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65 lines
2.3 KiB
65 lines
2.3 KiB
From: stewarts@ix.netcom.com (Bill Stewart)
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Newsgroups: sci.crypt
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Subject: Re: Diffie-Hellman key exchange
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Date: Wed, 11 Oct 1995 23:08:28 GMT
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Organization: Freelance Information Architect
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Lines: 32
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Message-ID: <45hir2$7l8@ixnews7.ix.netcom.com>
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References: <458rhn$76m$1@mhadf.production.compuserve.com>
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NNTP-Posting-Host: ix-pl4-16.ix.netcom.com
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X-NETCOM-Date: Wed Oct 11 4:09:22 PM PDT 1995
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X-Newsreader: Forte Free Agent 1.0.82
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Kent Briggs <72124.3234@CompuServe.COM> wrote:
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>I have a copy of the 1976 IEEE article describing the
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>Diffie-Hellman public key exchange algorithm: y=a^x mod q. I'm
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>looking for sources that give examples of secure a,q pairs and
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>possible some source code that I could examine.
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q should be prime, and ideally should be a "strong prime",
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which means it's of the form 2n+1 where n is also prime.
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q also needs to be long enough to prevent the attacks LaMacchia and
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Odlyzko described (some variant on a factoring attack which generates
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a large pile of simultaneous equations and then solves them);
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long enough is about the same size as factoring, so 512 bits may not
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be secure enough for most applications. (The 192 bits used by
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"secure NFS" was certainly not long enough.)
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a should be a generator for q, which means it needs to be
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relatively prime to q-1. Usually a small prime like 2, 3 or 5 will
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work.
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....
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Date: Tue, 26 Sep 1995 13:52:36 MST
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From: "Richard Schroeppel" <rcs@cs.arizona.edu>
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To: karn
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Cc: ho@cs.arizona.edu
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Subject: random large primes
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Since your prime is really random, proving it is hard.
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My personal limit on rigorously proved primes is ~350 digits.
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If you really want a proof, we should talk to Francois Morain,
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or the Australian group.
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If you want 2 to be a generator (mod P), then you need it
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to be a non-square. If (P-1)/2 is also prime, then
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non-square == primitive-root for bases << P.
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In the case at hand, this means 2 is a generator iff P = 11 (mod 24).
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If you want this, you should restrict your sieve accordingly.
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3 is a generator iff P = 5 (mod 12).
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5 is a generator iff P = 3 or 7 (mod 10).
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2 is perfectly usable as a base even if it's a non-generator, since
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it still covers half the space of possible residues. And an
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eavesdropper can always determine the low-bit of your exponent for
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a generator anyway.
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Rich rcs@cs.arizona.edu
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