Arithmetic modular

In Mathematical and more precisely in Algebraic theory of the numbers, the modular arithmetic is a whole of methods allowing the solution to problem on the integers. These methods derive from the study of the remainder obtained by a Euclidean Division.

If its origins go back to antiquity, the historians generally associate his birth with the year 1801, date of the publication of the book Disquisitiones arithmeticae of Carl Friedrich Gauss (1777 - 1855) . Its new approach makes it possible to establish famous conjectures and simplifies the demonstrations of important results, at the price of a greater abstraction. If the natural field of these methods is the Théorie of the numbers, the consequences of the ideas of Gauss are found in other fields of mathematics, like the Algèbre or the Géométrie.

The 20th century amends of arithmetic modular. On the one hand, other methods are necessary to progress in Théorie of the numbers. In addition, the development of many industrial applications imposes the settling of Algorithme S resulting from the modular techniques. They solve primarily questions raised by the Information theory. This branch is now especially regarded as Mathématiques applied.

the article Congruence on the entireties proposes a more mathematical introduction, Anneau Z/nZ covers the same subject in a less didactic and more exhaustive way.

Uses

In Mathematical pure, this term is used very little. The close term most frequent is Algebraic theory of the numbers , which indicates a broader field, container for example the algebraic whole concepts of or Théorie of Welshman.

In Mathematical applied, this expression is of a frequent use to describe the mathematical bases of various fields of the Information theory: Cryptology, Data-processing Theory of the codes and . Many tools and Algorithme S enter this field of study. One finds there the tests of primalities, the Décomposition in product of factors first, the use of the characters of a group for example for the discrete Transformée of Fourier or the study of others quotients that those of the entireties, like that of the Polynôme S.

According to the various authors and the scope of application, these extensions are considered, either like an integral part of arithmetic modular, or like applications, even are not quoted a whole. In its simple form, it takes sometimes the arithmetic name of of the clock . The term of modular system is used to indicate arithmetic modular on other units that the entireties.

History

Origins

To the III E Euclide formalizes, in its book the Éléments , the bases of the arithmetic one. One finds there the lemma bearing his name, a version dated from the fundamental Théorème of arithmetic the and a study on the perfect numbers in proposal 36 of his book IX. Diophante of Alexandria (approx. 250) written Arithmetica containing 130 equations. It deals with primarily problems having a single numerical solution and with fractional or whole value. The fact there is found that the naps numbers of two square perfect are never of form 4 N + 3. The equations, with whole coefficients and whose required solutions are whole take today the name of equations diophantiennes.

The China develops arithmetic modular in parallel. Sun Zi writes about the year 300 a treaty of mathematics whose problem 26 of chapter 3 is the following: Are objects of which one is unaware of the number. By counting them 3 by 3 there remain 2 about it, by counting them 5 by 5, he remains 3 about it and by counting them 7 by 7, there remain 2 about it. How much are there objects? .

Qin Jiushao (1202 - 1261) develops the Théorème of the Chinese remainders. Its treaty is remarkably advanced, it treats of a Système of linear equations of congruences if the modulo are not first between them two to two. Its work on the systems of Congruence S exceeds in sophistication those of Leonhard Euler (1707 - 1783) . One can quote George Sarton for which: Qin Jiushao was one of largest the mathematicians of Chinese race, its time and to tell the truth of all time .

The 14th century sees a progressive decline then a lapse of memory of these results, the knowledge of Qin Jiushao does not exceed the Chinese borders before the 20th century. It is redécouvert by work of the historian of sciences Joseph Needham. On the other hand, of many similarities between the Arab and Chinese notations let think of contacts during the previous periods.

The India has also a strong tradition into arithmetic. Âryabhata (476 - 550) search for manner systematic whole solutions of the linear equation with two unknown factors with whole coefficients. It uses for that an algorithm called Kuttaka published in its book called Aryabhatiya . The equations diophantiennes of degree two are studied by Brahmagupta (598 - 668) using the Méthode chakravala.

The Islamic civilization plays a double part into arithmetic: it conveys the knowledge acquired by the Greeks, Indiens, Chinese and Europeans, and it brings new knowledge in particular on the study of the properties of certain whole categories of , like the prime numbers, perfect, friendly or illustrated. In this context, Qusta ibn Lûqâ carries out a translation partial of the Arithmetica of Diophante of Alexandria; his/her colleague Al-Khuwārizmī (env 783 - env 850) writes a book on Indian numeration. If the book is lost, there remains known by a Latin translation Algoritmi of number Indorum . Thābit (836 - 901) studies the friendly Nombres and the perfect numbers. Alhazen (965 - 1039) continues its work on the perfect numbers and discovers the Théorème of Wilson.

