The Sapphire Stream Cipher

17 Dec 2003

      THE SAPPHIRE STREAM CIPHER

The Sapphire Stream Cipher is designed to have the following properties:

 * Be useful for generation of cryptographic check values as well as
   protecting message privacy.

 * Accept a variable length key.

 * Strong enough to justify _at least_ a 64 bit key for balanced security.

 * Small enough to be built into other applications with several keys active
   at once.

 * Key setup fast enough to support frequent key change operations but slow
   enough to discourage brute force attack on the key.

 * Fast enough to not significantly impact file read & write operations on
   most current platforms.

 * Portable among common computers and efficient in C, C++, and Pascal.

 * Byte oriented.

 * Include both ciphertext and plain text feedback (for both optimal data
   hiding and value in creation of cryptographic check values).

 * Acceptable performance as a pure pseudorandom number generator without
   providing a data stream for encryption or decryption.

 * Design in a little extra strength where there is doubt about what attacks
   might be a threat.

HISTORY AND RELATED CIPHERS

The Sapphire Stream Cipher is very similar to a cipher I started work on in
November 1993.  It is also similar in some respects to the alledged RC-4 that
was posted to sci.crypt recently.  Both operate on the principle of a
mutating permutation vector.  Alledged RC-4 doesn't include any feedback of
ciphertext or plain text, however.  This makes it more vulnerable to a known
plain text attack, and useless for creation of cryptographic check values. 
On the other hand, alledged RC-4 is faster.

The Sapphire Stream Cipher is used in the shareware product Quicrypt, which
is available at ftp://ftp.csn.net/mpj/qcrypt10.zip and on the Colorado
Catacombs BBS (303-772-1062).  There are two versions of Quicrypt:  the
exportable version (with a session key limited to 32 bits but with strong
user keys allowed) and the commercial North American version (with a session
key of 128 bits).  A variant of the Sapphire Stream Cipher is also used in
the shareware program Atbash, which has no weakened exportable version.

I don't recall ever reading anything about using a stream cipher like this
for the generation of cryptographic check values, but it seems like it should
be a fast technique compared to some existing hash functions.

OVERVIEW

The Sapphire Stream Cipher is based on a state machine.  The state consists
of 5 index values and a permutation vector.  The permutation vector is simply
an array containing a permutation of the numbers from 0 through 255.  Five of
the bytes in the permutation vector are moved to new locations (which may be
the same as the old location) for every byte output.  The output byte is a
nonlinear function of all 5 of the index values and 7 of the bytes in the
permutation vector, thus frustrating attempts to solve for the state
variables based on past output.  On initialization, the index variables are
set (somewhat arbitrarily) to 1, 3, 5, 7, and 11.  The permutation vector
(called the cards array in the source code below) is shuffled based on the
user key.  This shuffling is done in a way that is designed to minimize the
bias in the destinations of the bytes in the array.  The biggest advantage in
this method is not in the elimination of the bias, per se, but in slowing
down the process slightly to make brute force attack more expensive. 
Eliminating the bias (relative to that exhibited by RC-4) is nice, but this
advantage is probably of minimal cryptographic value.

KEY SETUP

Key setup (illustrated by the function initialize(), below) consists of three
parts:

    1.  Initialize the index variables.
    2.  Set the permutation vector to a known state (a simple counting
        sequence).
    3.  Starting at the end of the vector, swap each element of the
        permutation vector with an element indexed somewhere from 0
	to the current index (chosen by the function keyrand()).

The keyrand() function returns a value between 0 and some maximum number
based on the user's key, the current state of the permutation vector, and an
index running sum called rsum.  Note that the length of the key is used in
keyrand(), too, so that a key like "abcd" will not result in the same
permutation as a key like "abcdabcd".

ENCRYPTION

Each encryption involves updating the index values, moving (up to) 5 bytes
around in the permutation vector, selecting an output byte, and adding the
output byte bitwise modulo-2 (exclusive-or) to the plain text byte to produce
the cipher text byte.  The index values are incremented by different rules. 
The index called rotor just increases by one (modulo 256) each time.  Ratchet
increases by the value in the permutation vector pointed to by rotor. 
Avalanche increases by the value in the permutation vector pointed to by 
another byte in the permutation vector pointed to by the last cipher text
byte.  The last plain text and the last cipher text bytes are also kept as
index variables.  See the function called encrypt(), below for details.

