From: Jim Gillogly <jim@rand.org>
Date: Wed, 20 Apr 94 13:16:14 PDT
To: fnerd@smds.com (FutureNerd Steve Witham)
Subject: Re: What the heck is this? Optical noise encryption? [and RNG probs]
In-Reply-To: <9404201756.AA07286@smds.com>
Message-ID: <9404202015.AA22973@mycroft.rand.org>
MIME-Version: 1.0
Content-Type: text/plain



> fnerd@smds.com (FutureNerd Steve Witham) writes:
> I think that's a very clear handle on chaos for people who know about
> PRNGs, or vice-versa.  Chaos = PRNG.  I wonder if there's any good
> cross-fertilization of the two fields beyond the obvious.

Here's an exchange from sci.crypt in 1991 that's relevant to chaos and
cryptography.  A guy was using the logistic function as his RNG.  If you
look at the output from my decryptions, that might explain why the Keno
guy was getting only 19 out of 20 right instead of 20 out of 20.  If the
Keno RNG is based on some fractal-type function, then pieces are
self-similar, which means you don't need to find the right piece of it...
just a piece that's similar to the right seed, which need not be close
at all.

Total speculation, of course, without knowing what the RNG really was.

	Jim Gillogly
	Sterday, 29 Astron S.R. 1994, 20:12

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From: 2fmnsilly@kuhub.cc.ukans.edu (Otter)
Newsgroups: sci.crypt
Subject: Cryption system based on chaos mathmatics
Message-Id: <1991Mar18.234703.29145@kuhub.cc.ukans.edu>
Date: 19 Mar 91 05:47:03 GMT
Organization: University of Kansas Academic Computing Services
Lines: 50

Here is a simple (can you tell I just learned 'C') en/decryption algorithm
based on the mathematics of chaos.  It uses the magic number of the 'strange
attractor' to produce the 'random' cipher characters for the data to be
XOR'd with.  It is written for Turbo C.  Feedback is encouraged.


/* crypt.c                             */
/* CHAOS encryption/decryption routine */
/*-------------------------------------*/
/* Written by Chris Raile 1989         */
/* 2fmnsilly@kuhub.cc.ukans.edu        */
/* 2fmnsilly@ukanvax.bitnet            */
/*-------------------------------------*/
/* Implementation:                     */
/*                                     */
/* 'in'    File to be en/decrypted     */
/* 'out'   Resulting en/decrypted file */
 
#include "stdio.h"
 
main()
{
        FILE *fptrin;
        FILE *fptrout;
        int i, ch;
        double r = 3.56994571869;
        double j, x=.31379412;           /* <-- change numbers after 1st '3' */
        fptrin = fopen("in","rb");       /* to alter encryption scheme (key) */        fptrout = fopen("out","wb");
        while ( (ch=getc(fptrin)) != EOF)
                {
                x=(r*x)*(1-x);
                j=x*100;
                i=(int)j;
                ch=i^ch;
                putc(ch,fptrout);
                }
        fclose(fptrin);
        fclose(fptrout);
}


-- 
+------------------------------------+------------------------------+
| Reverend Chris "Otter" Raile from  | 2fmnsilly@kuhub.cc.ukans.edu |
| 'The Slackmeisters Of The Holy     | 2fmnsilly@UKANVAX.BITNET     |
| Evaporated Milk' -- A division     |                              |
| of the Church of the SubGenius(TM) | "A CornNut could not drive   |
|                                    | me to insanity--it's just    |
|  *The best damned Amway salesman*  | toasted corn."  - Me         |
+------------------------------------+------------------------------+

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From: jim@rand.org (Jim Gillogly)
Newsgroups: sci.crypt
Subject: Re: Cryption system based on chaos mathmatics
Summary: No good
Keywords: chaos, index of coincidence
Message-Id: <1991Mar19.172839.881@rand.org>
Date: 19 Mar 91 17:28:39 GMT
References: <1991Mar18.234703.29145@kuhub.cc.ukans.edu>
Sender: news@rand.org
Organization: Banzai Institute
Lines: 99

Chris Raile suggests an encryption routine based on the logistic function.
I won't state categorically that chaos isn't useful in cryptography, but
this particular routine isn't cryptographically effective.  If you use it
to encrypt an input file of all a's, for example, you'll see some striking
repetitions.

