android / platform / external / openssl.git / f42d491ab90c82302b0054c62014c1ee9b638aff / . / crypto / dh / generate

From: stewarts@ix.netcom.com (Bill Stewart) | |

Newsgroups: sci.crypt | |

Subject: Re: Diffie-Hellman key exchange | |

Date: Wed, 11 Oct 1995 23:08:28 GMT | |

Organization: Freelance Information Architect | |

Lines: 32 | |

Message-ID: <45hir2$7l8@ixnews7.ix.netcom.com> | |

References: <458rhn$76m$1@mhadf.production.compuserve.com> | |

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Kent Briggs <72124.3234@CompuServe.COM> wrote: | |

>I have a copy of the 1976 IEEE article describing the | |

>Diffie-Hellman public key exchange algorithm: y=a^x mod q. I'm | |

>looking for sources that give examples of secure a,q pairs and | |

>possible some source code that I could examine. | |

q should be prime, and ideally should be a "strong prime", | |

which means it's of the form 2n+1 where n is also prime. | |

q also needs to be long enough to prevent the attacks LaMacchia and | |

Odlyzko described (some variant on a factoring attack which generates | |

a large pile of simultaneous equations and then solves them); | |

long enough is about the same size as factoring, so 512 bits may not | |

be secure enough for most applications. (The 192 bits used by | |

"secure NFS" was certainly not long enough.) | |

a should be a generator for q, which means it needs to be | |

relatively prime to q-1. Usually a small prime like 2, 3 or 5 will | |

work. | |

.... | |

Date: Tue, 26 Sep 1995 13:52:36 MST | |

From: "Richard Schroeppel" <rcs@cs.arizona.edu> | |

To: karn | |

Cc: ho@cs.arizona.edu | |

Subject: random large primes | |

Since your prime is really random, proving it is hard. | |

My personal limit on rigorously proved primes is ~350 digits. | |

If you really want a proof, we should talk to Francois Morain, | |

or the Australian group. | |

If you want 2 to be a generator (mod P), then you need it | |

to be a non-square. If (P-1)/2 is also prime, then | |

non-square == primitive-root for bases << P. | |

In the case at hand, this means 2 is a generator iff P = 11 (mod 24). | |

If you want this, you should restrict your sieve accordingly. | |

3 is a generator iff P = 5 (mod 12). | |

5 is a generator iff P = 3 or 7 (mod 10). | |

2 is perfectly usable as a base even if it's a non-generator, since | |

it still covers half the space of possible residues. And an | |

eavesdropper can always determine the low-bit of your exponent for | |

a generator anyway. | |

Rich rcs@cs.arizona.edu | |