PSUEDORANDOM BYTE GENERATION

If you want to generate random numbers without encrypting any particular
ciphertext, simply encrypt 0.  There is still plenty of complexity left in
the system to ensure unpredictability (if the key is not known) of the output
stream when this simplification is made.

DECRYPTION

Decryption is the same as encryption, except for the obvious swapping of the
assignments to last_plain and last_cipher and the return value.  See the
function decrypt(), below.

C++ SOURCE CODE FRAGMENT

The original implimentation of this cipher was in Object Oriented Pascal, but
C++ is available for more platforms.

/* sapphire.h -- Interface for the Saphire stream cipher.

   Dedicated to the Public Domain the author and inventor
   (Michael Paul Johnson).  This code comes with no warranty.
   Use it at your own risk.
   Ported from the Pascal implementation of the Sapphire Stream
   Cipher 9 December 1994.

   unsigned char is assumed to be 8 bits.  If it is not, the
   results of assignments need to be reduced to 8 bits with
   & 0xFF or % 0x100, whichever is faster.
*/

class sapphire
    {
    // These variables comprise the state of the state machine.

    unsigned char cards[256];       // A permutation of 0-255.
    unsigned char rotor,            // Index that rotates smoothly
        ratchet,                    // Index that moves erratically
        avalanche,                  // Index heavily data dependent
        last_plain,                 // Last plain text byte
        last_cipher;                // Last cipher text byte

    // This function is used by initialize(), which is called by the
    // constructor.

    unsigned char keyrand(int limit, unsigned char *user_key,
                          unsigned char keysize,
                          unsigned char *rsum,
                          unsigned *keypos);

    public:

    sapphire(unsigned char *key = NULL, // Calls initialize if a real
        unsigned char keysize=0);       // key is provided.  If none
                                // is provided, call initialize
                                // before encrypt or decrypt.
    ~sapphire();                // Destroy cipher state information.
    void initialize(unsigned char *key, // User key is used to set
        unsigned char keysize);         // up state information.
    unsigned char encrypt(unsigned char b = 0);   // Encrypt byte
                                        // or get a random byte.
    unsigned char decrypt(unsigned char b);       // Decrypt byte.
    void burn(void);            // Destroy cipher state information.
    };

/* sapphire.cpp -- the Saphire stream cipher class.
   Dedicated to the Public Domain the author and inventor:
   (Michael Paul Johnson).  This code comes with no warranty.
   Use it at your own risk.
   Ported from the Pascal implementation of the Sapphire Stream
   Cipher 9 December 1994.
*/

#include <mem.h>
#include "sapphire.h"

unsigned char sapphire::keyrand(int limit,
                                unsigned char *user_key,
                                unsigned char keysize,
                                unsigned char *rsum,
                                unsigned *keypos)
    {
    unsigned u,             // Value from 0 to limit to return.
        retry_limiter,      // No infinite loops allowed.
        mask;               // Select just enough bits.

    retry_limiter = 0;
    mask = 1;               // Fill mask with enough bits to cover
    while (mask < limit)    // the desired range.
        mask = (mask << 1) + 1;
    do
        {
        *rsum = cards[*rsum] + user_key[(*keypos)++];
        if (*keypos >= keysize)
            {
            *keypos = 0;            // Recycle the user key.
            *rsum += keysize;   // key "aaaa" != key "aaaaaaaa"
            }
        u = mask & *rsum;
        if (++retry_limiter > 11)
            u %= limit;     // Prevent very rare long loops.
        }
    while (u > limit);
    return u;
    }

void sapphire::initialize(unsigned char *key, unsigned char keysize)
    {
    // Key size may be up to 256 bytes.
    // Pass phrases may be used directly, with longer length
    // compensating for the low entropy expected in such keys.
    // Alternatively, shorter keys hashed from a pass phrase or
    // generated randomly may be used. For random keys, lengths
    // of from 4 to 16 bytes are recommended, depending on how
    // secure you want this to be.

    int i;
    unsigned char toswap, swaptemp, rsum;
    unsigned keypos;

    // Initialize the indices and data dependencies.
    // Indices are set to different values instead of all 0
    // to reduce what is known about the state of the cards
    // when the first byte is emitted.

    rotor = 1;
    ratchet = 3;
    avalanche = 5;
    last_plain = 7;
    last_cipher = 11;

    // Start with cards all in order, one of each.

