Cohen, Henri
Overview
Works:  18 works in 201 publications in 2 languages and 3,304 library holdings 

Genres:  Conference papers and proceedings Handbooks and manuals Textbooks 
Roles:  Author, Editor, Translator, Interviewee 
Classifications:  QA241, 512.74028551 
Publication Timeline
.
Most widely held works by
Henri Cohen
A course in computational algebraic number theory by
Henri Cohen(
Book
)
49 editions published between 1993 and 2015 in English and Undetermined and held by 755 WorldCat member libraries worldwide
Describes 148 algorithms that are fundamental for numbertheoretic computations including computations related to algebraic number theory, elliptic curves, primality testing, and factoring. A complete theoretical introduction is given for each subject, reducing prerequisites to a minimum. The detailed description of each algorithm allows immediate
49 editions published between 1993 and 2015 in English and Undetermined and held by 755 WorldCat member libraries worldwide
Describes 148 algorithms that are fundamental for numbertheoretic computations including computations related to algebraic number theory, elliptic curves, primality testing, and factoring. A complete theoretical introduction is given for each subject, reducing prerequisites to a minimum. The detailed description of each algorithm allows immediate
Advanced topics in computational number theory by
Henri Cohen(
Book
)
17 editions published between 1999 and 2013 in English and held by 441 WorldCat member libraries worldwide
The present book addresses a number of specific topics in computational number theory whereby the author is not attempting to be exhaustive in the choice of subjects. The book is organized as follows. Chapters 1 and 2 contain the theory and algorithms concerning Dedekind domains and relative extensions of number fields, and in particular the generalization to the relative case of the round 2 and related algorithms. Chapters 3, 4, and 5 contain the theory and complete algorithms concerning class field theory over number fields. The highlights are the algorithms for computing the structure of (Z_K/m)^*, of ray class groups, and relative equations for Abelian extensions of number fields using Kummer theory. Chapters 1 to 5 form a homogeneous subject matter which can be used for a 6 months to 1 year graduate course in computational number theory. The subsequent chapters deal with more miscellaneous subjects. Written by an authority with great practical and teaching experience in the field, this book together with the author's earlier book will become the standard and indispensable reference on the subject
17 editions published between 1999 and 2013 in English and held by 441 WorldCat member libraries worldwide
The present book addresses a number of specific topics in computational number theory whereby the author is not attempting to be exhaustive in the choice of subjects. The book is organized as follows. Chapters 1 and 2 contain the theory and algorithms concerning Dedekind domains and relative extensions of number fields, and in particular the generalization to the relative case of the round 2 and related algorithms. Chapters 3, 4, and 5 contain the theory and complete algorithms concerning class field theory over number fields. The highlights are the algorithms for computing the structure of (Z_K/m)^*, of ray class groups, and relative equations for Abelian extensions of number fields using Kummer theory. Chapters 1 to 5 form a homogeneous subject matter which can be used for a 6 months to 1 year graduate course in computational number theory. The subsequent chapters deal with more miscellaneous subjects. Written by an authority with great practical and teaching experience in the field, this book together with the author's earlier book will become the standard and indispensable reference on the subject
Number theory by
Henri Cohen(
Book
)
4 editions published between 2007 and 2010 in English and held by 339 WorldCat member libraries worldwide
"A unique collection of topics centered on a unifying topic. Includes more than 350 exercises. Text is largely selfcontained. The central theme of this graduatelevel number theory textbook is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three aspects. The first is the local aspect: one can do analysis in padic fields, and here the author starts by looking at solutions in finite fields, then proceeds to lift these solutions to local solutions using Hensel lifting. The second is the global aspect: the use of number fields, and in particular of class groups and unit groups. This classical subject is here illustrated through a wide range of examples. The third aspect deals with specific classes of equations, and in particular the general and Diophantine study of elliptic curves, including 