Math 480 (Spring 2007): Homework 4
Due: Monday, April 23
There are 5 problems. Each problem is worth 6 points and parts of multipart prob-
lems are worth equal amounts. You may work with other people and use a computer,
unless otherwise stated. Acknowledge those who help you.
1. (Work by hand alone on this.) Find all four solutions xwith 0 ≤x < 55 to the
equation
x2−1≡0 (mod 55).
2. (Work by hand alone on this.) How many solutions (with 0 ≤x < 15015) are
there to the equation
x2−1≡(mod 15015).
You may use that 15105 = 3 ·5·7·11 ·13.
3. Find the first prime p > 19 such that the smallest primitive root modulo pis 19.
(This requires a computer.)
4. You and Nikita wish to agree on a secret key using the Diffie-Hellman key ex-
change. Nikita announces that p= 3793 and g= 7. Nikita secretly chooses a
number n < p and tells you that gn≡454 (mod p). You choose the random
number m= 1208. What is the secret key?
5. In this problem you will digitally sign the number 2007. The grader will verify
your digital signature.
(a) Choose primes pand qwith 5 digits each, but do not write them down on
your homework assignment. Instead, write down n=pq. (Your answer to
this problem is n. The grader will factor nusing a computer and verify that
indeed n=pq with p,qboth prime.)
(b) Let e= 3. Compute the decryption key dsuch that ed ≡1 (mod ϕ(n)).
Do not write down d. Instead encrypt the number 2007 using (d, n), i.e.,
digitally sign 2007. Your answer is the number mmodulo n. (The grader
will encrypt musing your public key (3, n); if the grader gets 2007 as the
encryption, you get full credit; otherwise no credit.)
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