WORKSHOP ON NUMBER THEORY AND MODERN CRYPTOGRAPHY

WORKSHOP ON NUMBER THEORY AND MODERN CRYPTOGRAPHY

EMANUEL EDUARDO PIRES VAZ

139 pages

Intended Audience: This is a book of pure mathematics which results in practice in concrete. It can be used to advantage students of mathematics the second and third cycles of universities and is a base of support for those who recognize the encryption as a partner  in their work.
Contents: Encryption.  Decryption.  Cryptography. Job appointment. Software Program. Classical cryptography. Properties. Euclidean division and fundamental theorem of arithmetic. Proposition. Lemma. Corollary.Theorem.Corollary.Theorem. ongruences.Proposition.Corollary. Theorem 1.10. (theorema of Chinese rests). Corollary. Euler`s function and the Fermat`s small theorem. Proposition. Theorem (Euler). Corollary. Little Fermat`s theorem. Corollary. Corollary. Lemma. Theorem. (Wilson). The function of Möbius. Theorem. Lemma.  Theorem. (Reversal formula of Möbius) . Theorem. (Second formula reversal of Möbius). Conjecture. Bases.  Theorem.  Theorem.  On the distribution of prime numbers.  Theorem. (Theorem of prime numbers). Proposition. Corollary.  Theorem. (Dirichlet). RSA: Public-key cryptography.  Fermat`s little theorem.  Fermat`s last theorem. Greatest common divisor.  Euclidean algorithm.  Public-key cryptography. Applications.  How does the method of public key work? Definition (notation of order). Definition.  Euclid`s algorithm.  Theorem. Corollary.  Theorem.  Example. Theorem.  Cost of the extended algorithm of Euclid.  Theorem. . Modular rapid exponentiation. Example.
2.8.1.2. Classes of complexity. Definitions. Estimation. The problem of the millennium. Example. Definitions. Example. Generation of public and private keys. Encryption. Decryption.  (RSA). Examples. Multiplicative arithmetic functions. Example. Theorem. Definition. Example. Definition.  Corollary. Theorem. Lemma. Euler`s theorem. Primitive roots. Definition.  Lemma. Lemma. Gauss`s theorem. Lemma. Lemma. Lemma. Corollary.  Corollary. Theorem of primitive roots. Definition. Law of quadratic reciprocity. Legendre symbol. Definition. Theorem. (Euler`s criterion).  Theorem. Theorem. (Lemma of Gauss). Lemma. Theorem. Theorem. Eisenstein`s theorem. Definition. Theorem. Lemma. Solution of the congruence  . Theorem. Chinese theorem of the rests.  Problem. Problem. Proof of RSA theorem. Example. Accelerated RSA method. Calculation of the decryption exponent  . Definition.  Diffie- Hellman.  Theorem. Optimization of deciphering. Definition. Theorem. Safety in RSA. Calculating    without  factoring  . Calculating   without knowing  .Problem of the sum pf subsets. System for encrypting messages of Merckle-Hellman. Example.  Generation of public and private keys. Knapsack encryption system.  Encryption. Decryption.  Example. Parameters and operation of the system. Diffie-Hellman algotithm. Exploring Diffie-Hellman encryption. Password authenticated key agreement. Diffie-Hellman example. Diffie-Hellman conclusion.  Crypto analyse. RSA algotithm I. RSA algorithm II. RSA algorithm III.  Example RSA. Security of RSA. Digital signature using RSA.  Schem of digital signature using RSA.  Cryptography and digital signature.  Distribution of keys. Certification authority. Example of content for digital certificate. Authentication of message.  Still the “message authentication”. Unidirectional Hash algorithms. Elliptic curve cryptography.  A little story. Elliptic curves. Generating an elliptic curve public key.  Encrypting and decrypting messages. Cracking the elliptic curve key.  Elliptic curves over the real numbers. Graphs of curves. The group law.  Elliptic curves over the complex numbers. Elliptic curves over the rational numbers. The structure of rational points. The Birch and Swinnerton-Dyer conjecture (BSD). The modularity theorem and its application to Fermat`s last theorem. Modularity theorem. Integral points. Generalization to number fields. Elliptic curves over a general field. Isogeny. Elliptic curves over finite fields. The Sato-Tate conjecture. Algorithms that use elliptic.  Elliptic curve DSA. Key and signature size comparison to DSA. Signature verification algorithm. The idea behind the attack. Timing attacks are practical in many cases. Examples. Bibliography.

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