Math 110 Homework 2 Solutions
Bowei Liu
Spring 2014
Question 1. (10 points) In base 30 using 0-9 to represent the digits 0-9 and A-T to represent the digits
10-29 evaluate:
2 BE + NOT 2 BE.
Please show your work at least at the level of detail of that in
Homework #3
Math 110
Due: Thursday 29 January 2015
Suggested reading: Trappe-Washington 3.53.6, 1.11.2, 6.1
1. (a) Describe the method in Section 3.5 for efficiently computing exponentials ab (mod n), and verify
the books claim that this can be done in at
Math 110 Homework 9 Solutions
March 12, 2015
1. For this question, refer to your handout on Field Axioms.
(a) State which of the examples in Section 2 are elds, and for each of the non-elds, cite at least one
axiom that fails. No proof needed.
(b) Using t
Math 110 Homework 8 Solutions
March 5, 2015
1. (a) Dene the bound Aq (n, d) on the set of (n, M, d) codes.
(b) Show that Aq (n, 1) = q n and Aq (n, n) = q.
(c) State the GilbertVarshamov bound (18.3.2).
Solution: (a) The integer Aq (n, d) is the largest M
Math 110 Homework 4 Solutions
February 5, 2015
1. (a) Let p be a positive prime. Dene a primitive root modulo p.
(b) Identify all primitive roots modulo 11. Is your solution consistent with the claim that there are
(p) primitive roots modulo p?
(c) We sta
Math 110 Homework 2 Solutions
January 22, 2015
1. Let a, n Z, n > 0.
(a) Suppose that a is a unit modulo n. Show that the multiplicative inverse of the congruence class [a]
is unique. This justies referring to the multiplicative inverse of [a] and using t
Math 110 Homework 5 Solutions
February 12, 2015
1. (a) Dene the Legendre symbol and the Jacobi symbol.
(b) Enumerate the key properties of the Jacboi symbol (as on page 91 of Chapter 3.10). What additional
properties hold when n is prime?
(c) Suppose that
Math 110 Homework 1 Solutions
January 15, 2015
1. (a) Dene the phrase m divides n.
(b) Given integers m and n, state the denition of the greatest common divisor of m and n.
(c) Suppose that m and n are two integers such that m | n. Find gcd(m, n), and pro
Math 110 Homework 3 Solutions
January 29, 2015
1. (a) Describe the method in Section 3.5 for eciently computing exponentials ab (mod n), and verify
the books claim that this can be done in at most 2 log2 (b) multiplications.
(b) Use this method to compute
Math 110 Homework 7 Solutions
February 26, 2015
1. (a) Describe the Exponent Factorization Method (Chapter 6.4). What additional information do we
need to use this factorization method that is usually prohibitively dicult to obtain?
(b) Suppose I know tha
Math 110
Homework #9
Due: Thursday 12 March 2015
Suggested reading: handout on Field Axioms; Trappe-Washington 3.11, 16.13.
1. For this question, refer to your handout on Field Axioms.
(a) State which of the examples in Section 2 are fields, and for each
Math 110
Homework #0
Due: Thursday 8 January 2015
1. Read the course webpage: http:/web.stanford.edu/~jchw/2015Math110 and familiarize yourself
with the course policy and the available resources.
If you have questions or find any errors, contact Jenny at
Math 110 Homework 1 Solutions
Bowei Liu
Spring 2014
Question 2. (10 points)
Write a few complicated equations. In particular, show that you know how to typeset fractions, square
roots, subscripts, superscripts, properly sized parentheses, and a few specia
Math 110 Homework 3 Solutions
Bowei Liu
Spring 2014
This homework is due Thursday May 1st at the start of class. Remember to justify your work even if the
A
problem does not explicitly say so. Writing your solutions in L TEXis recommend though not require
Math 110 Homework 4 Solutions
Bowei Liu
Spring 2014
This homework is due Thursday May 15th at the start of class. Remember to justify your work even if
A
the problem does not explicitly say so. Writing your solutions in L TEXis recommend though not requir
Math 110 Homework 5 Solutions
Bowei Liu
Spring 2014
This homework is due Thursday May 29th at the start of class. Remember to justify your work even if
A
the problem does not explicitly say so. Writing your solutions in L TEXis recommend though not requir
Math 110
Homework #6
Due: Thursday 19 February 2015
Suggested reading: Trappe-Washington 6.36.4, Koblitz V.2.
For this assignment, please write out your steps to show how you are applying the alogrithms. You
are welcome to use computer software for the co
Homework #4
Math 110
Due: Thursday 5 February 2015
Suggested reading: Trappe-Washington 3.7, 7.1, 7.2 (stopping before 7.2.1), 7.4, 7.5, 3.9
1. (a) Let p be a positive prime. Define a primitive root modulo p.
(b) Identify all primitive roots modulo 11. Is
Homework #5
Math 110
Due: Thursday 12 February 2015
Suggested reading: Trappe-Washington 3.10, 6.3
1. (a) Define the Legendre symbol and the Jacobi symbol.
(b) Enumerate the key properties of the Jacboi symbol (as on page 91 of Chapter 3.10). What additio
Math 110
Homework #8
Due: Thursday 5 March 2015
Suggested reading: Trappe-Washington 18.34.
1. (a) Define the bound Aq (n, d) on the set of (n, M, d) codes.
(b) Show that Aq (n, 1) = q n and Aq (n, n) = q.
(c) State the GilbertVarshamov bound (18.3.2).
2.
Math 110
Homework #1
Due: Thursday 15 January 2015
Suggested reading: Trappe-Washington Ch 3.1 3.3. Note: this assignment does not need to be typed.
This assignment includes four optional challenge problems. These are not for credit, though you may
do a c
Math 110
Homework #7
Due: Thursday 26 February 2015
Suggested reading: Trappe-Washington 6.4, 7.2, 18.12.
For this assignment, please write out your steps to show how you are applying the alogrithms. You are
welcome to use computer software for the comput
Math 110 Homework 6 Solutions
February 19, 2015
1. (a) Describe the Solovay-Strassen primality test (Chapter 6.3) and explain why it works.
(b) Use the test on n = 804 509. Is n composite, prime, or inconclusive?
Solution: The Solovay-Strassen primality t