Use Boolean logic to program a Web site function that requires users to confirm their identities to gain access to their online accounts. Then, solve three proof problems related to networking and security.
The Assessment 3 Context document contains information about mathematical logic, specifically set theory and proofs.
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Imagine that you are programming a Web site where users must confirm their identities to attain access to their online accounts. The users are able to confirm their identities by supplying the following information: UserID, SS#, MothersName, and Password. Specifically, a user will be able to gain access by correctly answering at least 3 of the 4 above queries. For example, a user who supplies the correct UserID, SS#, and Password, but supplies an incorrect MothersName, will gain access.Note that these 4 input variables/conditions are Boolean variables: the information is supplied either correctly (true) or incorrectly (false). We will also use a 5th Boolean variable (output variable) named Access, which is true if the user supplies the necessary information correctly and is false otherwise.
Answer each of the 3 following problems that require the synthesis of logical proofs. Be sure to answer all the parts of each problem.
Assume you have been tasked with assessing the “load” or “user traffic” that users might impose on your company’s servers. Specifically, you wish to bound the total number of threads opened up on the servers given the number of users active. Given initial analysis, the number of threads, t, can be estimated in terms of the number of users, n, within a reasonable error using the following equation:Σ^{n}_{x}= 1^{x }= (1 + 2 + 3 + … + n) = tYou believe n^{2} is a reasonable bound for (1 + 2 + 3 +…+ n). To justify this, you must first prove the following inequality holds for all integers n ≥ 1.
(1 + 2 + 3+ …+ n) = ≤ n^{2}
Security and encryption is also a concern of IT personnel. Many simple encryption schemes rely on the use of the modulus function. The modulus function is also popularly used as a hash function, which is used in the construction of hash tables. The modulus function returns the remainder value resulting from a division operation. For example, 6 mod 5 = 1 and 13 mod 7 = 6.
2 ^{n} mod 2 m = 4 ^{n} mod 2( m+ 1)
Proof 3: Logical Reasoning and Proofs: Encryption and SecurityNetwork security and encryption is also a concern of IT personnel. 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, 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. Seemingly this strategy would work best if there are many large prime numbers, so that one could not easily guess the prime divisor of z.
CRITERIA | DISTINGUISHED | |||
---|---|---|---|---|
Define Boolean variables. | Defines Boolean variables, explains their use in the IT/computing scenario, and relates Boolean variables to propositional logic. | |||
Construct truth table using Boolean algebra and logic. | Constructs truth table using Boolean algebra and logic. Designs truth table in an orderly manner and has columns for intermediate results. | |||
Compose Boolean equation using Boolean operators. | Composes Boolean equation using Boolean operators, arranges equation in a standardized form, and reduces the size of the expression. | |||
Design a logical circuit diagram using correct symbols for gates. | Designs a logical circuit diagram using correct symbols for gates, and the number of logic gates used is minimal. | |||
Select an appropriate proof type for proof problems. | Selects an appropriate proof type for proof problems. Investigates or rejects other proof types for each problem. | |||
Construct a proof using mathematical reasoning, deductive logic, or inductive logic to assess truth of mathematical or logical statements. | Constructs a proof using mathematical reasoning, deductive logic, or inductive logic to assess truth of mathematical or logical statements. Proofs are formal: orderly, organized, and flawless. |
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