    for (i=0;i<256;i++)
        cards[i] = i;

    // Swap the card at each position with some other card.

    toswap = 0;
    keypos = 0;         // Start with first byte of user key.
    rsum = 0;
    for (i=255;i>=0;i--)
        {
        toswap = keyrand(i, key, keysize, &rsum, &keypos);
        swaptemp = cards[i];
        cards[i] = cards[toswap];
        cards[toswap] = swaptemp;
        }
    toswap = swaptemp = rsum = 0;
    keypos = 0;
    }

sapphire::sapphire(unsigned char *key, unsigned char keysize)
    {
    if (key && keysize)
        initialize(key, keysize);
    }

void sapphire::burn(void)
    {
    // Destroy the key and state information in RAM.
    memset(cards, 0, 256);
    rotor = ratchet = avalanche = last_plain = last_cipher = 0;
    }

sapphire::~sapphire()
    {
    burn();
    }

unsigned char sapphire::encrypt(unsigned char b)
    {
    // Picture a single enigma rotor with 256 positions, rewired
    // on the fly by card-shuffling.

    // This cipher is a variant of one invented and written
    // by Michael Paul Johnson in November, 1993.

    unsigned char swaptemp;

    // Shuffle the deck a little more.

    ratchet += cards[rotor++];
    swaptemp = cards[last_cipher];
    cards[last_cipher] = cards[ratchet];
    cards[ratchet] = cards[last_plain];
    cards[last_plain] = cards[rotor];
    cards[rotor] = swaptemp;
    avalanche += cards[swaptemp];

    // Output one byte from the state in such a way as to make it
    // very hard to figure out which one you are looking at.

    last_cipher = b^cards[cards[(cards[ratchet] + cards[rotor] +
                                 cards[last_plain] +
                                 cards[last_cipher] +
                                 cards[avalanche])&0xFF]];
    last_plain = b;
    return last_cipher;
    }

unsigned char sapphire::decrypt(unsigned char b)
    {
    unsigned char swaptemp;

    // Shuffle the deck a little more.

    ratchet += cards[rotor++];
    swaptemp = cards[last_cipher];
    cards[last_cipher] = cards[ratchet];
    cards[ratchet] = cards[last_plain];
    cards[last_plain] = cards[rotor];
    cards[rotor] = swaptemp;
    avalanche += cards[swaptemp];

    // Output one byte from the state in such a way as to make it
    // very hard to figure out which one you are looking at.

    last_plain = b^cards[cards[(cards[ratchet] + cards[rotor] +
                                cards[last_plain] +
                                cards[last_cipher] +
                                cards[avalanche])&0xFF]];
    last_cipher = b;
    return last_plain;
    }

GENERATION OF CRYPTOGRAPHIC CHECK VALUES (HASH VALUES)

For a fast way to generate a cryptographic check value (also called a hash or
message integrity check value) of a message of arbitrary length, simply
generate a set of 20 bytes (160 bits) by encrypting zeroes.  The output so
generated is the cryptographic check value.  To generate a cryptographic
check value when message integrity is desired but encryption is not (for
example, as part of a digital signature process), either use a "standard" key
(like four bytes of zero) or simply bypass the "card shuffling" part of the
key setup (for even more speed).  The plain text is still fed to the encrypt
function, but the ciphertext is discarded until the check value is generated.

SECURITY ANALYSIS

There are several security issues to be considered.  Some are easier to
analyze than others.  The following includes more "hand waving" than
mathematical proofs, and looks more like it was written by an engineer than a
mathematician.  The reader is invited to improve upon or refute the
following, as appropriate.

KEY LENGTH

There are really two kinds of user keys to consider: (1) random binary keys,
and (2) pass phrases.  Analysis of random binary keys is fairly straight
forward.  Pass phrases tend to have much less entropy per byte, but the
analysis made for random binary keys applies to the entropy in the pass
phrase.  The length limit of the key (255 bytes) is adequate to allow a pass
phrase with enough entropy to be considered strong.