2 and 3descent and the Heegner point method. These subjects form the first two parts, forming Volume I. The study of Bernoulli numbers, the gamma function, and zeta and Lfunctions, and of padic analogues is treated at length in the third part of the book, including many interesting and original applications. Much more sophisticated techniques have been brought to bear on the subject of Diophantine equations, and for this reason, the author has included five chapters on these techniques forming the fourth part, which together with the third part forms Volume II. These chapters were written by Yann Bugeaud, Guillaume Hanrot, Maurice Mignotte, Sylvain Duquesne, Samir Siksek, and the author, and contain material on the use of Galois representations, points on highergenus curves, the superfermat equation, Mihailescu's proof of Catalan's Conjecture, and applications of linear forms in logarithms. The book contains 530 exercises of varying difficulty from immediate consequences of the main text to research problems, and contain many important additional results."Publisher's website
4 editions published between 2007 and 2010 in English and held by 339 WorldCat member libraries worldwide
"A unique collection of topics centered on a unifying topic. Includes more than 350 exercises. Text is largely selfcontained. The central theme of this graduatelevel number theory textbook is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three aspects. The first is the local aspect: one can do analysis in padic fields, and here the author starts by looking at solutions in finite fields, then proceeds to lift these solutions to local solutions using Hensel lifting. The second is the global aspect: the use of number fields, and in particular of class groups and unit groups. This classical subject is here illustrated through a wide range of examples. The third aspect deals with specific classes of equations, and in particular the general and Diophantine study of elliptic curves, including 2 and 3descent and the Heegner point method. These subjects form the first two parts, forming Volume I. The study of Bernoulli numbers, the gamma function, and zeta and Lfunctions, and of padic analogues is treated at length in the third part of the book, including many interesting and original applications. Much more sophisticated techniques have been brought to bear on the subject of Diophantine equations, and for this reason, the author has included five chapters on these techniques forming the fourth part, which together with the third part forms Volume II. These chapters were written by Yann Bugeaud, Guillaume Hanrot, Maurice Mignotte, Sylvain Duquesne, Samir Siksek, and the author, and contain material on the use of Galois representations, points on highergenus curves, the superfermat equation, Mihailescu's proof of Catalan's Conjecture, and applications of linear forms in logarithms. The book contains 530 exercises of varying difficulty from immediate consequences of the main text to research problems, and contain many important additional results."Publisher's website
Algorithmic number theory : second internati[o]nal symposium, ANTSII, Talence, France, May 1823, 1996 : proceedings by
Henri Cohen(
Book
)
21 editions published in 1996 in English and Undetermined and held by 312 WorldCat member libraries worldwide
"This book constitutes the refereed postconference proceedings of the Second International Algorithmic Number Theory Symposium, ANTSII, held in Talence, France in May 1996. The 35 revised full papers included in the book were selected from a variety of submissions. They cover a broad spectrum of topics and report stateoftheart research results in computational number theory and complexity theory. Among the issues addressed are number fields computation, Abelian varieties, factoring algorithms, finite fields, elliptic curves, algorithm complexity, lattice theory, and coding."PUBLISHER'S WEBSITE
21 editions published in 1996 in English and Undetermined and held by 312 WorldCat member libraries worldwide
"This book constitutes the refereed postconference proceedings of the Second International Algorithmic Number Theory Symposium, ANTSII, held in Talence, France in May 1996. The 35 revised full papers included in the book were selected from a variety of submissions. They cover a broad spectrum of topics and report stateoftheart research results in computational number theory and complexity theory. Among the issues addressed are number fields computation, Abelian varieties, factoring algorithms, finite fields, elliptic curves, algorithm complexity, lattice theory, and coding."PUBLISHER'S WEBSITE
Number theory by
Henri Cohen(
Book
)
49 editions published between 2007 and 2011 in English and held by 141 WorldCat member libraries worldwide
