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Math Help - Decoding a message

  1. #1
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    Question Decoding a message

    I could use all of the help I can get with this...it did not make any sense to me when my teacher explained it. Ok so here's the problem

    Let p=13 and q=23. Let e=5. Decode the message: When I think about my professor, I think about 69,214,69.

    1. First find d with de congruent to 1 mod((p-1)(q-1)).

    2. Now, find 69^d mod((13)(23)) and 214^d mod((13)(23)). I'l give you 214^d is congruent to 15 mod((13)(23)).

    3. Write decoded message in terms of letters.

    Thank you SO MUCH for any help with this.

    ***UPDATE: Ok, so I think that d=53. Verify this if you'd like. So now, what I need to do is figure out 69^(53)=____ mod((13)(23)). I am not sure how to do this...the large exponent is throwing me off, so it'd be nice if someone could explain that.
    Last edited by steph3824; November 9th 2009 at 11:06 AM. Reason: Update
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  2. #2
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    Quote Originally Posted by steph3824 View Post
    I could use all of the help I can get with this...it did not make any sense to me when my teacher explained it. Ok so here's the problem

    Let p=13 and q=23. Let e=5. Decode the message: When I think about my professor, I think about 69,214,69.

    1. First find d with de congruent to 1 mod((p-1)(q-1)).

    2. Now, find 69^d mod((13)(23)) and 214^d mod((13)(23)). I'l give you 214^d is congruent to 15 mod((13)(23)).

    3. Write decoded message in terms of letters.

    Thank you SO MUCH for any help with this.

    ***UPDATE: Ok, so I think that d=53. Verify this if you'd like. So now, what I need to do is figure out 69^(53)=____ mod((13)(23)). I am not sure how to do this...the large exponent is throwing me off, so it'd be nice if someone could explain that.
    What you need is modular exponentiation

    Essentially, you use 69^{a+b}=69^{a}69^b and substitute each term with its residue modulo 13*23, so you have smaller numbers. Optimally, using decomposition in base 2, you reduce to power 2 (and powers of 2, by induction). Read the wikipedia (or google for modular exponentiation if it is not clear enough).
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