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Course Digest for MATH 601 Spring 2022. Date Tuesday, March 8. Tasks Watch Lecture 13. Read , , in HSP. Practice Problem , (a), replacing the k there with the one furnished by Diffie-Hellman, as in lecture. Synopsis We started today's lecture with some remarks on Team Project 2, and on the recent Short Quiz 3. After this, we recalled the basics of the Diffie-Hellman Key Exchange, and the ElGamal cryptosystem. Note that there are many ways to tweak the ElGamal cryptosystem scheme that we presented in class, and this might be what you run across if you investigate it online, or in HSP. We went over some numerical examples, and then turned our attention to the Discrete Logarithm problem. We started by proving a simple theorem Let g be a primitive roots modulo an odd prime p, and suppose that x = a is a solution to the equation g x = A mod p, where A is nonzero modulo p. (1) x = a is even if and only if A(p 1)/2 1 mod p. (2) x = a is odd if and only if A(p 1)/2 1 mod p.

Hint: Apply Euler’s Theorem. (2)Simplify 5454 mod 151 Hint: 151 is prime, so apply Fermat’s Little Theorem. Synopsis We started today’s lecture with another standard quiz, and then presented a lengthy overview of the rst group project, including giving a demonstration of the basics of Sage, conducted on SageMathCloud.

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  Overview, Theorem, S theorem, Fermat

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