Diophantine equations are equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This book provides information on Diophantine equation, a theme that is the central motivation for the modern theory of arithmetic algebraic geometry
49 editions published between 2007 and 2011 in English and held by 141 WorldCat member libraries worldwide
Diophantine equations are equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This book provides information on Diophantine equation, a theme that is the central motivation for the modern theory of arithmetic algebraic geometry
Number theory by
Henri Cohen(
Book
)
22 editions published between 2007 and 2011 in English and held by 107 WorldCat member libraries worldwide
The central theme of this graduatelevel number theory textbook is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three aspects. The first is the local aspect: one can do analysis in padic fields, and here the author starts by looking at solutions in finite fields, then proceeds to lift these solutions to local solutions using Hensel lifting. The second is the global aspect: the use of number fields, and in particular of class groups and unit groups. This classical subject is here illustrated through a wide range of examples. The third aspect deals with specific classes of equations, and in particular the general and Diophantine study of elliptic curves, including 2 and 3descent and the Heegner point method. These subjects form the first two parts, forming Volume I. The study of Bernoulli numbers, the gamma function, and zeta and Lfunctions, and of padic analogues is treated at length in the third part of the book, including many interesting and original applications. Much more sophisticated techniques have been brought to bear on the subject of Diophantine equations, and for this reason, the author has included five chapters on these techniques forming the fourth part, which together with the third part forms Volume II. These chapters were written by Yann Bugeaud, Guillaume Hanrot, Maurice Mignotte, Sylvain Duquesne, Samir Siksek, and the author, and contain material on the use of Galois representations, points on highergenus curves, the superfermat equation, Mihailescu's proof of Catalan's Conjecture, and applications of linear forms in logarithms. The book contains 530 exercises of varying difficulty from immediate consequences of the main text to research problems, and contain many important additional results
22 editions published between 2007 and 2011 in English and held by 107 WorldCat member libraries worldwide
The central theme of this graduatelevel number theory textbook is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three aspects. The first is the local aspect: one can do analysis in padic fields, and here the author starts by looking at solutions in finite fields, then proceeds to lift these solutions to local solutions using Hensel lifting. The second is the global aspect: the use of number fields, and in particular of class groups and unit groups. This classical subject is here illustrated through a wide range of examples. The third aspect deals with specific classes of equations, and in particular the general and Diophantine study of elliptic curves, including 2 and 3descent and the Heegner point method. These subjects form the first two parts, forming Volume I. The study of Bernoulli numbers, the gamma function, and zeta and Lfunctions, and of padic analogues is treated at length in the third part of the book, including many interesting and original applications. Much more sophisticated techniques have been brought to bear on the subject of Diophantine equations, and for this reason, the author has included five chapters on these techniques forming the fourth part, which together with the third part forms Volume II. These chapters were written by Yann Bugeaud, Guillaume Hanrot, Maurice Mignotte, Sylvain Duquesne, Samir Siksek, and the author, and contain material on the use of Galois representations, points on highergenus curves, the superfermat equation, Mihailescu's proof of Catalan's Conjecture, and applications of linear forms in logarithms. The book contains 530 exercises of varying difficulty from immediate consequences of the main text to research problems, and contain many important additional results
Handbook of elliptic and hyperelliptic curve cryptography by
Henri Cohen(
Book
)
20 editions published between 2005 and 2006 in English and held by 85 WorldCat member libraries worldwide
