculars of cyclic quadrilaterals
    'Elsevier BV', 2012
    Co-Authors: Kichenassamy Satyanad
    Abstract:

    International audienceWe continue a recent analysis of Propositions XII.21-28 of Brahmagupta's Brāhma-sphuṭa-siddhānta (India, 628 A.D.), on the area and diagonals of the cyclic quadrilateral, by examining Propositions XII.29-32, that explain how to determine the perpendiculars as well as all the portions of diagonals and perpendiculars. These results include the result nowadays referred to as "Brahmagupta's theorem" (XII.30-31). Brahmagupta describes both the geometric situation and the key elements of the derivation of his results. We analyze the expression of hypotheses and derivations, using only Brahmagupta's conceptual framework, that does not include the notion of angle, and uses proportion only in a standard form (XII.25)

  • Brahmagupta’s derivation of the area of a cyclic quadrilateral
    Elsevier Inc., 2010
    Co-Authors: Kichenassamy Satyanad
    Abstract:

    AbstractThis paper shows that Propositions XII.21–27 of Brahmagupta’s Bra¯hmasphuṭasiddha¯nta (628 a.d.) constitute a coherent mathematical discourse leading to the expression of the area of a cyclic quadrilateral in terms of its sides. The radius of the circumcircle is determined by considering two auxiliary quadrilaterals. Observing that a cyclic quadrilateral is split by a diagonal into two triangles with the same circumcenter and the same circumradius, the result follows, using the tools available to Brahmagupta. The expression for the diagonals (XII.28) is a consequence. The shortcomings of earlier attempts at reconstructing Brahmagupta’s method are overcome by restoring the mathematical consistency of the text. This leads to a new interpretation of Brahmagupta’s terminology for quadrilaterals of different types

  • Brahmagupta's derivation of the area of a cyclic quadrilateral
    'Elsevier BV', 2010
    Co-Authors: Kichenassamy Satyanad
    Abstract:

    International audienceThis paper shows that Propositions XII.21–27 of Brahmagupta's Br¯ ahma-sphut. asiddh¯ anta (628 a.d.) constitute a coherent mathematical discourse leading to the expression of the area of a cyclic quadrilateral in terms of its sides. The radius of the circumcircle is determined by considering two auxiliary quadrilaterals. Observing that a cyclic quadrilateral is split by a diagonal into two triangles with the same circumcenter and the same circumradius, the result follows, using the tools available to Brahmagupta. The expression for the diagonals (XII.28) is a consequence. The shortcomings of earlier attempts at reconstructing Brahmagupta's method are overcome by restoring the mathematical consistency of the text. This leads to a new interpretation of Brahmagupta's terminology for quadrilaterals of different types. Résumé. On montre que les propositions XII.21–27 du Br¯ ahmasphut. asiddh¯ anta (628 ap. J.-C.) forment un discours cohérent conduisantàconduisantà l'expression de l'aire d'un quadrilatère cyclique en termes de ses côtés. Le rayon du cercle circonscrit est déterminé en considérant deux quadrilatères auxiliaires. Exprimant que le quadrilatère cyclique est partagé par une diagonale en deux triangles ayant en commun le centre et le rayon de leur cercle circonscrit, on obtient l'aire du quadrilatère, ` a l'aide des outils connus de Brahmagupta. L'expression des diagonales (XII.28) en découle. Les difficultés des tentatives antérieures en vue de retrouver la démarche de Brahmagupta sont résolues en restituant la cohérence mathématique du texte. On est ainsi conduitàconduità une nouvelle interprétation des termes qu'utilise Brahmagupta pour désigner des quadrilatères de différentes classes