Appearance in Europe

Into 1621 Claude-Gaspard Bachet de Méziriac (1581 - 1638) translates the book of Diophante into Latin. The raised questions interest the mathematicians of the time, particularly the French. Pierre de Fermat (1601 - 1665) proposes a great number of statements, three more famous being probably its great theorem, its theorem of the two squares and its small theorem. The scientific community launches out challenges on this subject, thus Fermat asks: a square number which, added to the sum of its aliquot parts (IE its dividers), makes a cube. He concludes by: I await the solution of these questions; if it is provided neither by England, nor by Gaulle Belgium or Celtic, it will be it by the Narbonnese . Marine Mersenne (1588 - 1648) research of the prime numbers private individuals. Fermat writes to him: If I then once to hold the fundamental reason that 3,5,7,17,257,65 537,…, are prime numbers, it seem to me that I will find very beautiful things in this matter, because I already found things marvellous of which I will make you share . These numbers are now called numbers of Fermat and its sentence proves to be the single false conjecture suggested by the author. Rene Descartes (1596 - 1650) research without reaching that point, to show that if division by eight of a Prime number gives for remainder one or three, he is written form X ² + 2 there ².

On this type of problems, two elements are remarkable:

* the equations diophantiennes fascinate, the title of one of the books of Bachet de Méziriac is evocative: Problèmes plaisans and delectable which is done by the numbers, one can still quote the remark of Fermat in connection with its great theorem: I found a solution marvellous…

* the problems arising are difficult. In spite of some successes like the Identity of Bézout probably due to Bachet de Méziriac, the main part of the questions remain unanswered, like the theorem of the two squares of Fermat, or with answers at the very least not very convincing, like that of Fermat for its great theorem: … but the place misses to me here to develop it (the first recognized proof will appear into 1994,1995). Very often, Fermat finishes its theorems by comments acknowledging its failure: I acknowledge you any Net (because by advance I you informed that I am not able to allot to me more than I do not know, I say with the same frankness what I do not know) that I could not yet show… this beautiful proposal that I sent to you…

Methods used

The next century sees the resolution of some of these questions, often by Leonhard Euler: he contradicts Fermat by showing that its numbers are not always first, and proves the theorem of the two squares. He makes also errors, his attempt at demonstration of the great theorem of Fermat for N equal to three is a failure, his first demonstration proves to be false. He raises of another questions like the quadratic Loi of reciprocity in 1782. There still, and in spite of an attempt at Adrien-Marie Legendre (1752 - 1833) , the solution remains out of reach.

To the paddle of the 19th century the methods used, if they indicate a great easy way in the mathematicians, are finally very few and of simple principles. The following example, drawn from the problem of the two squares, into famous three: does there exist a number whose division by four gives for remainder three and who is nap of two squares? Either has ² and B ² these two squares. Only one is even because if not their amount would be even and its division by 4 would have either 0 or 2 as remains. Let us suppose has even and B odd. has is even thus its square is a multiple of 4. The entirety B is odd thus is written 2 C + 1, its square is equal to 4 C ² + 4 C + 1, the division of B ² by four gives for remainder a and summons it of the two squares also gives for remainder a .

• the first tool used is that of the prime numbers, this reasoning is exact because 2 is a prime number.

• the second is the algebraic transformation, B is transformed into 2 C + 1. Used with virtuosity, it makes it possible to the mathematicians of the time to solve certain equations diophantiennes. All the easy way resides to choose with skill the good transformations.
• the third is the Euclidean Division, the squares and their nap is systematically divided by 4.
• the fourth is not illustrated in the example, it corresponds to the infinite descent used in the demonstration of Euler of the theorem of the two squares. It consists in finding starting from a positive whole solution another positive and smaller whole solution. The continuation of the solutions goes down in an infinite way as a whole from the positive entireties, which is not possible. This method was already used by Fermat to show its great theorem if '' N '' is equal to 4 .

The rustic character of the tools results in long and technical demonstrations, such as for example the proof of Euler for the theorem of the two squares. Moreover, and in spite of more than one century of efforts, the main part of the equations diophantiennes resist such an approach.

The contribution of Carl Friedrich Gauss

At the 17 years age, Carl Friedrich Gauss (1777 - 1855) shows the quadratic law of reciprocity. One year later, on March 30th 1796 it becomes aware that its arithmetic calculations allow to build with the rule and the compass the Heptadécagone, i.e. the regular polygon with seventeen with dimensions, problem remained open since Antiquity. Finally in 1801, it publishes Disquisitiones arithmeticae ( arithmetic Recherches French ) and is called the prince of the mathematicians .

These two great discoveries proceed of the same step, the development of tools more sophisticated than those Fermat or Euler had, simplifying the demonstrations at the price of a larger abstraction. Its step founds the arithmetic modular one.

It is applied initially to the entireties then with the polynomials then with a new whole of entireties, now called whole of Gauss. Johann Peter Gustav Lejeune Dirichlet (1805 - 1859) discovers a similar unit, that of the whole of Dirichlet. It enables him to initiate the proof of the theorem of Fermat for N = 5 qu ' it sends to the Academy of Science in 1825. It is analyzed by Legendre which spends a few months to complete it.

All the equations diophantiennes of Fermat are now solved except for its great theorem. New conjectures appear. For example, if has and B is first between them, the arithmetic Suite of initial value has and of reason B contain does prime numbers, so yes how much? Gauss and other mathematician as Legendre imagined well that there is an infinity but does not manage to show it. In the same way the quadratic law of reciprocity must have equivalents for the higher orders.