"The Handbook of Elliptic and Hyperelliptic Curve Cryptography introduces the theory and algorithms involved in curvebased cryptography. After a very detailed exposition of the mathematical background, it provides readytoimplement algorithms for the arithmetic of elliptic and hyperelliptic curves and the computation of pairings. It explores methods for point counting and constructing curves with the complex multiplication method. It also surveys generic methods to compute discrete logarithms and details index calculus methods for hyperelliptic curves as well as transfers of discrete logarithm problems for special curves. It ends up with concrete realizations of cryptosystems in smart cards, including efficient implementation in hardware and sidechannel attacks as well as countermeasures."BOOK JACKET
20 editions published between 2005 and 2006 in English and held by 85 WorldCat member libraries worldwide
"The Handbook of Elliptic and Hyperelliptic Curve Cryptography introduces the theory and algorithms involved in curvebased cryptography. After a very detailed exposition of the mathematical background, it provides readytoimplement algorithms for the arithmetic of elliptic and hyperelliptic curves and the computation of pairings. It explores methods for point counting and constructing curves with the complex multiplication method. It also surveys generic methods to compute discrete logarithms and details index calculus methods for hyperelliptic curves as well as transfers of discrete logarithm problems for special curves. It ends up with concrete realizations of cryptosystems in smart cards, including efficient implementation in hardware and sidechannel attacks as well as countermeasures."BOOK JACKET
Class field theory : from theory to practice by
Georges Gras(
Book
)
3 editions published in 2003 in English and French and held by 16 WorldCat member libraries worldwide
"Global class field theory is a major achievement of algebraic number theory, based on the functorial properties of the reciprocity map and the existence theorem. The author works out the consequences and the practical use of these results by giving detailed studies and illustrations of classical subjects (classes, ideles, ray class fields, symbols, reciprocity laws, Hasse's principles, the GrunwaldWang theorem, Hilbert's towers ...). This book, intermediary between the classical literature published in the sixties and the recent computational literature, gives much material in an elementary way, and is suitable for students, researchers, and all who are fascinated by this theory."Jacket
3 editions published in 2003 in English and French and held by 16 WorldCat member libraries worldwide
"Global class field theory is a major achievement of algebraic number theory, based on the functorial properties of the reciprocity map and the existence theorem. The author works out the consequences and the practical use of these results by giving detailed studies and illustrations of classical subjects (classes, ideles, ray class fields, symbols, reciprocity laws, Hasse's principles, the GrunwaldWang theorem, Hilbert's towers ...). This book, intermediary between the classical literature published in the sixties and the recent computational literature, gives much material in an elementary way, and is suitable for students, researchers, and all who are fascinated by this theory."Jacket
Handbook of elliptic and hyperelliptic curve crytography(
Book
)
2 editions published between 2011 and 2014 in English and held by 11 WorldCat member libraries worldwide
2 editions published between 2011 and 2014 in English and held by 11 WorldCat member libraries worldwide
Formes modulaires à une et deux variables by
Henri Cohen(
Book
)
4 editions published in 1976 in French and Undetermined and held by 10 WorldCat member libraries worldwide
FORMES MODULAIRES SUR GAMMA O(4). APPLICATIONS DU CALCUL DES COEFFICIENTS DE FOURIER DES SERIES D'EISENSTEINHECKE. COMBINATOIRE ET DERIVEES DE FORMES MODULAIRES. IDENTITES FAISANT INTERVENIR LES VALEURS AUX ENTIERS NEGATIFS DES FONCTIONS L DE CARACTERES QUADRATIQUES. FORMES MODULAIRES DE HILBERT A DEUX VARIABLES ASSOCIEES A UNE FORME A UNE VARIABLE. DESCRIPTION DES GROUPES GAMMA N
4 editions published in 1976 in French and Undetermined and held by 10 WorldCat member libraries worldwide
FORMES MODULAIRES SUR GAMMA O(4). APPLICATIONS DU CALCUL DES COEFFICIENTS DE FOURIER DES SERIES D'EISENSTEINHECKE. COMBINATOIRE ET DERIVEES DE FORMES MODULAIRES. IDENTITES FAISANT INTERVENIR LES VALEURS AUX ENTIERS NEGATIFS DES FONCTIONS L DE CARACTERES QUADRATIQUES. FORMES MODULAIRES DE HILBERT A DEUX VARIABLES ASSOCIEES A UNE FORME A UNE VARIABLE. DESCRIPTION DES GROUPES GAMMA N
Théorie algorithmique des nombres et équations diophantiennes by
Henri Cohen(
Book
)
1 edition published in 2005 in French and held by 6 WorldCat member libraries worldwide
1 edition published in 2005 in French and held by 6 WorldCat member libraries worldwide
L'état des maths(
Visual
)
3 editions published between 2008 and 2010 in French and held by 3 WorldCat member libraries worldwide
