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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!. Required Math - Summations.

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required math summations
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!

CS 4953 The Hidden Art of Steganography

required math summations1
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 + Xj; }
  • How many times will this loop?

CS 4953 The Hidden Art of Steganography

required math summations2
Required Math - Summations
  • n = total number of elements
  • P = probability of occurrence of a particular element
  • X = element
  • j = index
  • If pj 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

CS 4953 The Hidden Art of Steganography

required math logarithms
Required Math - Logarithms
  • Definition: logbN = x is such that bX = 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
    • 10X, eX, yx
  • 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 

CS 4953 The Hidden Art of Steganography

required math logarithms1
Required Math - Logarithms
  • logbN = logaN/logab
  • 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: 10x * 10y = 10(x + y)
    • log (N / M) = log N – log M
    • log NM = M log N
    • logbb = 1 (base matches number)
    • log 1 = 0
    • log 0 = undefined
  • For log base 2, we will use the notation “lg”
    • log2N = lg N
  • Unless otherwise indicated, log N will imply a base of 10

CS 4953 The Hidden Art of Steganography

required math logarithms2
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 10356?
    • lg 1?
    • lg 2?
    • lg 256?
    • lg 1024?
    • lg 21024?
  • Don’t wait, learn the properties now!

CS 4953 The Hidden Art of Steganography

required math logarithms3
Required Math - Logarithms
  • A few other properties:
    • Log N > 0 if N > 1
    • Log N = 0 if N = 1
    • Log N < 0 if 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?

CS 4953 The Hidden Art of Steganography

background probability
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?)

CS 4953 The Hidden Art of Steganography

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