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21 produkter
21 produkter
Inbunden, Engelska, 2015
539 kr
Skickas inom 10-15 vardagar
The theory of elliptic curves involves a pleasing blend of algebra, geometry, analysis, and number theory. This volume stresses this interplay as it develops the basic theory, thereby providing an opportunity for advanced undergraduates to appreciate the unity of modern mathematics. At the same time, every effort has been made to use only methods and results commonly included in the undergraduate curriculum. This accessibility, the informal writing style, and a wealth of exercises make Rational Points on Elliptic Curves an ideal introduction for students at all levels who are interested in learning about Diophantine equations and arithmetic geometry.Most concretely, an elliptic curve is the set of zeroes of a cubic polynomial in two variables. If the polynomial has rational coefficients, then one can ask for a description of those zeroes whose coordinates are either integers or rational numbers. It is this number theoretic question that is the main subject of Rational Points on Elliptic Curves. Topics covered include the geometry and group structure of elliptic curves, the Nagell–Lutz theorem describing points of finite order, the Mordell–Weil theorem on the finite generation of the group of rational points, the Thue–Siegel theorem on the finiteness of the set of integer points, theorems on counting points with coordinates in finite fields, Lenstra's elliptic curve factorization algorithm, and a discussion of complex multiplication and the Galois representations associated to torsion points. Additional topics new to the second edition include an introduction to elliptic curve cryptography and a brief discussion of the stunning proof of Fermat's Last Theorem by Wiles et al. via the use of elliptic curves.
Del 106 - Graduate Texts in Mathematics
Arithmetic of Elliptic Curves
Inbunden, Engelska, 2009
539 kr
Skickas inom 10-15 vardagar
The theory of elliptic curves is distinguished by the diversity of the methods used in its study. This book treats the arithmetic theory of elliptic curves in its modern formulation, through the use of basic algebraic number theory and algebraic geometry.
Del 151 - Graduate Texts in Mathematics
Advanced Topics in the Arithmetic of Elliptic Curves
Inbunden, Engelska, 1994
960 kr
Skickas inom 5-8 vardagar
In the introduction to the first volume of The Arithmetic of Elliptic Curves (Springer-Verlag, 1986), I observed that "the theory of elliptic curves is rich, varied, and amazingly vast," and as a consequence, "many important topics had to be omitted."
Del 151 - Graduate Texts in Mathematics
Advanced Topics in the Arithmetic of Elliptic Curves
Häftad, Engelska, 1994
699 kr
Skickas inom 10-15 vardagar
This book continues the treatment of the arithmetic theory of elliptic curves begun in the first volume. The book begins with the theory of elliptic and modular functions for the full modular group r(1), including a discussion of Hekcke operators and the L-series associated to cusp forms. This is followed by a detailed study of elliptic curves with complex multiplication, their associated Grossencharacters and L-series, and applications to the construction of abelian extensions of quadratic imaginary fields. Next comes a treatment of elliptic curves over function fields and elliptic surfaces, including specialization theorems for heights and sections. This material serves as a prelude to the theory of minimal models and Neron models of elliptic curves, with a discussion of special fibers, conductors, and Ogg's formula. Next comes a brief description of q-models for elliptic curves over C and R, followed by Tate's theory of q-models for elliptic curves with non-integral j-invariant over p-adic fields. The book concludes with the construction of canonical local height functions on elliptic curves, including explicit formulas for both archimedean and non-archimedean fields.
