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session_03_number_theory_&_public_key_090708

session_03_number_theory_&_public_key_090708 -...

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0 M. Mogollon – 08/02 - 0 Cryptography and Network Security Cryptography and Network Security TECH 6350 Manuel Mogollon [email protected] Graduate School of Management Information Assurance University of Dallas Session 3 Session 3 Session 3 Number Theory, and Number Theory, and Public Key Ciphers Public Key Ciphers
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1 M. Mogollon – 01/08 - 1 Number Theory Exponentiation Ciphers Public-Key Ciphers Key Management DH / RSA Session 3 Contents Session 3 Contents Number Theory and Finite Arithmetic Counting in modulo p Arithmetic Congruence Arithmetic Fermat’s Theorem Euler’s Theorem Confidentiality using Public-Key Ciphers Pohlig-Hellman Algorithm The RSA Algorithm ElGamal Algorithm Key Management Using Exponentiation Ciphers The Diffie-Hellman Key Agreement RSA Key Transport When we read a book about the history of cryptography, we find out that all the advances in cryptography were made by individuals who, among other things, were great mathematicians. Number theory is an ancient and fascinating branch of mathematics that plays an important role in public-key crypto systems. Knowing certain basic concepts of number theory, such as modular arithmetic, and congruence, is necessary for an understanding of Public-Key cryptosystems. The mathematics of Public-Key is based on raising large numbers to a very large power. Microsoft Excel cannot perform the operation of raising 1000 to the power of 1000 because the result is too large. So how it is possible in the RSA Public-Key encryption algorithm to raise a large number, 200 digits or even larger, to the power of another 200 digit number? The only way is by using modular arithmetic. In this session, the basic concepts of number theory and congruence arithmetic are described to be able to understand Public-Key theory. Then, the most used Public-Key Ciphers, Pohlig-Hellman Algorithm, RSA Algorithm, ElGamal algorithm and Diffie-Hellman, will be presented.
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2 M. Mogollon – 01/08 - 2 Number Theory Exponentiation Ciphers Public-Key Ciphers Key Management DH / RSA The Set of Real Numbers The Set of Real Numbers -7, -2/5, 0, 1, ¾, 5.42, Set of all rational and irrational numbers. Real R -7, -2/5, 0, ¾, 5.42 Any number that can be represented as a/b, where and a and b are integers and b 0. Rational Q .., -2, -1, 0, 1, 2, … Set of natural numbers, their negatives, and zero. Integers Z 1, 2, 3, 4, 5, ….. Counting numbers (also called positive integers). Natural Numbers N Examples Description Number System Symbol π , 5 , 2 The set of real numbers shown in the table is not generally applicable to cryptography because in arithmetic, information is lost through round-off errors, or truncation in integer division, and, also, because real numbers are infinite fields.
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