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| 1 From: stewarts@ix.netcom.com (Bill Stewart) | |
| 2 Newsgroups: sci.crypt | |
| 3 Subject: Re: Diffie-Hellman key exchange | |
| 4 Date: Wed, 11 Oct 1995 23:08:28 GMT | |
| 5 Organization: Freelance Information Architect | |
| 6 Lines: 32 | |
| 7 Message-ID: <45hir2$7l8@ixnews7.ix.netcom.com> | |
| 8 References: <458rhn$76m$1@mhadf.production.compuserve.com> | |
| 9 NNTP-Posting-Host: ix-pl4-16.ix.netcom.com | |
| 10 X-NETCOM-Date: Wed Oct 11 4:09:22 PM PDT 1995 | |
| 11 X-Newsreader: Forte Free Agent 1.0.82 | |
| 12 | |
| 13 Kent Briggs <72124.3234@CompuServe.COM> wrote: | |
| 14 | |
| 15 >I have a copy of the 1976 IEEE article describing the | |
| 16 >Diffie-Hellman public key exchange algorithm: y=a^x mod q. I'm | |
| 17 >looking for sources that give examples of secure a,q pairs and | |
| 18 >possible some source code that I could examine. | |
| 19 | |
| 20 q should be prime, and ideally should be a "strong prime", | |
| 21 which means it's of the form 2n+1 where n is also prime. | |
| 22 q also needs to be long enough to prevent the attacks LaMacchia and | |
| 23 Odlyzko described (some variant on a factoring attack which generates | |
| 24 a large pile of simultaneous equations and then solves them); | |
| 25 long enough is about the same size as factoring, so 512 bits may not | |
| 26 be secure enough for most applications. (The 192 bits used by | |
| 27 "secure NFS" was certainly not long enough.) | |
| 28 | |
| 29 a should be a generator for q, which means it needs to be | |
| 30 relatively prime to q-1. Usually a small prime like 2, 3 or 5 will | |
| 31 work. | |
| 32 | |
| 33 .... | |
| 34 | |
| 35 Date: Tue, 26 Sep 1995 13:52:36 MST | |
| 36 From: "Richard Schroeppel" <rcs@cs.arizona.edu> | |
| 37 To: karn | |
| 38 Cc: ho@cs.arizona.edu | |
| 39 Subject: random large primes | |
| 40 | |
| 41 Since your prime is really random, proving it is hard. | |
| 42 My personal limit on rigorously proved primes is ~350 digits. | |
| 43 If you really want a proof, we should talk to Francois Morain, | |
| 44 or the Australian group. | |
| 45 | |
| 46 If you want 2 to be a generator (mod P), then you need it | |
| 47 to be a non-square. If (P-1)/2 is also prime, then | |
| 48 non-square == primitive-root for bases << P. | |
| 49 | |
| 50 In the case at hand, this means 2 is a generator iff P = 11 (mod 24). | |
| 51 If you want this, you should restrict your sieve accordingly. | |
| 52 | |
| 53 3 is a generator iff P = 5 (mod 12). | |
| 54 | |
| 55 5 is a generator iff P = 3 or 7 (mod 10). | |
| 56 | |
| 57 2 is perfectly usable as a base even if it's a non-generator, since | |
| 58 it still covers half the space of possible residues. And an | |
| 59 eavesdropper can always determine the low-bit of your exponent for | |
| 60 a generator anyway. | |
| 61 | |
| 62 Rich rcs@cs.arizona.edu | |
| 63 | |
| 64 | |
| 65 | |
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