Inbunden, Engelska, 2000
1 071 kr
Skickas inom 10-15 vardagar
This volume contains expanded versions of lectures given at an instructional conference on number theory and arithmetic geometry held August 9 through 18, 1995 at Boston University. The purpose of the conference, and of this book, is to introduce and explain the many ideas and techniques used by Wiles in his proof that every (semi- stable) elliptic curve over Q is modular, and to explain how Wiles' result can be combined with Ribet's theorem and ideas of Frey and Serre to show, at long last, that Fermat's Last Theorem is true. Contributors to this volume include: B. Conrad, H. Darmon, E. de Shalit, B. de Smit, F. Diamond, S.J. Edixhoven, G. Frey, S. Gelbart, K. Kramer, H.W. Lenstra, Jr., B. Mazur, K. Ribet, D.E. Rohrlich, M. Rosen, K. Rubin, R. Schoof, A. Silverberg, J.H. Silverman, P. Stevenhagen, G. Stevens, J. Tate, J. Tilouine, and L. Washington. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions, modular curves, Galois cohomology, and finite group schemes.Representation theory, which lies at the core of Wiles' proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serre's conjectures, Galois deformations, universal deformation rings, Hecke algebras, complete intersections and more, as the reader is led step-by-step through Wiles' proof. In recognition of the historical significance of Fermat's Last Theorem, the volume concludes by looking both forward and backward in time, reflecting on the history of the problem, while placing Wiles' theorem into a more general Diophantine context suggesting future applications. Students and professional mathematicians alike will find this volume to be an indispensable
Inbunden, Engelska, 1992
534 kr
Skickas inom 5-8 vardagar
The theory of elliptic curves involves a pleasing blend of algebra, geometry, analysis, and number theory. "Rational Points on Elliptic Curves" streses this interplay as it develops the basic theory, thereby providing an opportunity for advance undergraduates to appreciate the unity of modern mathematics. At the same time, every effort has been made to use only methods and results commonly included in the undergraduate curriculum. This accessibility, the informal writing style, and a wealth of exercises make "Rational Points on Elliptic Curves" an ideal introduction for students at all levels who are interested in learning about Diophantine equations and arithmetic geometry.
Del 201 - Graduate Texts in Mathematics
Diophantine Geometry
An Introduction
Inbunden, Engelska, 2000
1 071 kr
Skickas inom 10-15 vardagar
This is an introduction to diophantine geometry at the advanced graduate level. The book contains a proof of the Mordell conjecture which will make it quite attractive to graduate students and professional mathematicians. In each part of the book, the reader will find numerous exercises.
Häftad, Engelska, 2000
931 kr
Skickas inom 5-8 vardagar
This volume contains expanded versions of lectures given at an instructional conference on number theory and arithmetic geometry held August 9 through 18, 1995 at Boston University. Contributor's includeThe purpose of the conference, and of this book, is to introduce and explain the many ideas and techniques used by Wiles in his proof that every (semi-stable) elliptic curve over Q is modular, and to explain how Wiles' result can be combined with Ribet's theorem and ideas of Frey and Serre to show, at long last, that Fermat's Last Theorem is true. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions, modular curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of Wiles' proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serre's conjectures, Galois deformations, universal deformation rings, Hecke algebras, complete intersections and more, as the reader is led step-by-step through Wiles' proof.In recognition of the historical significance of Fermat's Last Theorem, the volume concludes by looking both forward and backward in time, reflecting on the history of the problem, while placing Wiles' theorem into a more general Diophantine context suggesting future applications. Students and professional mathematicians alike will find this volume to be an indispensable resource for mastering the epoch-making proof of Fermat's Last Theorem.
Häftad, Engelska, 2021
469 kr
Skickas inom 5-8 vardagar
In New York City during the winter of 1922 and the spring of 1923, Mair Jose Benardete recorded the texts of the thirty-nine traditional ballads published in this volume. His collection, the beginning of Judeo-Spanish ballad research in America, was assembled when the oral tradition was still rich and vigorous among immigrants to New York from the Sephardic settlements of the Eastern Mediterranean and North Africa. Among the ballads are a number of rare text types, some never again recorded in the Sephardic communities of the United States, In addition, many of the texts provide new insights into the origins of the thematic traditions they represent. Samuel G. Armistead and Joseph H. Silverman have edited the ballads collected by Benardete, offering an English abstract and exhaustive bibliography for each ballad. In addition to placing each ballad within the context of its Sephardic variants, the bibliographies refer to the most important collections in the modern Castilian, Portuguese, Catalan, and Hispano-American traditions, to earlier (fifteenth- to seventeenth-century) evidence, and to any known analogs in other European traditions. The volume also includes a general bibliography, a thematic classification of the ballads, several indexes, and a glossary of exotic lexical elements. In an introduction, professors Armistead and Silverman present a documented survey of Judeo-Spanish ballad scholarship with particular attention to fieldwork in teh United States and elsewhere. Benardete himself attributed the decline of ballad singing among the Sephardim to a growing preference for phonographic recordings over traditional family singers. The need for further field-work increases as "Sephardic folkspeech and folklore retreat before the irresistible onslaught of the English language and modern American mass-media culture" (from the Introduction). This title is part of UC Press's Voices Revived program, which commemorates University of California Press's mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1981.
