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Summer 2004CS 4953 The Hidden Art of Steganography Required Math - Summations The summation symbol should be one of the easiest math symbols for you computer science majors to understand – if you don’t now, you will!
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Summer 2004CS 4953 The Hidden Art of Steganography Required Math - Summations The summation symbol should be one of the easiest math symbols for you computer science majors to understand – if you don’t now, you will! This symbol simply means to add up a series of numbers The equivalent in C is: –int j, sum = 0; –for(j = 0; j < n; j++) –{ sum = sum + X j ; } How many times will this loop?
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Summer 2004CS 4953 The Hidden Art of Steganography Required Math - Summations n = total number of elements P = probability of occurrence of a particular element X = element j = index If p j is the same for all elements (this is called equiprobable), then what is a simple, everyday name for ‘a’? Sometimes, ‘n’ is not shown so it just means to SUM over the entire set, whatever the size
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Summer 2004CS 4953 The Hidden Art of Steganography Required Math - Logarithms Definition: log b N = x is such that b X = N Read as “the logarithm base b of N equals x” You have two logarithm buttons on your calculator –“log” and “ln” –log is base 10 and ln is base e (e ~ 2.71) –ln is called the natural log and has numerous scientific applications You also have three inverse log buttons –10 X, e X, y x Unfortunately, we will deal with log base 2, and you do not (likely) have this buttons available However, there is a formula to help you out
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Summer 2004CS 4953 The Hidden Art of Steganography Required Math - Logarithms log b N = log a N/log a b The value of ‘a’ is arbitrary, so you can use either log base 10 or log base ‘e’ Some other properties of logarithms (base is irrelevant) –log (N * M) = log N + log M note: 10 x * 10 y = 10 (x + y) –log (N / M) = log N – log M –log N M = M log N –log b b = 1 (base matches number) –log 1 = 0 –log 0 = undefined For log base 2, we will use the notation “lg” –log 2 N = lg N Unless otherwise indicated, log N will imply a base of 10
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Summer 2004CS 4953 The Hidden Art of Steganography Required Math - Logarithms Examples – should be able to answer by inspection –3 + 5 = ? You see how quickly you can answer the above question? That’s how fast you need to be able to answer these: –log 1? –log 10? –log 100? –log 1000? –log 10 356 ? –lg 1? –lg 2? –lg 256? –lg 1024? –lg 2 1024 ? Don’t wait, learn the properties now!
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Summer 2004CS 4953 The Hidden Art of Steganography Required Math - Logarithms A few other properties: –Log N > 0 if N > 1 –Log N = 0 if N = 1 –Log N 0 && N < 1 What about the inverse lg? Example, what is the inverse lg of 8? inv log 6? inv log 0? inv lg 12?
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Summer 2004CS 4953 The Hidden Art of Steganography Background - Probability Probability may be defined in several ways It may be the likelihood of outcomes of a certain set of events, for example, the flipping of a coin –How many possible outcomes are there? Another way of looking at probability is that it is a measure of human surprise at the outcome More surprise, more information, lower probability Probability is always between zero and one –Probability = zero means it will NOT happen (is this EVER true?) –Probability = one means it will happen (is this ever true?)
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