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EECS 70 Discrete Mathematics and Probability Theory Fall 2021 1 Modular Arithmetic In several settings, such as error-correcting codes and cryptography, we sometimes wish to work over a smaller range of numbers. Modular arithmetic is useful in these settings, since it limits numbers to a prede- fined range {0, 1, . . . , N […]
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CS 70 Discrete Mathematics and Probability Theory Fall 2021 1 RSA Warm-Up Consider an RSA scheme with modulus N = pq, where p and q are distinct prime numbers larger than 3. (a) What is wrong with using the exponent e = 2 in an RSA public key? (b) Recall that e must be relatively
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CS 70 Discrete Mathematics and Probability Theory Fall 2021 1 Stable Matching Consider the set of jobs J = {1, 2, 3} and the set of candidates C = {A, B, C} with the following preferences. Jobs Candidates 1 A>B>C 2 B>A>C 3 A>B>C Candidates Jobs A 2>1>3 B 1>3>2 C 1>2>3 Run the traditional
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EECS 70 Discrete Mathematics and Probability Theory Fall 2021 The next major topic of the course is probability theory. Suppose you toss a fair coin a thousand times. How likely is it that you get exactly 500 heads? And what about 1000 heads? It turns out that the chances of 500 heads are roughly 2.5%,
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CS 70 Discrete Mathematics and Probability Theory Fall 2021 1 Berlekamp- Up Let P(i), a polynomial applied to the input i, be the original encoded polynomial before sent, and let ri be the received info for the input i which may or may not be corrupted. (a) When does ri = P(i)? When does ri
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EECS 70 Discrete Mathematics and Probability Theory Fall 2021 Error Correcting Codes In this note, we will discuss the problem of transmitting messages across an unreliable communication chan- nel. The channel may cause some parts of the message (“packets”) to be lost, or dropped; or, more seriously, it may cause some packets to be corrupted.
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EECS 70 Discrete Mathematics and Probability Theory Fall 2021 Polynomials Polynomials constitute a rich class of functions which are both easy to describe and widely applicable in topics ranging from Fourier analysis, cryptography and communication, to control and computational geom- etry. You’ve seen them earlier in many contexts like Taylor approximation and other contexts in
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CS 70 Discrete Mathematics and Probability Theory (Optional) HW 7 Due: Saturday 10/16, 4:00 PM Grace period until Saturday 10/16, 5:59 PM Before you start writing your final homework submission, state briefly how you worked on it. Who else did you work with? List names and email addresses. (In case of homework party, you can
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CS 70 Discrete Mathematics and Probability Theory Fall 2021 Due: Friday 9/17, 10:00 PM Grace period until Friday 9/17 11:59 PM Before you start writing your final homework submission, state briefly how you worked on it. Who else did you work with? List names and email addresses. (In case of homework party, you can just
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CS 70 Discrete Mathematics and Probability Theory Fall 2021 1 Polynomial Practice (a) If f and g are non-zero real polynomials, how many roots do the following polynomials have at least? How many can they have at most? (Your answer may depend on the degrees of f and g.) (i) f+g (ii) f·g (iii) f/g,assumingthatf/gisapolynomial
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