As it happens, this implementation isn't particularly sensitive to initial
conditions.  The program below tests about 1000 key values in the given
range (.3 to .4) and looks at the result.

Here's a sample crypto file (hex dump from "od"):

0000000  0751 2746 3102 245d 3b49 2010 2c51 7043
0000020  3044 2711 2b43 3e5c 384b 7441 2c4f 3353
0000040  2a04 3557 3747 2259 234a 2143 7951 3858
0000060  2d49 3711 374d 2418 3543 744c 314d 2253
0000100  3604 3346 7950 3156 334a 3e01 5300

The analysis program is mildly instructive -- if you haven't used the
Index of Coincidence to test for a successful decryption, you should.
The I.C. for English is around 0.066, so the program prints out all the
results it finds above .06 for our amusement.  Here's the result:

Key 0.3136: (IC 0.061)
  Knuuh tells us that random number generatoul shoumd#not be chosen at randoj.
Key 0.3137: (IC 0.066)
  Knuth tells us that random number generatoul shoumd not be chosen at random1
Key 0.3138: (IC 0.066)
  Knuth tells us that random number generators should not be chosen at randoj.
Key 0.3139: (IC 0.062)
  Knuth tellt?us th`t random number generators should not be chosen at randoj.
Key 0.3169: (IC 0.061)
  Jotwh tellt?us th`t random number generators should not be chosen at randoj.
Key 0.3170: (IC 0.061)
  Jotwh tellt?us th`t random number generators should not be chosen at randoj.
Key 0.3171: (IC 0.065)
  Jotwh tells us that random number generatorl should not be chosen at random1
Key 0.3172: (IC 0.061)
  Jotwh tells us that random number generatoul shoumd#not be chosen at random1

None of these is perfect, but it certainly tells us where to try refining
our key.  It's interesting that the decryptions get back on track after
initial derailments in some case... no butterfly effect here.

Jim Gillogly
Banzai Institute

------- program follows -----
/* CHAOS encryption analysis hack, Jim Gillogly, 19 Mar 91 */
/* Tries a spread of initial keys and sees what comes close. */
/* Addresses the following program: */
	/* CHAOS encryption/decryption routine */
	/* Written by Chris Raile 1989         */

#include <stdio.h>
#include <ctype.h>
 
double english_like();

main()
{
	FILE *in;
	char ct[200], pt[200], *s, *t;  /* Room for a line of ciphertext */
        double r = 3.56994571869;
	double x, x0, e;
	int len, i;

	in = fopen("out","rb");     /* Read the ciphertext */
	for (s = ct; (*s = getc(in)) != EOF; s++);
	len = s - ct;

	for (x0 = .3; x0 < .4; x0 += .0001) /* Try about a thousand keys */
	{       /* Decrypt using key x0 */
		for (x = x0, i = 0, s = ct, t = pt; i < len; s++, t++, i++)
		{       x *= r * (1 - x);
			if (! isprint(*t = ( (int) (100 * x)) ^ *s)) break;
		}
		*t = 0;
		if ((e = english_like(pt)) > .060)
			printf("Key %5.4f: (IC %5.3f)\n  %s\n", x0, e, pt);
	}
}

double english_like(s)  /* Do index of coincidence on a string */
char *s;                /* 26-letter English comes out around .066 */
{       int n, i;
	char freqs[256];
	double sum;

	if ((n = strlen(s)) <= 1) return 0.;
	bzero(freqs, 256);              /* Clear the counters */
	while (*s) freqs[*s++]++;       /* Frequency count */
	for (i = sum = 0; i < 256; i++)
		sum += freqs[i] * (freqs[i] - 1);
	return sum / n / (n - 1);
}
-- 
 Jim Gillogly
 jim@rand.org
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