Häftad, Engelska, 2022
662 kr
Skickas inom 5-8 vardagar
Folk Literature of the Sephardic Jews: Judeo-Spanish Ballads from Oral Tradition offers a comprehensive exploration of the rich oral traditions within the Sephardic Jewish community, focusing on ballads that trace their roots to medieval epic narratives. Drawing from a collection of ballads recorded between 1957 and 1980 across Eastern and North African Sephardic communities, the book examines five key narrative types, particularly those connected to the French medieval epic Chanson de Roland and the historical event of Roncesvalles. The volume provides a deep dive into the relationship between these modern Sephardic ballads and their epic antecedents, comparing them with sixteenth-century romancero versions, and placing them within the wider context of Pan-European balladry. Through synthetic texts and an analysis of variants, it uncovers the creative evolution of these ballads within the oral tradition, offering a fresh perspective on their narrative structures and thematic elements.The book addresses ongoing debates in scholarship, particularly the contested connection between epic poetry and ballads. While some critics have downplayed or denied this link, the study presents ample evidence to confirm the genetic and oral-traditional relationship between the romancero and its epic origins. It highlights the preservation of medieval poetic forms and narrative motifs across centuries, particularly the survival of specific poetic features in the Sephardic ballads. The volume also emphasizes the importance of these ballads in understanding the continuity of Hispanic epic poetry within the Sephardic diaspora. Through detailed analysis and scholarly commentary, the book sheds light on the dynamic, living tradition of balladry, demonstrating the creative and evolving nature of these oral narratives across generations.This title is part of UC Press's Voices Revived program, which commemorates University of California Press’s mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1994.
Inbunden, Engelska, 2021
775 kr
Skickas inom 5-8 vardagar
In New York City during the winter of 1922 and the spring of 1923, Mair Jose Benardete recorded the texts of the thirty-nine traditional ballads published in this volume. His collection, the beginning of Judeo-Spanish ballad research in America, was assembled when the oral tradition was still rich and vigorous among immigrants to New York from the Sephardic settlements of the Eastern Mediterranean and North Africa. Among the ballads are a number of rare text types, some never again recorded in the Sephardic communities of the United States, In addition, many of the texts provide new insights into the origins of the thematic traditions they represent. Samuel G. Armistead and Joseph H. Silverman have edited the ballads collected by Benardete, offering an English abstract and exhaustive bibliography for each ballad. In addition to placing each ballad within the context of its Sephardic variants, the bibliographies refer to the most important collections in the modern Castilian, Portuguese, Catalan, and Hispano-American traditions, to earlier (fifteenth- to seventeenth-century) evidence, and to any known analogs in other European traditions. The volume also includes a general bibliography, a thematic classification of the ballads, several indexes, and a glossary of exotic lexical elements. In an introduction, professors Armistead and Silverman present a documented survey of Judeo-Spanish ballad scholarship with particular attention to fieldwork in teh United States and elsewhere. Benardete himself attributed the decline of ballad singing among the Sephardim to a growing preference for phonographic recordings over traditional family singers. The need for further field-work increases as "Sephardic folkspeech and folklore retreat before the irresistible onslaught of the English language and modern American mass-media culture" (from the Introduction). This title is part of UC Press's Voices Revived program, which commemorates University of California Press's mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1981.
