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I'm taking a class on Cryptography and the professor mentioned that given mod N and base g with a certain order, you should pick powers m and n such that they create a certain property with the order of g. I wasn't completely sure of what he said and cant seem to find much info on it. I believe it was something along the lines of if the exponents are larger than the order of g, it will make it more difficult for eve to recover the common key. Can someone confirm that for me and explain why?

An example I had from my notes was - N=101 g=6, which has an order of 10 so pick m, n larger than 10, ie m=14 n=84 Thus solving 6^x congruent to 14 gets 3 and solving 6&y congruent to 84 ets 4 So they obtain a common key of 6^3*4

Which I think gives them an incorrect key?

J Queen
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    "I'm taking a class on Cryptography and the professor mentioned [...]. I wasn't completely sure of what he said and cant seem to find much info on it." So why don't you ask him? He's supposed to teach you. – Luc Apr 10 '13 at 17:55
  • Because the next class is on Monday.... sorry for wanting to learn faster. – J Queen Apr 10 '13 at 20:30
  • That actually sounds like good motivation ;). +1 your post. – Luc Apr 10 '13 at 20:32
  • Is "The order of G should be prime or have a large prime factor" the property you want? http://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange#Security – armb Apr 18 '13 at 08:25
  • In practice what you actually do is pick a well known group from RFC 2412 or RFC 3526. (Or 2409, section 6.) See "APPENDIX E The Well-Known Groups" in RFC 2412 for some of the properties. – armb Apr 18 '13 at 08:35

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Exponent is just an integer; but exponents make sense only modulo the group order. If g has order q (q is necessarily a divisor of N-1 where N is the prime modulus), and x and y are two integers such that x = y mod q (i.e. q divides y-x), then gx = gy mod q. Thus, for exponent x, only x mod q matters. So you can just choose each exponent in the 1..q-1 range (it won't change anything to get an exponent outside of this range).

Tom Leek
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