1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
|
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
|