Inbunden, Engelska, 2022
891 kr
Skickas inom 5-8 vardagar
Folk Literature of the Sephardic Jews: Judeo-Spanish Ballads from Oral Tradition offers a comprehensive exploration of the rich oral traditions within the Sephardic Jewish community, focusing on ballads that trace their roots to medieval epic narratives. Drawing from a collection of ballads recorded between 1957 and 1980 across Eastern and North African Sephardic communities, the book examines five key narrative types, particularly those connected to the French medieval epic Chanson de Roland and the historical event of Roncesvalles. The volume provides a deep dive into the relationship between these modern Sephardic ballads and their epic antecedents, comparing them with sixteenth-century romancero versions, and placing them within the wider context of Pan-European balladry. Through synthetic texts and an analysis of variants, it uncovers the creative evolution of these ballads within the oral tradition, offering a fresh perspective on their narrative structures and thematic elements.The book addresses ongoing debates in scholarship, particularly the contested connection between epic poetry and ballads. While some critics have downplayed or denied this link, the study presents ample evidence to confirm the genetic and oral-traditional relationship between the romancero and its epic origins. It highlights the preservation of medieval poetic forms and narrative motifs across centuries, particularly the survival of specific poetic features in the Sephardic ballads. The volume also emphasizes the importance of these ballads in understanding the continuity of Hispanic epic poetry within the Sephardic diaspora. Through detailed analysis and scholarly commentary, the book sheds light on the dynamic, living tradition of balladry, demonstrating the creative and evolving nature of these oral narratives across generations.This title is part of UC Press's Voices Revived program, which commemorates University of California Press’s mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1994.
Inbunden, Engelska, 2012
1 313 kr
Skickas inom 5-8 vardagar
This monograph studies moduli problems associated to algebraic dynamical systems. It is an expanded version of the notes for a series of lectures delivered at a workshop on Moduli Spaces and the Arithmetic of Dynamical Systems at the Bellairs Research Institute, Barbados, in 2010. The author's goal is to provide an overview, with enough details and pointers to the existing literature, to give the reader an entry into this exciting area of current research. Topics covered include: (1) Construction and properties of dynamical moduli spaces for self-maps of projective space. (2) Dynatomic modular curves for the space of quadratic polynomials. (3) The theory of canonical heights associated to dynamical systems. (4) Special loci in dynamical moduli spaces, in particular the locus of post-critically finite maps. (5) Field of moduli and fields of definition for dynamical systems.
Del 106 - Graduate Texts in Mathematics
Arithmetic of Elliptic Curves
Häftad, Engelska, 2010
539 kr
Skickas inom 10-15 vardagar
The theory of elliptic curves is distinguished by the diversity of the methods used in its study. This book treats the arithmetic theory of elliptic curves in its modern formulation, through the use of basic algebraic number theory and algebraic geometry.
Häftad, Engelska, 2010
549 kr
Skickas inom 5-8 vardagar
In 1961 the second author deliv1lred a series of lectures at Haverford Col lege on the subject of "Rational Points on Cubic Curves. " These lectures, intended for junior and senior mathematics majors, were recorded, tran scribed, and printed in mimeograph form. Since that time they have been widely distributed as photocopies of ever decreasing legibility, and por tions have appeared in various textbooks (Husemoller [1], Chahal [1]), but they have never appeared in their entirety. In view of the recent inter est in the theory of elliptic curves for subjects ranging from cryptogra phy (Lenstra [1], Koblitz [2]) to physics (Luck-Moussa-Waldschmidt [1]), as well as the tremendous purely mathematical activity in this area, it seems a propitious time to publish an expanded version of those original notes suitable for presentation to an advanced undergraduate audience. We have attempted to maintain much of the informality of the orig inal Haverford lectures. Our main goal in doing this has been to write a textbook in a technically difficult field which is "readable" by the average undergraduate mathematics major. We hope we have succeeded in this goal. The most obvious drawback to such an approach is that we have not been entirely rigorous in all of our proofs. In particular, much of the foundational material on elliptic curves presented in Chapter I is meant to explain and convince, rather than to rigorously prove.