The arithmetic modular one is enriched. Dirichlet, a pupil of Gauss finds a demonstration of the Théorème of the arithmetic progression by developing the concept of the characters and by formalizing the bases of the analytical Théorie of the numbers. Its reasoning is a jewel of arithmetic modular, C.G.J. Jacobi (1804 - 1851) written, in a letter with his/her brother: By applying Fourier series to the theory of the numbers, Dirichlet recently found results reaching the tops of human perspicacity. Dirichlet is not the first to use tools which are now described as consequence of the harmonic Analyze on an abelian group finished. Legendre to try to show the quadratic Loi of reciprocity developed similar calculations on the real , formalizing what is now called the Symbole of Legendre. Gauss finally, generalizes this approach with the complex numbers in its book of 1801. Its calculations bear the name of Somme of Gauss and Period of Gauss.

At the 19th century, this mathematics is called arithmetic transcendent . If the term of Arithmétique remains largely used, Legendre considers, in 1830, this branch like sufficiently developed to deserve the term of Théorie of the numbers. Appearance of novel methods, different from that of Gauss, introduced a subdivision with the Algebraic theory of the numbers and the analytical Theory of the numbers. The term of arithmetic transcendent falls then in disuse with the appearance of the transcendent numbers.

20th century

Cryptology

This field leaves that of pure mathematics. On the other hand, an industrial application called, during time, more and more upon the mathematical concepts developed per Gauss: the science of the secret codes called Cryptology. In 1883, Auguste Kerckhoffs (1835 - 1903) states that: the safety of a system of cryptography should not rest on the secrecy of the Algorithme. Safety rests only on the secrecy of the key . This approach is at the origin of a profound change of this science. In the middle of the XXe century, it becomes a branch of the Mathématiques applied.

At the beginning of the Years 1930, the office of the Polish figure calls upon the mathematician Marian Rejewski (1905 - 1980) for to bore the code of the system Enigma, used by the Germans. The old codes, like the Figure of César, are reinterpreted like a mathematical transformation in the whole of the modulos of Gauss on the integers. The term of arithmetic modular is used to describe these techniques. During the Years 1970, Horst Feistel (1915 - 1990) develops a system with key private, the Data Encryption Standard or FROM, which becomes the standard of the not classified applications. The cryptanalystes of, and more generally of the symmetrical codings, will use mathematics resulting from work of Dirichlet on the characters, within the framework of a vector Space on a Corps finished with two elements.

In 1978 a new family of codes is discovered, founded on a public key . Industrial solutions are quickly developed, most famous is called R.S.A.. It is based on work of Fermat and Euler. The term arithmetic modular is, in this context, used to describe not only the structure of the modulos on the entireties, but also the theorems treating of the prime numbers like the Décomposition in product of factors first, the Chinese Théorème, the Petit theorem of Fermat and its generalization by Euler.

The arithmetic modular one is an active field of research at the beginning of the 21e century. A Mise in effective work requires the use of operations on great whole of modulo. The approach of Gauss is used on Polynôme S with coefficients in a finished body. The arithmetic modular one spreads with other units that the quotients of entireties, for example for the Cryptanalyse.

Information theory

See also: Information theory

Cryptography is not the single field using the term arithmetic modular . The end of the Second world war sees the appearance of a new science: the Information theory. In 1948, under the impulse of Claude Shannon (1916 - 2001) , it becomes a branch of the Mathématiques applied.

If the Confidentialité is one of the tackled subjects, reliability is also a major topic. Richard Hamming (1915 - 1998) proposes a first Algorithme as of 1950. Once again, the modulos on the entireties are used for one of the simplest techniques of code: the checksums. In 1960, of new more powerful codes are developed, on the basis of polynomial to coefficients in a finished body. The arithmetic one used often takes the name of modular .

Data processing becomes a university subject at the beginning of the Années 1960. The constraints inherent in the structure of the Processeur S impose the representation of the numbers in the form of a finished continuation of information, justifying the use of the modulos. The term of arithmetic modular often appears, one finds there even the whole Gauss or the polynomials, for example, for calculations on great entireties.

The techniques developed for cryptography, the theory of the codes and arithmetic data processing rest on the same concepts, offering a relative unit to mathematics of the information theory.

Tools of arithmetic modular

Congruence and entireties

See also: Congruence on the entireties

The historical example of arithmetic modular rest on the integers. An entirety N being fixed, calculation modulo N consists in identifying all the entireties with their remainder in Euclidean division by N ; this can be illustrated for the example of arithmetic of the clock , which corresponds to n=12 : the small needle is in the same position at two times distant twelve hours, one identifies for example 13:00 with 1:00 to obtain a calculation on such a unit, one checks that the addition and the multiplication are compatible with the identifications. This formalization is the work of Legendre, which gives the name of residue to the various elements.

The contribution of Gauss consists in analyzing the structure of this unit, now qualified name of ring of congruences and noted '' Z ''/'' nZ ''. It is divided initially into the study of the addition, which defines a cyclic Groupe of generator 1 ; then of the multiplication, which depends on the properties of the modulo. If this one is first, one obtains a body. This approach simplifies the arithmetic demonstrations. The two historical examples of the book Disquisitiones arithmeticae are the Théorème of Wilson and the small theorem of Fermat.

Modular calculation, if the modulo is not first, is more complex. The Théorème of the Chinese remainders makes it possible to elucidate the structure. The ring is not just, it exists dividing of zero, in fact numbers, multiplied by certain an other number, give zero. The number of invertible elements is in general given by the indicating function of Euler. It allows, for example, to generalize the small theorem of Fermat.

