Due Thu, Sept. 30 at 3:29PM (283 Soda)
1. (5 pts.) Any questions?
What’s the one thing you’d most like to see explained better in lecture or discussion sections? Are
lectures too fast, too slow, or just right? Are discussion sections too fast, too slow, or just right? Any
comments about what we could have done better in the past week would be appreciated.
2. (15 pts.) Chinese remaindering
You want to teach two kids to count up to 15. Unfortunately, they can only count up to five. Here’s
how you do it: You tell one of them to count up to three and go back to one when the number counted
exceeds that. Tell the other to do the same thing up to five. Then you can put together the answers
(say, 2 and 4) to find the unique number between 1 and 15 that has these remainders modulo 3 and 5,
respectively. (In this example, the only such number is 14.)
Here we show that this method always works.
(a) Let N=p1···pkbe the product of kdistinct primes. Now consider two integers x,ybetween 1
and N, and their remainders xi≡x(mod pi),yi≡y(mod pi). Show that, if yi≡xi(mod pi)
for all i=1,...,k, then x≡y(mod N). [Hint: What does xi≡yi(mod pi)imply for x−y?]
(b) Show that there is a one-to-one correspondence between integers between 1 and Nand their
k-tuple of remainders modulo pi,i=1,...,k. In other words, show that there is a bijective map
between these two sets. [Hint: How many k-tuples are there? Then use the previous part.]
(c) (Bonus problem) Find an efficient way of going back from the k-tuple of remainders xi≡x
(mod pi)to the unique integer between 1 and Nthat has these remainders.
3. (20 pts.) RSA
Let p=17, q=11 be a RSA private key and n=pq,e=3 be the public key. Show your work. Feel
free to use a computer to assist with the computations, but do show intermediate results.
(a) What is the result of encrypting the message m=86 with these keys?
(b) What is d?
(c) How would the encrypted message be decrypted?
4. (20 pts.) Polynomials in one indeterminate
We will now prove a fundamental result about polynomials: every nonzero polynomial of degree n
(over a field F) has at most nroots. If you don’t know what a field is, you can assume in the following
that F=R(the real numbers).
(a) Show that for any
α
∈F,P(x) = (x−
α
)Q(x) + b, where Q(x)is a polynomial of degree n−1
and b∈F.
(b) Show that if
α
is a root of P(x), then P(x) = (x−
α
)Q(x).
CS 70, Fall 2004, HW 4 1
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