Overview

Refresh a company’s computer network memory with respect to number representation conversions, decimal to binary and hexadecimal (and vice versa), using your ability to apply number representation and theory. Then, use discrete probability to assess the risk of a hacker foiling the company network’s RSA encryption.

Context

The Assessment 2 Context document contains additional information about set and probability theory, permutations and combinations, and cryptography.

• Assessment 2 Context.
  
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  Resources

  Suggested Resources

  The following optional resources are provided to support you in completing the assessment or to provide a helpful context. For additional resources, refer to the Research Resources and Supplemental Resources in the left navigation menu of your courseroom.

  Capella Resources

Capella Multimedia

Click the links provided below to view the following multimedia pieces:

• The Counting Principle | Transcript.
  • This presentation introduces the following topics:
    • Multiplication and addition principles.
    • Permutations and combinations.
    • Catalan numbers.
    • Discrete probability.
    • Conditional probability.
    • Pigeonhole principle.

Library Resources

The following e-book from the Capella University Library is linked directly in this course:

• Koshy, T. (2004). Discrete mathematics with applications. Burlington, MA: Elsevier Academic Press.
  • Chapter 6.

Bookstore Resources

The resource listed below is relevant to the topics and assessments in this course and is not required. Unless noted otherwise, this resource is available for purchase from the Capella University Bookstore. When searching the bookstore, be sure to look for the Course ID with the specific –FP (FlexPath) course designation.

• Johnsonbaugh, R. (2018). Discrete mathematics (8th ed.). New York, NY: Pearson.
  • Chapter 6, “Counting Methods and the Pigeonhole Principle,” sections 6.1, 6.2, 6.3, 6.5, and 6.6, are particularly useful for your work in this assessment. Topics in these sections include permutations, combinations, discrete probabilities, and discrete probability theory.

Assessment Instructions

Assume you help to oversee your company’s computer network. As such, it is important for you to understand and be able to apply number representation and number theory, as well as other IT concepts.

Part 1: Number Representation (application to binary encoding) and Combinatorics (application to IP network addressing)

Note: For each of the following, you must show your work for credit.

1. What is the decimal representation of 10001101₂ ?
2. What is the decimal representation of FFC6₁₆ ?
3. What is the binary representation of 17C6₁₆ ?
4. What is the hexadecimal representation of 11111000₂ ?

According to the IP internet protocol, each IP address is represented as a binary string. In IPv4 (Internet protocol version 4), a 32-bit binary string is used. For example, 00000011.00000111.00000000.11111111, which is often presented in dotted decimal: 3.7.0.255.

1. In mathematics, the study of combinations refers to the number of ways one can select items from a group disregarding order; the study of permutations refers to the number of ways one can permute, or arrange, items into a sequence. Given that each entry in a binary string must be either a 1 or a 0, what is the total number of addresses that can be encoded using a 32-bit binary string? Is this a combination or permutation problem? Justify your answer.
2. In IPv6, 128 bit, binary strings are used for addressing. How many addresses can be encoded using 128 bits? Is this a combination or permutation problem? Justify your answer.
3. In IPv4, how many addresses contain exactly eight 1s?

Part 2: Number Theory and Discrete Probability (application to encryption)

Note: For each of the following, you must show your work for credit. Some questions also require you to justify your answer.

1. Choose an example of p and q and compute their product z. Justify your selection.
2. Assume that you wish to make a risk assessment and you wish to determine how probable it may be for a hacker to determine p and q from z. You wish to use discrete probability for this computation. For the sake of example, you choose to assess z = 502,560,410,469,881. Say that a hacker will attempt to find p and thus q by randomly selecting a potential divisor and testing to see if it divides 502,560,410,469,881. (You know that p = 15,485,867 and q = 32,452,843, but the hacker does not.) For example, the hacker may choose 1021; however, upon inspection the hacker will see that 1021 does not divide z.
   

   For all questions below, please show all your work and/or justify your answers.

   1. Given this problem, what is the sample space of the problem? Hint: In this context, the sample space is the set of all possible values that the hacker may select.
   2. Given the sample space defined above, what events correspond to a successful guess by the hacker? Hint: An event is a subset of the sample space.
   3. Given the above, what is the probability that the hacker will successfully guess p and/or q?
   4. Assume the hacker selects five numbers to test.
      1. What is the probability that all five attempts will fail? Show your work.
      2. What is the probability that one of the five attempts will succeed? Show your work.
         

         Counting, Combinations, and Permutations Scoring Guide

         Criteria				DISTINGUISHED
         Convert numbers to different representations.				Converts numbers to different number representations and analyzes the utility of these number representations within the context of computing.
         Compute combinations and/or permutation.				Computes and explains combinations and/or permutation and integrates IT scenario into explanation.
         Apply number theory to encryption.				Applies number theory to encryption and evaluates prime number selections within the context of encryption.
         Compute discrete probability.				Analyzes computational design within context of encryption by constructing a sample space of minimal size, and computes discrete probability.
         Combine discrete probability.				Combines discrete probability and assesses the result within the context of encryption.