Residue and polynomial

See also: cyclotomic Polynomial

Gauss notices that the whole of the Polynôme S with rational coefficients can be seen applying the logic of modular calculation, since it has of an addition, a multiplication, and an Euclidean division. Congruences are the remainders of the Euclidean division of the polynomials by a given polynomial.

It applies this approach to the polynomial X ⁿ - 1 and finds its decomposition in product of irreducible factors, which take the cyclotomic name of polynomial . Gauss uses these results to find a new polygon regular constructible with the rule and the compass the heptadécagone.

He hesitates to regard this work as the arithmetic one; he writes: The theory of the division of the circle, or the regular polygons,…, does not belong by it even to the Arithmetic one, but its principles can be drawn only from the Arithmetic transcendent one . The term of arithmetic “transcendent” of Gauss is now replaced by that of arithmetic “modular”. The logic of this argument is always of topicality.

Algebraic entirety

See also: Whole of Gauss

The case of the polynomials with whole coefficients differs: the property of division functions only for polynomials whose greatest coefficient is equal to more or less a . The case of the modulos of the polynomial X ² + 1 is considered: the modular structure obtained is still that of a ring, being identified with the whole of the numbers of the form α + i.β where α and β are entireties and I indicates the Imaginary number, corresponding to the students' rag procession X . This unit is that of the entireties of Gauss.

It can be provided with a standard. With the entirety of Gauss = α has + i.β is associated the standard α² + β², which comes from the module of the complex numbers. This standard makes it possible to define a Euclidean Division, like illustrates it the figure of right-hand side. The entireties are represented by the intersections of the squaring. The value has / B exists if B is different from zero, however this value is not necessarily an entirety of Gauss. It is represented by the black spot of the figure. To say that an Euclidean division exists, amounts saying that there exists an entirety of Gauss to a standard strictly lower than a of this black spot. The illustration shows that, in this case, there exist at least three candidates. In the general case, there are some between one and four and in this context only the existence counts.

This result of Euclidean division implies properties on this ring of entireties: the Identity of Bézout, the existence of prime numbers of Gauss and an analog of the fundamental Theorem of arithmetic the. These prime numbers allow Richard Dedekind (1831 - 1916) to propose a simple resolution theorem of the two squares. The geometrical illustration is given on the figure of left. A prime number p is expressed as summons of two squares if the circle of radius the root of p crosses at least a Gauss entirety.

Ferdinand Eisenstein (1823 1852) , a pupil of Gauss, discovers new a ring of entireties; the arithmetic one on this ring offers a rigorous demonstration of the great theorem of Fermat for '' N '' equal to three, justifying, once again, the theoretical need for such a generalization of arithmetic modular.

Character of Dirichlet

See also: Character of Dirichlet

Dirichlet is interested in the prime numbers form N + λ .m where N and m is two entireties Premiers between them and λ a variable which describes the whole of the positive entireties. It indeed wishes to show that there exists an infinity of prime numbers of this nature.

The arithmetic modular one is a good tool for these problems, which are equivalent to find the cardinal of the whole of the prime numbers in a class of modulo.

Dirichlet considers the group of the invertible elements modulo m , and studies the whole of the functions of the group in the complex numbers nonnull which check, if has and B is two residues: F ( a.b ) = F ( has ). F ( B ). Such functions are called characters of Dirichlet. There exists about it φ ( N ), the product of two characters is still a character, and their multiplication table is exactly same the as that of the studied Groupe of the units.

Calculations on these functions are formally similar to those carried out previously by Joseph Fourier (1768 - 1830) . It is necessary nevertheless to reach the 20th century to see appearing a theory unifying the two approaches. It takes the harmonic name of Analyze.

Theoretical developments

It is frequent that mathematical concepts, developed in a context, are re-used in other fields. Thus the Théorie of the groups applies to the Arithmétique and the Géométrie. It is the same for the tools of arithmetic modular, whose tools feed from vast fields of the pure Mathématiques, like the general algebra or the Théorie of Welshman. These theories nevertheless are not regarded as particular cases of arithmetic modular because they call also upon good of other concepts.

Structure quotient

See also: Relation of equivalence

In modern language, the arithmetic modular one formalizes by the concept of quotient of Euclidean rings. The concept of Relation of equivalence makes it possible to generalize this concept with principal the algebraic structures. The quotient, of a group by a normal Sub-group, is, for example, a basic tool of the classification of the Groupes finished, through the Théorème of Jordan-Hölder. The quotients groups are also used in algebraic Topologie to classify the varieties. In the Theory of the rings, the concept of Idéal plays a part similar to that of the concept of sub-group normal in theory of the groups. It makes it possible to build rings quotients in a context more general than that of arithmetic modular. The Theorem of the zeros of Hilbert, bases bond between the commutative Algèbre and the algebraic Géométrie, is expressed in term of ideal.

The terms of congruence and modulo nevertheless are reserved for the quotients on an Euclidean ring.

Residues of polynomials and Welshman theory

See also: Theory of Welshman

The arithmetic modular one applies to the ring of the polynomials with coefficients in a body. It is the starting point of the theory of Welsh Evariste (1811 1832) and consists of the systematic study of the whole of irreducible modulos of polynomials, the equivalent of the prime numbers. These units are now called algebraic extensions.