Häftad, Engelska, 2022
568 kr
Skickas inom 5-8 vardagar
This abstract algebra textbook takes an integrated approach that highlights the similarities of fundamental algebraic structures among a number of topics. The book begins by introducing groups, rings, vector spaces, and fields, emphasizing examples, definitions, homomorphisms, and proofs. The goal is to explain how all of the constructions fit into an axiomatic framework and to emphasize the importance of studying those maps that preserve the underlying algebraic structure. This fast-paced introduction is followed by chapters in which each of the four main topics is revisited and deeper results are proven. The second half of the book contains material of a more advanced nature. It includes a thorough development of Galois theory, a chapter on modules, and short surveys of additional algebraic topics designed to whet the reader's appetite for further study. This book is intended for a first introduction to abstract algebra and requires only a course in linear algebra as a prerequisite. The more advanced material could be used in an introductory graduate-level course.
Inbunden, Engelska, 2014
753 kr
Skickas inom 10-15 vardagar
An Introduction to Mathematical Cryptography
Häftad, Engelska, 2016
753 kr
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This self-contained introduction to modern cryptography emphasizes the mathematics behind the theory of public key cryptosystems and digital signature schemes. The book focuses on these key topics while developing the mathematical tools needed for the construction and security analysis of diverse cryptosystems. Only basic linear algebra is required of the reader; techniques from algebra, number theory, and probability are introduced and developed as required. This text provides an ideal introduction for mathematics and computer science students to the mathematical foundations of modern cryptography. The book includes an extensive bibliography and index; supplementary materials are available online.The book covers a variety of topics that are considered central to mathematical cryptography. Key topics include:classical cryptographic constructions, such as Diffie–Hellmann key exchange, discrete logarithm-based cryptosystems, the RSA cryptosystem, anddigital signatures;fundamental mathematical tools for cryptography, including primality testing, factorization algorithms, probability theory, information theory, and collision algorithms;an in-depth treatment of important cryptographic innovations, such as elliptic curves, elliptic curve and pairing-based cryptography, lattices, lattice-based cryptography, and the NTRU cryptosystem.The second edition of An Introduction to Mathematical Cryptography includes a significant revision of the material on digital signatures, including an earlier introduction to RSA, Elgamal, and DSA signatures, and new material on lattice-based signatures and rejection sampling. Many sections have been rewritten or expanded for clarity, especially in the chapters on information theory, elliptic curves, and lattices, and the chapter of additional topics has been expanded to include sections on digital cash and homomorphic encryption. Numerous new exercises have been included.
Inbunden, Engelska, 2016
1 004 kr
Skickas inom 10-15 vardagar
The Judeo-Spanish folk literature of the Sephardic Jews of Bosnia, and with it their uncommonly rich balladry, has remained largely unknown to Western scholars. Since their move to Sarajevo in the sixteenth century, Serob-Croatian has displaced their original Spanish, and the entire culture is rapidly approaching extinction.This book preserved for posterity three fundamentally important groups of these rare ballads: Kalmi Baruch's Spanski romanse; ballads collected from the readers of the Sarajevo newspaper Jevrejski Glas; and five previously unedited eighteenth-century Bosnian ballads from a manuscript in the Jewish National and University Library in Jerusalem.Notes, abstracts in English, reproductions of the music itself, and other scholarly aids serve to make this colorful and strangely modern literature fully accessible to Hispanists, folklorists, and all students of comparative literature and Judaic culture.
Häftad, Engelska, 2015
539 kr
Skickas inom 10-15 vardagar
The theory of elliptic curves involves a pleasing blend of algebra, geometry, analysis, and number theory. This volume stresses this interplay as it develops the basic theory, thereby providing an opportunity for advanced undergraduates to appreciate the unity of modern mathematics. At the same time, every effort has been made to use only methods and results commonly included in the undergraduate curriculum. This accessibility, the informal writing style, and a wealth of exercises make Rational Points on Elliptic Curves an ideal introduction for students at all levels who are interested in learning about Diophantine equations and arithmetic geometry.Most concretely, an elliptic curve is the set of zeroes of a cubic polynomial in two variables. If the polynomial has rational coefficients, then one can ask for a description of those zeroes whose coordinates are either integers or rational numbers. It is this number theoretic question that is the main subject of Rational Points on Elliptic Curves. Topics covered include the geometry and group structure of elliptic curves, the Nagell–Lutz theorem describing points of finite order, the Mordell–Weil theorem on the finite generation of the group of rational points, the Thue–Siegel theorem on the finiteness of the set of integer points, theorems on counting points with coordinates in finite fields, Lenstra's elliptic curve factorization algorithm, and a discussion of complex multiplication and the Galois representations associated to torsion points. Additional topics new to the second edition include an introduction to elliptic curve cryptography and a brief discussion of the stunning proof of Fermat's Last Theorem by Wiles et al. via the use of elliptic curves.
