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Math 430
Midterm II Information
Spring 2015
WHEN: Wednesday, April 22, in class (no notes or books)
SPECIAL NOTE: You will be allowed to use a calculator.
COVERAGE: The midterm will focus on the material discussed in lecture from
Sections 7.2-7.4,
8.1-8.4,
9.1-9.3,
10.1;
however, you will of course need to rely on knowledge gained in the material covered by the first
midterm.
Note, you are only responsible for material that we explicitly discussed in lecture from the sections
above. Consult your class notes on this. For instance, we didn’t discuss the Lemmas on page 160 at
all.
EXTRA OFFICE HOURS:
Sunday, April 19th from 7 PM – MIDNIGHT
Monday, April 20th from 3 – 4 PM and 7:30 – 9:30 PM
Tuesday, April 21st from 10 – 11 AM, 1:30–3:00 PM, and 7:30–10:00 PM
Wednesday, April 22nd: most of the day from 10 AM onward (with breaks)
STUDYING: Here is an overview of the topics we have covered. You should be comfortable with
all of the following words below:
Chapter 7: Euler’s φ-function, Formulas and properties for φ(n), Euler’s generalization of Fermat’s
Little Theorem, Gauss’ Theorem
Chapter 8: Order of an integer, order of ah , primitive root (and which numbers have them, and
how many), Lagrange’s Theorem on polynomial congruences, index of a relative to r
Chapter 9: Quadratic congruence, quadratic residue, quadratic non-residue, Euler’s Criterion,
Legendre symbol, Gauss’ Law, Quadratic Reciprocity Law
Chapter 10: Cryptography: Caesar cipher, Polyalphabetic cipher (e.g. Vigenere cipher), Hill
cipher, RSA
THEOREMS FOR THE TEST: I would like you to carefully learn and understand the proofs
of some of the major results in number theory in this course. Below are listed 5 major theorems in
this part of the course that we have discussed. On the second midterm, four of these theorems will
appear, and you will be asked to supply a proof of any choice of two of them. Any correct proof is
acceptable (i.e. proof given in class, proof in book, or any other correct proof you would like to use).
• Theorem 7.2
• Theorem 7.6
• Theorem 8.3
• Theorem 8.5
• Theorem 8.6
IMPORTANT THEOREMS: In addition to the above theorems, here are some additional results
in this material that are important. You should know them well, and be able to apply them in solving
problems:
• Formula for φ(pk ) (Theorem 7.1)
• General formula for φ(n) (Theorem 7.3)
• φ(n) is even (Theorem 7.4)
• Euler’s Theorem (Theorem 7.5)
• Gauss’ Theorem (Theorem 7.6)
• Which powers of a are congruent to 1? (Theorem 8.1)
• Which powers of a are equal? (Theorem 8.2)
• The order of a power of a (Theorem 8.3)
• Powers of a primitive root (Theorem 8.4)
• Number of primitive roots (Corollary to Theorem 8.4)
• Lagrange’s Theorem (Theorem 8.5)
• Number of elements of order d (Theorem 8.6)
• Integers with no primitive roots (Theorems 8.7 and 8.8 and Corollary)
• Integers with primitive roots (Theorem 8.10)
• Properties of index (Theorem 8.11)
• Euler’s criterion for quadratic residues (Theorem 9.1 and Corollary)
• Properties of Legendre Symbol (Theorem 9.2)
• Sum of quadratic residues and non-residues (Theorem 9.4)
• Gauss’ test for quadratic residues (Theorem 9.5)
THINGS TO BE ABLE TO DO:
• Calculate φ(n) for any integer n.
• Apply Euler’s generalization of Fermat’s Little Theorem to solve congruences involving powers
of an unknown.
• Calculate the order of an element, and determine all primitive roots of a given number n.
• Determine how many primitive roots, if any, a given number n has, and to find them (e.g. by
taking powers of a given primitive root and applying Theorem 8.3)
• Find elements of a given order, modulo n, and find out how many elements have that order.