These extensions allow the analysis of the resolvability of the algebraic equations, i.e. equations being written in polynomial form. If the polynomial is irreducible, its whole of modulo is the smallest body containing at least a root. It is called Corps of rupture. By reiterating this process, a body containing all the roots, the Body of decomposition, is built. The modular logic of the quotient provides the algebraic structure adapted to these problems.

The Welshman theory calls upon good of other concepts. The study of the resolvability of the equation is possible via the study of the group of the Automorphisme S of the body, called Groupe of Welshman, thanks to the Correspondance of Welshman enter subfield and sub-groups. Beyond the study of the resolvability of the algebraic equations, the Welshman theory became a natural framework of resolution of many problems into arithmetic, arithmetic Géométrie or algebraic Géométrie, and especially makes it possible to formulate new more general problems in these various fields.

If this theory uses the concept of quotient of an Euclidean ring, the variety of tools specific to this field in fact a clean field, quite distinct from the subject of this article. It should be noted that one of the fruits of this theory: the Body finished S, still called body of Welshman, provide a natural framework to many applications into arithmetic modular.

Algebraic entirety and algebraic theory of the numbers

See also: Algebraic theory of the numbers

Arithmetic the modular offer a good conceptual framework for the resolution of the great theorem of Fermat. However, if N is larger than ten , the algebraic whole rings of , built according to the method of Gauss, present what Dirichlet calls a obstruction . It shows that the Groupe of the units of this ring, i.e. elements having a reverse for the multiplication, is not any more a cyclic Groupe or abelian finished as that which Gauss studied. It contains also copies of the ring of the entireties and is thus infinite. This result takes the name of Théorème of the units of Dirichlet. The obstruction comes from this new configuration. It prevents the application of the modular techniques used for the entireties of Gauss because the associated ring is not Euclidean any more.

Ernst Kummer (1810 - 1893) uses a tool related to the generalization of the quotient now formalized by the ideals. They replace the prime numbers absent. The algebraic theory of the numbers takes over then, with different techniques. The basic tool is a ring whose elements are called whole algebraic and who has a structure known as of Anneau of Dedekind. Kummer thus manages to show the great theorem for certain value of N first, i.e. for the regular prime numbers. The only untreated values lower than 100 are 37,59 and 67.

Other tools and objects of study appear, like the adelic Anneau, those of the Welshman theory, the elliptic curved , the series L of Dirichlet or the modular forms. Some come from almost direct consequences from arithmetic modular, as the finished bodies, used in an intensive way at the 20th century. The algebraic theory of the numbers is largely vaster than the framework of arithmetic modular, while resting in fine on sometimes similar techniques.

Character of Dirichlet and analytical theory of the numbers

See also: analytical Theory of the numbers

The discovery by Euler of an infinite produced, built using prime numbers and equal to the sixth of the square of the surface of a circle of radius a , opens the way with an approach different for comprehension from the numbers. Dirichlet uses it to show that each modulo of entireties of the group of the units contains an infinity of prime numbers. This result bears now the name of Théorème of the arithmetic progression.

To conclude its demonstration, Dirichlet develops a specific tool, the series L of Dirichlet. One of its series corresponds to a famous function which will take the name of ζ of Riemann. Its greater difficulty consists in proving than its functions do not have a root at the point a . For that purpose, it uses the analyzes harmonic on the group of the units of a class of modulo.

Nevertheless, once again, the arithmetic modular one is insufficient to come to end from the theorem. Dirichlet uses many analytical techniques, like the whole series and the Analyze complexes. The fruit of this work gives rise to a new branch of mathematics: the analytical theory of the numbers. One of the crucial points of this theory comes from the single article of Bernhard Riemann (1826 - 1866) in theory of the numbers: On the number of prime numbers lower than a given size . He conjectures a localization of the roots of his function ζ. The research of the position of the roots, initiated by Dirichlet, becomes a central concern and remains one of the conjectures had a presentiment of like most difficult of mathematics of our time ( 2007 ).

Cryptography

See also: Cryptography

The object of cryptography is to ensure the Confidentialité in the transmission of the messages and the Authentification of those. One can quote two examples: the protection of the messages which an army uses to prevent an anticipation of the enemy, or the credit card proposed by the banking system, offering to a user a good safety.

Into more mathematical terms, the operation of coding results in a Algorithme, i.e. a function F which, with a message in light m and a key K , associates a coded message F ( K , m ). The knowledge of the coded message and the algorithm must be insufficient to reconstitute the message in light without a key of deciphering. In the case of cryptography traditional, known as symmetrical Cryptography, the key of deciphering is identical to the key of coding or from of easily deduced. This key must then remain secret.

The asymmetrical Cryptographie is based on the report that only the key of deciphering must remain secret, and known only receiver of the message. It does not need to be communicated to its correspondents. Alice uses the key of coding of Bob, that this one made public, to send a message to him. Seul Bob can decipher it, even Alice, if ever it had lost the message in light, would be unable. Bob must answer by using the key of coding of Alice.

