| 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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| X-NETCOM-Date: Wed Oct 11 4:09:22 PM PDT 1995 |
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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 |
| |
| |
| |