JSFH Exam 2 As you know, you are not allowed to discuss this exam with any human other than me. The exam is open book, open notes. Make sure you show all of your work. Send the completed exam to me via Digital Dropbox. The exam is due on Friday, April 24, at 5:00. Enjoy, Dr Dan

I have included the following programs: num2mess(n) esum(ec, pt1, pt2, p) nsum(ec, n, pt, p) oncurve(ec, pt, p) mess2pt(a, ec, p) pt2mess(pt) I have also included the material for Microsoft's elliptic curve: msp - The prime that they use. msec - The elliptic curve. msP - The base point used. msord - The order of the base point msP. 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JSFH

Alice and Bob are using RSA. Bob came up with some really great primes, but alas, he used primes that were too close together, making his number N vulnerable to the difference of squares factoring method. Make Bob feel small and insignificant by factoring his number N. 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 JSFH

Ok, Bob feels really bad about getting his code broken, and so he switched his N to one that will not fall easily to the difference of squares factoring method. Unfortunately, Eve attacked his number with the Quadratic Sieve. Through lots of sweat and hard work (plus an extreme amount of luck), Eve found the numbers a1, a2, and a3 to be interesting. Finish up the Quadratic Sieve and factor N. Poor Bob. 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

Ok, now Bob is really starting to freak out. Somehow Eve seems to know just what to do to factor his numbers. So he and Alice decide to switch to a Diffie-Hellman key exchange instead. Alice and Bob picked a great value for p, but it turns out that they picked a terrible value for their primitive root g. Why is it bad? Try raising it to 5005 mod p. Then Alice and Bob picked their secret values a and b and computed their public values A and B. Alas, Alice made another bad descision, which you can see by working out 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 Alice encrypted her message m by the simple addition code c = m + k mod p, where k is the key they shared through Diffie-Hellman. Use Index Calculus to break their code. In the end, you should be able to get m and run it through num2mess and get a coherent message. 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

a) Use the Gaussian integer version of the simplified Number Field Sieve to factor the following number. QyQ+SSJORzYiLCYqJi1JIl5HJSpwcm90ZWN0ZWRHNiQiIiMiJFkjIiIiLUYpNiQiIzoiIztGLkYuRi5GLkYu b) Give a general form of a number that can be factored using the Gaussian integer version of the simplified Number Field Sieve.

You are using ElGamal with Microsoft's ellpitic curve to receive messages. You're private key nA is given below. I've used your public key to encrypt a message, and the result is C1 and C2 below. Decrypt my message. As usual, if you did things correctly, you will get a message back. 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

We are still with Microsoft's elliptic curve, and this time we are doing a key exchange. I'm Alce, and my QA is listed below. You are Bob, and your nB is listed below. Determine the value of our key. QyY+SSNRQUc2IjckIlFpL11AJUhOXGZYOWsjZVc4WWonZSpmNTxZdSJROlBiZHBNLmBRKGUpZUBEZi4mZSd6X0QocE4lIiIiPkkjbkJHRiUiLCYqUiQqPiIqKkYp JSFH