The objective is thus to define a function simple to evaluate but difficult to reverse without the knowledge of a secrecy. The arithmetic modular one was the first to offer solutions, and it is always at the base of many commercial solutions. For example the exchange of keys Diffie-Hellman, first historical example, exploits the practical difficulty to reverse the modular Exponentiation. The latter, or its generalizations with others groups, remains fundamental in the field.

Asymmetrical cryptography solves in particular the delicate problem of the distribution of the keys in symmetrical cryptography. If several correspondents communicate, in asymmetrical crypotographe, a different key proves to be necessary for each couple of speakers, whereas in symmetrical cryptography each correspondent has a key which it keeps secret, and of a key which it makes public. However it did not make disappear the symmetrical codes, which offer algorithms much more effective. For an equivalent safety, the symmetrical codes have the advantage of requiring keys definitely smaller, 128 Bit S for the current version of AES, against more than one thousand for RSA, but especially coding as the deciphering are from one hundred to thousand times the faster. The modern cryptographic systems, like those used by the bank cards, or the communication protocol encrypted SSL/TLS very much used on Internet, use only at the beginning of communication asymmetrical cryptography, to exchange the keys of a symmetrical coding which will take then the relai.

Asymmetrical cryptography

See also: asymmetrical Cryptography

The code RSA is a largely widespread example of asymmetrical cryptography. It is described in the following way:

Alice (the choice of the first names is traditional) wishes to be able to receive messages of Bob without Eve being able to decipher them. Alice chooses two large prime numbers p and Q and an entirety E , first with the order G of the group of the units of Z / pqZ . Here G is equal to ( p - 1) ( Q - 1), that is to say the value of the function Indicatrice of Euler in n=pq . The messages are supposed to be elements of this ring. If Bob wishes to transmit the message m to Alice, it transmits the value of m ^e modulo N . Alice made as a preliminary public the values of N = p.q , E and thus the function F of Chiffrement, which is equal here to:

$F\; (m)\; \backslash \; \backslash \; equiv\; \backslash \; m^e\; \backslash \; quad\; \backslash \; pmod\; n$ Eve could intercept the knowledge of F , E and N , this information is indeed public. On the other hand, it cannot intercept the values of p and Q which was never communicated.

For Bob, the function of coding is easy: it is about simple a modular Exponentiation. For Alice the reading is also, it is enough for him to find a solution with the Identité of Bézout:

$x.e\; +\; Y.\; (p-1)\; (q-1)\; \backslash \; =\; \backslash \; 1\; \backslash ;$ If ( X ₀, there ₀) is a solution, then the theorem of Euler watch that: $m\; \backslash \; equiv\; \backslash \; F\; (m)\; ^\; \{x\_0\}\; \backslash \; MOD\; N$ On the other hand Eve does not know a priori the Décomposition in product of factors first of N . She must calculate the discrete Logarithme of F ( m ), which constitutes a difficult problem. The known least slow method corresponds to determine the values of p and Q . If N is large, there does not exist, in 2007, of effective algorithm to solve this question.

The key allowing coding is the data of E and N . The force of such a system lies in the fact that the knowledge of this key does not allow the decoding, it can thus be public. The values of p and Q constitute the key of decoding.

Symmetrical cryptography

See also: symmetrical Cryptography

The asymmetrical cryptograpie would not exist without the methods resulting from arithmetic modular, which, for obvious historical reasons, is not the case of symmetrical cryptography. It is divided into two big families whose one, the codings by flood, uses like basic component the linear recurring continuations on a finished body (see below). The other, that of the codings per block, includes/understands inter alia and its successor, the Standard of advanced coding called AES for Advanced Encryption Standard . The latter operate on storage blocks of a fixed size counted in Octet S, eight for OF for example. A succession of rather simple primitive operations is applied in a way repeated to code a block. A byte, or more generally a block of N bits, can be seen like the coefficients of a polynomial on the entireties modulo two, of maximum degree n-1 . That led the cryptologists to be interested in certain operations on the Corps finished S of characteristic 2. Thus it proves that the operation of inversion on the body finished F _2ⁿ, composed with a transformation closely connected, has good cryptographic properties to do of it one of the primitives of codings per block. This was exploited by the authors of the coding Rijndael, which became the AES. Besides the official publication of this last by NIST (American federal agency) contains some mathematical preliminaries on the subject. However it is not no need for algorithmic on the arithmetic one or the bodies finished for the implementation: these operations are represented by tables, like the similar operations of obtained they in a way much more heuristic. Certain cryptologists saw a potential weakness in the too algebraic characterization of Rijndael, which would make it more accessible to imagination mathematicians, which did not prevent its adoption for the AES.

Test of primality

See also: Test of primality

Codes RSA use as key the prime numbers p and Q of the preceding paragraph. To find the use, they, consist in choosing a number randomly almost , to test if it is first or not and to start again if it is not it.

The Crible of Ératosthène is a fast method for the small numbers. Used skilfully, 46 tests are enough to check the primality of a number lower than 39.000. On the other hand, it is ineffective for an industrial application employing of the numbers which are written with several hundreds of figures.

The majority of the tests of industry are based on alternatives of the Petit theorem of Fermat. If a number p is first, then for entire has , has ^p is adequate with has modulo p . The reciprocal one is false: there exist numbers not first, called numbers of Carmichaël (for example 1729), for which congruence is true for any value of has . However, if p is neither a number of Carmichaël, nor a prime number, congruence is false for at least half of the values of has ranging between a and p . That congruence is checked for a great number of values of has indicates a very strong probability of primality for p , if it is not a number of Carmichaël.

