2015

COMPUTATIONAL NUMBER THEORY Course code: 15CS2202

L 3

P C 0 3

Pre requisites: Number theory basics, Security issues. Course Outcomes: A student who successfully completes this course should, at a minimum, be able to CO1: Develop the mathematical skills to solve number theory problems and to develop the mathematical skills of divisions, congruence’s, and number functions. CO2: Learn the history of number theory and its solved and unsolved problems. CO3: Investigate applications of number theory and the use of computers in a Number theory. CO4: Estimate the time and space complexities of various Secure Algorithms. CO5: Learn various factorization and logarithmic methods. UNIT-I: (10-Lectures) TOPICS IN ELEMENTARY NUMBER THEORY: O and Ω notations – time estimates for doing arithmetic – divisibility and the Euclidean algorithm – Congruence’s: Definitions and properties – linear congruence’s , residue classes, Euler’s phi function UNIT II: (10-Lectures) FERMAT’S LITTLE THEOREM – Chinese Remainder Theorem – Applications to factoring – finite fields – quadratic residues and reciprocity: Quadratic residues – Legendre symbol – Jacobi symbol. Enciphering Matrices – Encryption Schemes – Symmetric and Asymmetric Cryptosystems – Cryptanalysis – Block ciphers –Use of Block Ciphers.

M.TECH- CYBER SECURITY

5

GVP COLLEGE OF ENGINEERING (A)

2015

UNIT-III: (10-Lectures) MULTIPLE ENCRYPTION – Stream Ciphers –Affine cipher – Vigenere, Hill, and Permutation Cipher – Secure Cryptosystem. Public Key Cryptosystems: The idea of public key cryptography – The Difﬁe–Hellman Key Agreement Protocol - RSA Cryptosystem – Bit security of RSA – ElGamal Encryption UNIT-IV: (10-Lectures) DISCRETE LOGARITHM – Knapsack problem – ZeroKnowledge Protocols – From Cryptography to Communication Security - Oblivious Transfer. Primality and Factoring: Pseudo primes – the rho (γ) method – Format factorization and factor bases. UNIT-V: (10-Lectures) THE CONTINUED FRACTION METHOD – the quadratic sieve method. Number Theory and Algebraic Geometry: Elliptic curves – basic facts – elliptic curve cryptosystems – elliptic curve primality test – elliptic curve factorization. TEXT BOOKS: 1. Neal Koblitz: “A Course in Number Theory and Cryptography”, 2nd Edition, Springer, 2002. 2. Johannes A. Buchman: “Introduction to Cryptography”, 2 nd Edition, Springer, 2004. REFERENCES: 1. Serge Vaudenay, “Classical Introduction to Cryptography Applications for Communication Security”, Springer, 2006. 2. Victor Shoup: “A Computational Introduction to Number Theory and Algebra”, Cambridge University Press, 2005. 3. A. Manezes, P. Van Oorschot and S. Vanstone: “Hand Book of Applied Cryptography”, CRC Press, 1996.

M.TECH- CYBER SECURITY

6