66 lines
2.3 KiB
Text
66 lines
2.3 KiB
Text
|
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>
|
||
|
|
NNTP-Posting-Host: ix-pl4-16.ix.netcom.com
|
||
|
|
X-NETCOM-Date: Wed Oct 11 4:09:22 PM PDT 1995
|
||
|
|
X-Newsreader: Forte Free Agent 1.0.82
|
||
|
|
|
||
|
|
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
|
||
|
|
|
||
|
|
|
||
|
|
|