The Test of primality of Solovay-Strassen and especially the Test of primality of Miller-Rabin are two examples largely used. They are based on a more thorough analysis of the small theorem of Fermat and do not admit analogues with the numbers of Carmichaël, which raises one of the problems of the test of Fermat. For these two methods, a deterministic test of primality consists in checking the property for a number of values of has guaranteeing an irrefutable proof. The number of calculations to be carried out being crippling, one is satisfied with a probabilistic test. It consists in checking congruence on a whole of values of has , chosen to ensure a probability of primality higher than a given value, often equal to 1 - (1/2) ¹⁰⁰.

Decomposition in product of factors first

See also: Decomposition in product of factors first

The Sécurité by the darkness is not setting for codes RSA. It is important to precisely know the state of the art of the decomposition of the entireties in terms of factors first. A contest called, Compétition of factorization RSA is permanently open, proposing a price for whoever able to factorize a number selected in a list made public.

The screen of Ératosthène is a test of primality which offers a method of decomposition. But once again, it is not applicable for great numbers because too slow.

The various methods currently used often rest on the quadratic residues. A Diviseur of zero is a quadratic residue containing like representatives at least two square perfect. The objective is to find these two squares. This approach is that of the quadratic Crible and the Algorithme of factorization by screen on the bodies of numbers generalized, fastest known in 2007. One can still quote the algorithm ρ of Pollard, it uses the Paradoxe of the birthdays.

Finished body

See also: Body finished

Just as for the arithmetic one of pure mathematics, other structures are necessary to exploit the capacities offered by the arithmetic modular one. In data processing, the numbers are coded on N bits, i.e. correspond to a continuation length N made up of zero and ones . Such a continuation can be regarded as a Vecteur of a vector Space of dimension N on the body finished F ₂ with two elements.

This structure is often regarded as space of the polynomials with coefficients in F ₂. To guarantee the stability of the multiplication, the logic of Gauss is applied, this space is quotienté by an irreducible polynomial (the equivalent of a prime number) of degree N . The structure obtained is the finished body of cardinal 2ⁿ. A number has modulo 2ⁿ and a polynomial P of the modular system is very similar, they are written indeed:

$a\; \backslash \; =\; \backslash \; \backslash \; sum\_\; \{i=0\}\; ^\; \{n-1\}\; a\_i.2^i\; \backslash \; quad\; and\; \backslash \; quad\; P\; \backslash \; =\; \backslash \; \backslash \; sum\_\; \{i=0\}\; ^\; \{n-1\}\; \backslash \; alpha\_i.\; X^i\; \backslash \; quad,\; \backslash \; quad\; a\_i,\; \backslash \; alpha\_i\; \backslash \; in\; \backslash \; mathbb\; F\_2$

An example of use is the creation of a Générateur of pseudo-random numbers in F ₂, for example for a Chiffrement of flow used in the context of an oral communication by Cellphone. An element of the algorithm is the constitution of a Shift register. Using a key, here an entirety C modulo modulo 2ⁿ, such a register provides a pseudo-random continuation. It is obtained by a linear recurring Suite. If it is noted ( U _j), then:

$Si\; \backslash \; quad\; C\; \backslash \; =\; \backslash \; \backslash \; sum\_\; \{i=0\}\; ^\; \{n-1\}\; c\_i.2^i\; \backslash \; quad\; then\; \backslash \; quad\; u\_j\; \backslash \; =\; \backslash \; \backslash \; sum\_\; \{i=1\}\; ^n\; c\_i.u\_\; \{j-i\}\; \backslash \; quad,\; \backslash \; quad\; c\_i\; \backslash \; in\; \backslash \; mathbb\; F\_2$

The continuation obtained is periodic, however if the key is quite selected, the period is very long: 2n - 1. This situation occurs if the polynomial of feedback R , given by the following formula, is the minimal Polynôme of a primitive element of the cyclic Groupe F 2n*.

$R\; \backslash \; =\; \backslash \; \backslash \; sum\_\; \{i=0\}\; ^\; \{n-1\}\; c\_\; \{n-i-1\}.\; X^i$

Analyzes harmonic on a finished abelian group

See also: harmonic Analysis on an abelian group finished

The ideas of Dirichlet apply to the modular system of the preceding paragraph. For the addition, the vector space V preceding is a abelian Groupe finished. The characters of this group form a orthonormal Base of the whole of the functions of V in that of the complex numbers. It should be noted that the selected whole of arrival is not always that the complexes but sometimes the body F ₂. The results are strictly identical. Such a structure has a harmonic analysis. If the whole of arrival is selected equal to F ₂ the Transformée of Fourier takes the name of Transformée of Walsh. This approach is used at the same time for the systems OF, RSA and some codings of flow.