Häftad, Engelska, 2001
556 kr
Skickas inom 10-15 vardagar
ThesearetheproceedingsofCaLC2001,the?rstconferencedevotedtocr- tographyandlattices. Wehavelongbelievedthattheimportanceoflattices andlatticereductionincryptography,bothforcryptographicconstructionand cryptographicanalysis,meritsagatheringdevotedtothistopic. Theenthusiastic responsethatwereceivedfromtheprogramcommittee,theinvitedspeakers,the manypeoplewhosubmittedpapers,andthe90registeredparticipantsamply con?rmedthewidespreadinterestinlatticesandtheircryptographicappli- tions. WethankeveryonewhoseinvolvementmadeCaLCsuchasuccessfulevent; inparticularwethankNatalieJohnson,LarryLarrivee,DoreenPappas,andthe BrownUniversityMathematicsDepartmentfortheirassistanceandsupport. March2001 Je?reyHo?stein,JillPipher,JosephSilverman VI Preface Organization CaLC2001wasorganizedbytheDepartmentofMathematicsatBrownUniv- sity. Theprogramchairsexpresstheirthankstotheprogramcommiteeandthe additionalexternalrefereesfortheirhelpinselectingthepapersforCaLC2001. TheprogramchairswouldalsoliketothankNTRUCryptosystemsforproviding ?nancialsupportfortheconference. Program Commitee DonCoppersmith IBMResearch Je?reyHo?stein(co-chair), BrownUniversityandNTRUCryptosystems ArjenLenstra Citibank,USA PhongNguyen ENS AndrewOdlyzko AT&TLabsResearch JosephH.Silverman(co-chair), BrownUniversityandNTRUCryptosystems External Referees AliAkhavi,GlennDurfee,NickHowgrave-Graham,DanieleMicciancio Sponsoring Institutions NTRUCryptosystems,Inc. ,Burlington,MA Table of Contents An Overveiw of the Sieve Algorithm forthe Shortest Lattice Vector Problem 1 Miklos Ajtai, Ravi Kumar, and Dandapani Sivakumar Low Secret Exponent RSA Revisited ::::::::::::::::::::::::::::::::: 4 Johannes Bl. omer and Alexander May Finding Small Solutions to Small Degree Polynomials::::::::::::::::::: 20 Don Coppersmith Fast Reduction of Ternary Quadratic Forms::::::::::::::::::::::::::: 32 Friedrich Eisenbrand and Gunt .. er Rote Factoring Polynomialsand 0-1 Vectors:::::::::::::::::::::::::::::::: 45 Mark van Hoeij Approximate Integer Common Divisors::::::::::::::::::::::::::::::: 51 Nick Howgrave-Graham Segment LLL-Reduction of Lattice Bases ::::::::::::::::::::::::::::: 67 Henrik Koy and Claus Peter Schnorr Segment LLL-Reduction with Floating Point Orthogonalization:::::::::: 81 Henrik Koy and Claus Peter Schnorr TheInsecurity ofNyberg-Rueppel andOther DSA-LikeSignatureSchemes with Partially Known Nonces:::::::::::::::::::::::::::::::::::::::: 97 Edwin El Mahassni, Phong Q.Nguyen, and Igor E. Shparlinski Dimension Reduction Methods for Convolution Modular Lattices :::::::: 110 Alexander May and Joseph H. Silverman Improving Lattice Based Cryptosystems Using the Hermite Normal Form : 126 Daniele Micciancio The Two Faces of Lattices in Cryptology:::::::::::::::::::::::::::::: 146 Phong Q.