• Understand a table of indices, and use indices to solve congruences. This means you have to
know how to apply the properties of indices.
• Determine the Legendre symbol for an integer a modulo an odd prime p.
• Use properties of the Legendre symbol, including the Quadratic Reciprocity Law, to determine
quadratic residues and non-residues.
• Solve quadratic congruences.
• Encrypt plaintext by using a Caesar, Vigenere, Hill, or RSA cipher.
• Decrypt ciphertext generated with Caesar, Vigenere, Hill, or RSA ciphers.
ADVICE: I suggest reviewing the group work very carefully (especially Group Work #6 and after).
Also, study the Practice Quizzes since the last midterm carefully. If you have time, it might be best
to try re-working those exercises from scratch and looking up the answers afterwards. You will not
have time to re-do all of the homework, but you might try some of them again, especially the ones
that will assist you with the “THINGS TO BE ABLE TO DO” listed above. Try to re-do problems
from scratch, rather than just looking up solutions that I’ve created. Focus more on the mainstream
stuff and the frequently applied techniques and methods for solving, NOT weird special cases or
tricks that don’t come up much . I will not write the test with the intention of making it tricky!
If you prefer, you can just try some extra problems from scratch. I’ve included a number of random
practice problems below that you can try to work. I will post solutions to them.
It may also help you to discuss solutions to problems with other people. Explaining yourself to
someone else can be a tremendous benefit to both of you, especially with this particular material.
Often, there is more than one way to look at things, so it is useful to hear each other’s perspective
on the problems at hand.
Finally, you can ask me questions as much as you want, and I will be happy to review or pop-quiz
a topic with you if you feel shaky. Basically, I’m here to help and I want everybody to do well, so
please don’t be shy :=) !!
SAMPLE PROBLEMS:
Problem 1. Compute φ(8800), φ(1512), and φ(1000000).
Problem 2.
(a): Find all values of n such that φ(n) = n/6.
(b): Find at least 5 different numbers n such that φ(n) = 160. How many more can you find?
(c): What can you say about n if the value of φ(n) is a prime number?
Problem 3. (USE A CALCULATOR AND TABLE 1 OF YOUR BOOK IF YOU LIKE)
Find all solutions x to the congruence
(a): 35x ≡ 113x+1 (mod 13)
(b): 5x23 ≡ 18 (mod 37)
Problem 4. (USE A CALCULATOR) Decode the following message, which was sent using
RSA with the modulus n = 7081 and the exponent e = 1789:
5192
2604
4222
Problem 5. Does the congruence
x2 + 14x − 35 ≡ 0
( mod 337)
have a solution? What about the congruence
x2 − 3x − 1 ≡ 0
( mod 31957)
Problem 6. Compute the following Legendre symbols (see if you can derive them in several ways!):
(a): 72
29
(b): 60
79
63
(c): 113
55
(d): 179
(e): 37603
48611
Problem 7. Find all positive integers less than 89 having order 11, modulo 89. Repeat for order 4.
Problem 8. (THIS ONE CAN BE SKIPPED IF YOU WANT) Prove that if p > 5 is prime,
then the product of all primitive roots of p is congruent to 1 modulo p. [Hint: How many primitive
roots does p have?]
Problem 9. Prove that if p ≡ 3 (mod 4) and a is a quadratic residue of p, then x = a(p+1)/4 is a
solution to the congruence
x2 ≡ a ( mod p).
Problem 10. Suppose that q is a prime number that is congruent to 1 modulo 4, and suppose that
the number p = 2q + 1 is also a prime number. Show that 2 is a primitive root modulo p.
Problem 11. If a and b satisfy the relation ab ≡ 1 (mod p), how are the indices indr (a) and indr (b)
related?
Problem 12. In a lengthy ciphertext message, sent using a linear cipher C ≡ aP + b (mod 26), the
most frequently occurring letter is Q and the second most frequent is J. Write out the plaintext for
the intercepted message WCPQ JZQO MX.