3 editions published between 2008 and 2010 in French and held by 3 WorldCat member libraries worldwide
A Course in Computational Algebraic Number Theory by
Henri Cohen(
)
1 edition published in 1993 in English and held by 3 WorldCat member libraries worldwide
1 edition published in 1993 in English and held by 3 WorldCat member libraries worldwide
Minoration de la hauteur de NéronTate sur les variétés abéliennes sur la conjecture de Lang et Silverman by
Fabien Mehdi Pazuki(
Book
)
1 edition published in 2008 in French and held by 2 WorldCat member libraries worldwide
Cette thèse est consacrée à l'étude d'une conjecture de Lang et Silverman de minoration de la hauteur de NéronTate sur les variétés abéliennes sur les corps de nombres. Le premier chapitre décrit le matériel nécessaire à l'étude des chapitres suivants et fixe les notations et normalisations. On montre dans le second chapitre que la conjecture est vraie pour certaines classes de variétés abéliennes de dimension 2, en particulier pour les jacobiennes ayant potentiellement bonne réduction et restant loin des produits de courbes elliptiques dans l'espace de modules. Le second chapitre renferme aussi des corollaires allant dans la direction de la conjecture de borne uniforme sur la torsion et de majoration uniforme du nombre de points rationnels d'une courbe de genre 2. Le troisième chapitre généralise les résultats de minoration du second chapitre aux jacobiennes de courbes huperelliptiques de genre g ≥ 2. Le quatrième chapitre contient une étude de la restriction des scalaires à la Weil et une étude asymptotique de la hauteur des points de Heegner sur les jacobiennes de courbes modulaires. Le cinquième chapitre est une annexe contenant des formules explicites utiles pour la dimension 2 et un paragraphe sur un raisonnement par isogénies
1 edition published in 2008 in French and held by 2 WorldCat member libraries worldwide
Cette thèse est consacrée à l'étude d'une conjecture de Lang et Silverman de minoration de la hauteur de NéronTate sur les variétés abéliennes sur les corps de nombres. Le premier chapitre décrit le matériel nécessaire à l'étude des chapitres suivants et fixe les notations et normalisations. On montre dans le second chapitre que la conjecture est vraie pour certaines classes de variétés abéliennes de dimension 2, en particulier pour les jacobiennes ayant potentiellement bonne réduction et restant loin des produits de courbes elliptiques dans l'espace de modules. Le second chapitre renferme aussi des corollaires allant dans la direction de la conjecture de borne uniforme sur la torsion et de majoration uniforme du nombre de points rationnels d'une courbe de genre 2. Le troisième chapitre généralise les résultats de minoration du second chapitre aux jacobiennes de courbes huperelliptiques de genre g ≥ 2. Le quatrième chapitre contient une étude de la restriction des scalaires à la Weil et une étude asymptotique de la hauteur des points de Heegner sur les jacobiennes de courbes modulaires. Le cinquième chapitre est une annexe contenant des formules explicites utiles pour la dimension 2 et un paragraphe sur un raisonnement par isogénies
Calculs effectifs des points entiers et rationnels sur les courbes by
Sylvain Duquesne(
Book
)
1 edition published in 2001 in French and held by 2 WorldCat member libraries worldwide
Cette thèse est constituée de trois parties indépendantes concernant la détermination des points rationnels et entiers sur les courbes algébriques. Nous nous sommes particulièrement intéressé aux aspects algorithmiques de ces problèmes. Dans la première partie, nous avons considéré la famille des courbes elliptiques définies par les "simplest cubic fields". Nous décrivons une méthode pour calculer exactement tous les points entiers sur ces courbes. Une étude expérimentale sur un grand nombre de courbes nous a amené à observer certains phénomènes que nous prouvons par la suite. En particulier, nous donnons explicitement tous les points entiers sur ces courbes lorsque leur rang vaut 1. La deuxième partie concerne le calcul des points rationnels sur les courbes de plus grand genre. Dans le but de calculer ces points lorsque le rang de la jacobienne est supérieur au genre de la courbe, Flynn et Wetherell utilisent le groupe formel d'une courbe elliptique de rang l. Nous développons une version explicite du théorème de préparation de Weierstrass pour généraliser cette méthode au cas des courbes elliptiques de rang supérieur à 1, ce qui permet de traiter de nouvelles équations diophantiennes. La dernière partie de cette thèse consiste à calculer les traces de la loi de groupe sur la variété de Kummer d'une courbe hyperelliptique de genre 3 définie par un polynôme de degré 7. Ces traces permettent de construire une fonction hauteur sur la jacobienne. Nous pouvons alors espérer d'une part en déduire une procédure effective de calcul du sousgroupe de torsion et d'autre part effectuer des descentes infinies
