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- Mathematics for Actuarial Science 9 1. For each of the following sets determine (i) the elements of the set and (ii) the subsets
- Ciphers and Number Theory 5 1. For n = 989 show how to convert the message abandon all hope into suitable blocks
- Ciphers and Number Theory 1 1. Encode the message the pirates are coming using a Caesar cipher with shift 7.
- Let us return to the RSA algorithm. We saw that to decrypt a message we need to calculate cd mod n. If we choose e to be small (so that
- Mathematics for Actuarial Science 3 1. Solve, if possible, the system of equations
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- Cantor's diagonal argument All of the infinite sets we have seen so far have been `the same size'; that is, we have
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- Ciphers and Number Theory CW2 Solutions, 2011 Here are model solutions to the coursework, together with comments (in
- Mathematics for Actuarial Science: Answer sheet 2 1. x = 3, y = -1
- 4.6 Tangents and normals to curves We have already defined the value of the derivative f of a function f at
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- Example 3.12: Solve x 2 mod 23
- Answer all six questions from this section. Each question carries 8 marks. 1. Give the definition of a necesssary and of a sufficient condition. For each of
- The RSA system relies on the difficulty of factorising a number into a product of primes. The basic idea is quite simple.
- 1.7 Sums of series We often want to sum a series of terms, for example when we look at
- In what follows it will be convenient to have an efficient way to calculate powers of an integer modulo n.
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- We have seen that monoalphabetic ciphers are susceptible to frequency analysis. In this lecture we will briefly consider some
- On some applications of infinitesimal methods to quantum groups and related algebras
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- Let us return to our examination of ciphers, and make things more mathematical.
- 2.4 Logarithm and exponential functions We first consider the function
- 4. Calculus I: Differentiation 4.1 The derivative of a function
- Example 4.7.4: Find the stationary points of the curve f(x) = 6 ln
- 5.6 Integration as a measure of content The area contained between the
- MA2609: Ciphers and Number Theory 1. Introduction to ciphers
- 2. The Euclidean algorithm and modular arithmetic We will be interested in properties of the integers, Z. Recall that a Z
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- Mathematical communication 2 This is an assessed coursework, and will count towards your final grade. Solutions
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- Mathematics for Actuarial Science 6 1. Calculate
- Ciphers and Number Theory Coursework 1 This is an assessed coursework, and will count towards your final grade. Solutions
- Given a, b N with b < a we have defined sequences ri and qi, with r-1 = a, r0 = b, and ri-2 = ri-1qi + ri for 1 i n. We have seen that
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- Before working out what restrictions we need on n to have a well-defined inverse to y mod n, we need the following consequence
- 3. Euler's function and the Chinese Remainder Theorem We saw in the last chapter that a has an inverse modulo n if and only if
- Last lecture we proved that any prime n = 2sd + 1 with d odd satisfied for all a with (n, a) = 1 either
- Ciphers and number theory: revision notes Here is a list of the main topics which were covered in the course, and an indication of the
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- Mathematical communication 1 This is an assessed coursework, and will count towards your final grade. Solutions should
- Mathematical communication 3 This is an assessed coursework, and will count towards your final grade. Solutions
- Mathematical communication 4 This is an assessed coursework, and will count towards your final grade. Solutions
- An introduction to Mathematical Communication The object of this course is to introduce you to some of the basic tools needed by any
- Predicate logic So far we have looked at propositional logic. There is a more general kind of logic called
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- Mathematics for Actuarial Science 1 1. Given that
- Mathematics for Actuarial Science 5 1. Find the gradient of the curve at the point for which x = 4 in each of the following
- 1.7 Sums of series We often want to sum a series of terms, for example when we look at
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- We saw in Example 5.4.2 that we sometimes need to apply integration by parts several times in the course of a single calculation.
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- 2.4 Logarithm and exponential functions We first consider the function
- Consider the function pn(x) = f(c) + f (c)(x -c) +
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- Ciphers and Number Theory 2 1. Encode the message one of our agents has defected using a Vigen`ere cipher with
- Ciphers and Number Theory Coursework 2 This is an assessed coursework, and will count towards your final grade. Solutions
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