Capella University Number Representation Number Theory & Discrete Probability Paper

Capella University Number Representation Number Theory & Discrete Probability Paper

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.

The assessment focuses on number theory, discrete probability theory, counting rules, permutations, and combinations.

Context

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

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Assessment 2 Context.

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.
• Koshy, T. (2004). Discrete mathematics with applications. Burlington, MA: Elsevier Academic Press.
  • Chapter 6.
• 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.
• Given your responsibilities, you decide to refresh your memory with respect to number representation conversions: decimal to binary and hexadecimal (and vice versa). In the following questions, the base is denoted as a subscript. For example, 1510 is 15 in decimal (base 10), 00112 is 3 in binary (base 2), and 1A16 is 26 in hexadecimal (base 16).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.

  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. 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 ?
  5. 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.
  6. 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.
  7. In IPv4, how many addresses contain exactly eight 1s?
• Network security and encryption is also a concern of a network administrator. Many encryption schemes are based on number theory and prime numbers; for example, RSA encryption. These methods rely on the difficulty of computing and testing large prime numbers. (A prime number is a number that has no divisor except for itself and 1.)For example, in RSA encryption, one must choose two prime numbers, p and q; these numbers are private but their product, z = pq, is public. For this scheme to work, it is important that one cannot easily find p or q given z, which is why p and q are generally large numbers.
  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.