A Shift register is too easily decipherable. The algorithm of Berlekamp-Massey makes it possible to determine the continuation grace the knowledge of N consecutive values with a quadratic complexity. Thus, if the key is made up of 128 bits, it is enough to a multiple of 128 X 128 = 16.384 stages to decipher it, which represents an insufficient safety. The adopted solution consists in using several shift registers. The various results are seen like an element of a new vector space on F ₂. A Boolean Fonction associates with each element of this space the value a or zero . If such a function is quite selected, the best algorithm of known deciphering requires about 2ⁿ stages to find the signal bringing confusion. The determination of this function is obtained using the tools of the harmonic analysis.

For a code RSA and at the end of the 20th century, the key is often a number exceeding 10³⁰⁸. It is important to have a fast multiplication on the great modulos. The technique consists in identifying the modulos with the polynomials on a finished body. The multiplication of two polynomials of degree N is an operation which, if it is carried out in a naive way, imposes a quadratic complexity. The characters of the additive group associated being orthogonal, complexity becomes linear if such a base is used. A faster calculation consists in carrying out a Transformée of fast Fourier, multiplying the two results and operating the transform of opposite Fourier. Total complexity is then in N log ( N ) where log indicates here the Logarithme basic two .

Code correct

See also: correct Code

A correct code does not have the vocation to ensure the safety, but the reliability of the transmission of a message. It makes it possible to restore the original text even if a random and moderate disturbance occurs during the transmission. The message encodé is transmitted in a form called word of the code . It contains not only information of the initial message but also the redundancies allowing the validation of a good communication and sometimes the autocorrection of possible errors.

A word of the code is made up of a family of N letters selected in a alphabet , in general binary. The industrial case most frequent is that of the linear Code, the value N is constant for each word of the code and is called dimension code. The whole of the words of the code is provided with a structure of vector Space of dimension N .

The linear codes use primarily the arithmetic modular one as bases mathematical. Many enriches the structure of vector space by that by a ring of polynomials on a Corps finished. This body is sometimes the binary body, often a body of cardinal a power of two. One then speaks about cyclic Code.

The linear codes are largely present in industry. The Télécommunication uses them for the Cellphone or Internet, data processing for in particular the communication between the memory and of Processeur, the audio-visual for the compact disks or other formats of comparable nature like DVD.

Summon of control

See also: Checksum

A sum of control is a linear Code particularly used. It corresponds to a correct Code whose only detection is automatizable, the correction is obtained by a request for repetition of the message.

To the initial message a data on a letter is added. The letter is selected in such manner so that the congruence of the sum of the letters of the word of the code is equal to a given value, often zero . In the example illustrated on the figure of right-hand side, the initial message is made up of two letters, a word of the code contains three of them, if the initial message is 00, the word of the transmitted code is 00 0 , if the message is 01, the word of the code becomes 01 1 . The four possible words of the code are illustrated by the green points of the figure.

If a single error occurs, on any of the three letters of the word of the code, the received message corresponds to a black spot. The receiver knows that it must ask the renewal of the transmission. This configuration is similar whatever the length of the word of the code. A length equalizes to eight is often chosen, it allows the transmission of a message of seven letters. The result is identical, each licit word of the code has for neighbor only words out of the code, a single error during the transmission is thus detected. On the other hand a double anomaly systematically overlooked.

Cyclic code

See also: cyclic Code

There exist certain situations where the request for repetition is not possible, for example for a DVD, when a dust masks information. It is necessary to imagine correct codes which, not only detect, but correct the errors automatically.

The method used consists in moving away the words from the code to a sufficient distance to locate the good message of origin. The distance between two points corresponds to the number of letters to modify to pass from the one to the other. The graph of left illustrates this situation. The green points correspond to the words of the code, by definition without error. The blue ones are those obtained when a single letter is faded in the transmission and the reds when two letters are modified. In the diagram, one notices that each blue point contains a single green point at a distance from a and to each red point corresponds a single green point located at a distance from two . If one or two errors occurred, the single green point nearest necessarily corresponds to the initial message. Such a code is able to protect up to two errors.

The arithmetic modular one provides optimal solutions to build the geometry of a linear Code correct. As a vector space does not constitute a sufficient structure to define modulos, it is enriched by a structure of ring of polynomials quotienté by X ⁿ - 1, where N indicates the dimension of the vector space. In this space of modulo, students' rag procession X ⁿ is identified with the constant polynomial a . If the chain (a₀, a₁,…, a_n) is a word of the code, then it is the same of the following one: (a_n, a₁, a₂,…, a_n-1). One speaks for this reason about cyclic code . Logic is the same one as that of a correct code, with the difference close congruence is not defined on an entirety but on a cyclotomic Polynôme with coefficients in a finished body.

The simplest example corresponds to the Hamming code whose messages to be transmitted comprise four letters and three additional letters describe the redundancies.

Identity of Mac Williams

See also: Identity of Mac Williams

The context of a linear code is similar to that of cryptography, one finds there also spaces vector of dimension N on a finished body and a system of modulo of polynomials, the selected polynomial being often: X ⁿ + 1. The characters of the group are used, as well as the analyzes harmonic associated.

The identity of Mac Williams is an archetypal example. It allows the determination of the polynomial enumerator of the weights of the dual code or the study of the characteristics of the Hamming code. It is obtained thanks to the harmonic analysis, using a Produit convolution.

Random links:

Malicious Kiki | Trac (software) | Völkerball | April 28th in sport | Sean Scully | Europeanization_des_écritures_saintes

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org