CS3626Sp01_HW01

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Kennesaw State University *

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Computer Science

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Apr 3, 2024

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- 1 - CS3626 Homework 01 Spring 2024 Numeric Theory Total Points: 25 Be as brief as possible and use your own words when describing concepts. SHOW ALL WORK for Questions requiring calculations and algorithms. Q-1: Calculate the following modular arithmetic values in the table below. (Completed example in row 1) ID Expression Dividend(a) Quotient(q) Modulus(b) Remainder(r ) (b | a) true? i 39mod45 39 0 45 39 FALSE ii 782mod-23 iii 13524mod127 iv -139mod45 v 0modN vi N mod 0 N can be any positive integer > 0. (ii) calculations (iii) calculations (iv) calculations

- 2 - 4 points

- 3 - Q-2: Using the Euclidian Algorithm, find the GCD of the following pairs, SHOW ALL WORK steps/logic. ID Expression GCD Is Relatively Prime? i GCD(81,21) ii GCD(-71,121) iii GCD(-2503,-6553) (i) a b q r (ii) a b q r (iii) a b q r 3 points

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- 4 - Q-3: Show all work for each step of the calculation: (i) Given the equation y = g x mod p calculate y for (g,x,p) equal to: ID Y G X P i 15 -1 23 ii 2 118250 13 iii 40 325 -17 iv -17 20 7 (i) Calculations (ii) Calculations (iii) Calculations (iv) Calculations 9 points

- 5 -

- 6 - Q-4: Using the Extended Euclidean calculate for the following: SHOW ALL WORK calculations/steps/graphs used to complete the answer. Find the values of d, x and y: (i) d = GCD(a = 1024, b = 192) and the (x, y) where d = ax + by = gcd(a,b) n a b q r sn-2 sn-1 sn-0 tn-2 tn-1 tn-0 Are the numeric pairs Relatively Prime? (ii) d = GCD(a = -357, b = -1809) and the (x,y) where d = ax + by = gcd(a,b) n a b q r sn-2 sn-1 sn-0 tn-2 tn-1 tn-0 Are the numeric pairs Relatively Prime? (iii) d = GCD(a = 117, b = -110) and the (x,y) where d = ax + by = gcd(a,b) n a b q r sn-2 sn-1 sn-0 tn-2 tn-1 tn-0 Are the numeric pairs Relatively Prime?

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- 7 - 9 points Submission Guidelines:  No handwritten submission is accepted, always submit answers as text within this or similar document file with any support images embedded in the file.  EXCEPTION : If asked for source code implementation you can submit those individually and as separate files in ASCII format in their original file format .cpp, .java, .py, .cs etc. or even as a .txt file will be acceptable. Do not insert code into the submission document file. It ruins spacing which makes .python and some languages (perl, awk etc.) difficult to test build.  Do not submit ZIP files… ever… for anything in D2L. The system is extremely unhelpful with regards to those filetypes and grading.  You may include your freehand drawing/image and handwritten scans in the submission. However, the writing and images must be clearly legible. Though, it is best to present non-handwritten submissions, generally, as is done in the professional setting.  If asked, show all work/calculations/graphs etc. in the determination of the problem.  Please complete your entire work in a single Word Document and Save the file as: yournetid_CS3502_Assignment01.docx (e.g. ogarcia5_CS3502_Assignment01.docx.) and upload your file in D2L.  Please observe the submission due date and time. After the due date there is a 50% penalty for the next 24 hours. Any submission after 24 hours of the due date will be graded at 0%.  If you include a reference or an image taken from other sources, please cite them appropriately. APA is preferred but cite them so they can be found. NOTE: verbatim copying or even paraphrasing is plagiarism so if the source used constitutes your answer rather than simply supporting the answer, it will be considered invalid. This is especially true of source code implementation answers.  If you resubmit, please make sure to attach the file again. Your latest submission before the due date will be the one graded.

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