1 edition published in 2001 in French and held by 2 WorldCat member libraries worldwide
Cette thèse est constituée de trois parties indépendantes concernant la détermination des points rationnels et entiers sur les courbes algébriques. Nous nous sommes particulièrement intéressé aux aspects algorithmiques de ces problèmes. Dans la première partie, nous avons considéré la famille des courbes elliptiques définies par les "simplest cubic fields". Nous décrivons une méthode pour calculer exactement tous les points entiers sur ces courbes. Une étude expérimentale sur un grand nombre de courbes nous a amené à observer certains phénomènes que nous prouvons par la suite. En particulier, nous donnons explicitement tous les points entiers sur ces courbes lorsque leur rang vaut 1. La deuxième partie concerne le calcul des points rationnels sur les courbes de plus grand genre. Dans le but de calculer ces points lorsque le rang de la jacobienne est supérieur au genre de la courbe, Flynn et Wetherell utilisent le groupe formel d'une courbe elliptique de rang l. Nous développons une version explicite du théorème de préparation de Weierstrass pour généraliser cette méthode au cas des courbes elliptiques de rang supérieur à 1, ce qui permet de traiter de nouvelles équations diophantiennes. La dernière partie de cette thèse consiste à calculer les traces de la loi de groupe sur la variété de Kummer d'une courbe hyperelliptique de genre 3 définie par un polynôme de degré 7. Ces traces permettent de construire une fonction hauteur sur la jacobienne. Nous pouvons alors espérer d'une part en déduire une procédure effective de calcul du sousgroupe de torsion et d'autre part effectuer des descentes infinies
Modular forms : a classical approach by
Henri Cohen(
Book
)
1 edition published in 2017 in English and held by 1 WorldCat member library worldwide
1 edition published in 2017 in English and held by 1 WorldCat member library worldwide
Number Theory : Volume I: Tools and Diophantine Equations(
)
1 edition published in 2007 in English and held by 0 WorldCat member libraries worldwide
1 edition published in 2007 in English and held by 0 WorldCat member libraries worldwide
Computability of Julia sets by
Mark Braverman(
)
1 edition published in 2009 in English and held by 0 WorldCat member libraries worldwide
"Computational hardness of Julia sets is the main subject of this book. By definition, a computable set in the plane can be visualized on a computerscreen with an arbitrarily high magnification. There are countless programs to draw Julia sets. Yet, as the authors have discovered, it is possible to constructively produce examples of quadratic polynomials, whose Julia sets are not computable. This result is striking it says that while a dynamical system can be described numerically with an arbitrary precision, the picture of the dynamics cannot be visualized. The book summarizes the present knowledge about the computational properties of Julia sets in a selfcontained way. It is accessible to experts and students with interest in theoretical computer science or dynamical systems."Jacket
1 edition published in 2009 in English and held by 0 WorldCat member libraries worldwide
"Computational hardness of Julia sets is the main subject of this book. By definition, a computable set in the plane can be visualized on a computerscreen with an arbitrarily high magnification. There are countless programs to draw Julia sets. Yet, as the authors have discovered, it is possible to constructively produce examples of quadratic polynomials, whose Julia sets are not computable. This result is striking it says that while a dynamical system can be described numerically with an arbitrary precision, the picture of the dynamics cannot be visualized. The book summarizes the present knowledge about the computational properties of Julia sets in a selfcontained way. It is accessible to experts and students with interest in theoretical computer science or dynamical systems."Jacket
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Algebra AlgebraData processing Algebraic number theory Algebraic number theoryData processing Algorithms Class field theory Combinatorial analysis Computational complexity Computer science Computer scienceMathematics Computer security Computer software Cryptography CryptographyMathematics Curves, Elliptic Curves, EllipticData processing Data encryption (Computer science) Diophantine analysis Diophantine equations Field theory (Physics) Forms, Modular Forms (Mathematics) Geometry, Algebraic Information theory Machine theory Mathematics Number theory Number theoryData processing Public key cryptography Smart cards