COT 3100 Take Home Portion
Please check the attached pdf fileCOT 3100.001 Fall 2020 Exam 2 Take Home Portion Name: U-Number: Instructions • Read all instructions carefully. • You may write in pencil, blue ink, black ink, but no other colors are permitted. • To receive full credit, correct answers must be sufficiently justified, well-organized, and printed legibly and large enough to read without straining. • Graders may take off points or assign a zero score for not following instructions or if your work is particularly sloppy and/or difficult to follow. • Due to the nature of this exam, it is important to precisely follow instructions so that your results are consistent with expected results. • You may use your notes and work with classmates, but all work submitted must be your own, i.e. it must be clear that you understand your solutions and did not merely copy them. • This portion of the exam has 30 points and will be added to the Canvas portion giving a total of 110 points possible. • You may use any calculator to help you with your calculations, such as an online calculator to simplify modular arithmetic so long as you provide the result of the requested calculations. • Tables are provided and must be completed as indicated for full credit. • Your work must be submitted on the pages provided, i.e. not on your own paper. If you can’t print this exam and scan, Microsoft Edge, for instance, allows you to write on a pdf and save. COT 3100.001 Fall 2020 Exam 2 Take Home Portion 1. (10 points) On Canvas, you will find in your gradebook, scores for p, q, and e. Find them and then complete each part of this question. (DO NOT MAKE UP YOUR OWN!) (a) Write down your p, q, and your e here. (b) Let n = pq. Compute φ(n) and write it here. (c) Let E be the function from Zn to Zn to be used for encryption Write down its definition, i.e. E(x) = ?, which should feature your modulus n and your exponent e. (See the RSA Definition in our class slides.) (d) Compute gcd(φ(n), e) using the Extended Euclidean Algorithm. Write down each step below, ensuring that the ri column has descending values. Stop once you reach the row where ri is the GCD. Reduce the inverse of e that you find modulo φ(n). The result d will be the multiplicative inverse of e modulo φ(n) so write it below. To check your work, you should verify that de ≡ 1 (mod φ(n)), but do not include this check. Use as many rows as needed below to complete your calculation. i qi ri xi yi −1 1 0 0 0 1 d = Page 2 COT 3100.001 Fall 2020 Exam 2 Take Home Portion 2. (10 points) We are going to find an inverse d of your e modulo φ(n) in another way. Let M = φ(n) and then compute φ(M) using any correct method, such as an online calculator. (You may also compute this using a general formula for φ that we have not covered). Note that in this case, φ(M) = φ(φ(n)). Now find e φ(M)−1 mod M using the method of repeated squaring. Each value listed after the first must be the square of the previous one reduced modulo M. You may stop once you reach the last square that will be needed in the final calculation, i.e. when the exponent of e is the largest power of 2 which is at most φ(M) − 1. Place a check in the cell next to each square to be used in the final calculation modulo M. Now use these squares to compute the final result modulo M. This result should be the same as the d you found earlier, in which case you should have some confidence that you did your work correctly. (Don’t forget to answer the second part of this question below.) e 1 mod M = e 2 mod M = = = = = = = = = = = = = = = φ(M) = e φ(M)−1 mod M = Which theorem allows us to conclude that de ≡ 1 (mod M)? (Substitute e φ(M)−1 mod M for d in this expression and simplify. This should all be symbolic, not with your numbers.) Page 3 COT 3100.001 Fall 2020 Exam 2 Take Home Portion 3. (10 points) Using the d you computed, write the definition of D as a function from Zn to Zn, as you wrote the definition of E for question 1. Again your definition should include the modulus n. Now, write down the eight digits of your U-number below, but for security reasons, change digits 0 to 10 and 1 to 11. We must do this since D(0) = 0 and D(1) = 1, which is not very secure. For example, if your U# is U0123-4567, then the first column below, in order from top to bottom should read 10, 11, 2, 3, 4, 5, 6, 7. For each number x written below, compute D(x). If you did all of your work correctly, then my numbers should match your numbers. What you have just done was to digitally sign your exam, with your unique private key. You can check your work by computing E(D(x)) for each value of D(x) that you find. Warning: Make sure that you use D(x) not E(x) below! You are digitally signing your U-#, not encrypting it! U # D(x) Page 4
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