To be real generous to a cryptanalyst, assume dedicated Sapphire Stream
Cipher cracking hardware.  The constant portion of the key scheduling can be
done in one cycle.  That leaves at least 256 cycles to do the swapping
(probably more, because of the intricacies of keyrand(), but we'll ignore
that, too, for now).  Assume a machine clock of about 256 MegaHertz (fairly
generous).  That comes to about one key tried per microsecond.  On average,
you only have to try half of the keys.  Also assume that trying the key to
see if it works can be pipelined, so that it doesn't add time to the
estimate.  Based on these assumptions (reasonable for major governments), and
rounding to two significant digits, the following key length versus cracking
time estimates result:

    Key length, bits    Time to crack
    ----------------    -------------
                  32    35 minutes (exportable in qcrypt)
		  33    1.2 hours (not exportable in qcrypt)
		  40    6.4 days
		  56    1,100 years (kind of like DES's key)
		  64    290,000 years (good enough for most things)
		  80    19 billion years (kind of like Skipjack's key)
		 128    5.4E24 years (good enough for the clinically paranoid)

Naturally, the above estimates can vary by several orders of magnitude based
on what you assume for attacker's hardware, budget, and motivation.

In the range listed above, the probability of spare keys (two keys resulting
in the same initial permutation vector) is small enough to ignore.  The proof
is left to the reader.

INTERNAL STATE SPACE

For a stream cipher, internal state space should be at least as big as the
number of possible keys to be considered strong.  The state associated with
the permutation vector alone (256!) constitutes overkill.

PREDICTABILITY OF THE STATE

If you have a history of stream output from initialization (or equivalently,
previous known plaintext and ciphertext), then rotor, last_plain, and
last_cipher are known to an attacker.  The other two index values, flipper
and avalanche, cannot be solved for without knowing the contents of parts of
the permutation vector that change with each byte encrypted.  Solving for the
contents of the permutation vector by keeping track of the possible positions
of the index variables and possible contents of the permutation vector at
each byte position is not possible, since more variables than known values
are generated at each iteration.  Indeed, fewer index variables and swaps
could be used to achieve security, here, if it were not for the hash
requirements.

CRYPTOGRAPHIC CHECK VALUE

The relatively large portion of the state altered with each byte encrypted
(relative to alledged RC-4) contributes to a rapid avalanche of generated
check values -- probably more than is needed.  A single bit change in a
message causes a radical change in the check value generated (about half of
the bits change).  This is one good feature of a cryptographic check value.

Another good property of a cryptographic check value is that it is too hard
to compute a message that results in a certain check value.  In this case, we
assume the attacker knows the key and the contents of a message that has the
desired check value, and wants to compute a bogus message having the same
check value.  There are two obvious ways to do this attack.  One is to solve
for a sequence that will restore the state of the permutation vector and
indices back to what it was before the alteration.  The other one is the
so-called "birthday" attack that is to cryptographic hash functions what
brute force is to key search.

To generate a sequence that restores the state of the cipher to what it was
before the alteration probably requires at least 256 bytes, since the index
"rotor" marches steadily on its cycle, one by one.  The values to do this
cannot easily be computed, due to the nonlinearity of the feedback, so there
would probably have to be lots of trial and error involve.  In practical
applications, this would leave a gaping block of binary garbage in the middle
of a document, and would be quite obvious, so this is not a practical attack,
even if you could figure out how to do it (and I haven't).  If anyone has a
method to solve for such a block of data, though, I would be most interested
in finding out what it is.  Please email me at m.p.johnson@ieee.org if you
find one.

The "birthday" attack just uses the birthday paradox to find a message that
has the same check value.  With a 20 byte check value, you would have to find
at least 80 bits to change in the text such that they wouldn't be noticed (a
plausible situation), then try the combinations until one matches.  2 to the
80th power is a big number, so this isn't practical either.  If this number
isn't big enough, you are free to generate a longer check value with this
algorithm.  Someone who likes 16 byte keys might prefer 32 byte check values
for similar stringth.

OTHER HOLES

Are there any?  Take you best shot and let me know if you see any.  I offer
no challenge text with this algorithm, but you are free to use it without
royalties to me if it is any good.

LEGAL STUFF

The intention of this document is to share some research results on an
informal basis.  You may freely use the algorithm and code listed above as
far as I'm concerned, as long as you don't sue me for anything, but there may
be other restrictions that I am not aware of to your using it.  The C++ code
fragment above is just intended to illustrate the algorithm being discussed,
and is not a complete application.  I understand this document to be
Constitutionally protected publication, and not a munition, but don't blame
me if it explodes or has toxic side effects.

                  ___________________________________________________________
                 |                                                           |
 |\  /| |        | Michael Paul Johnson  Colorado Catacombs BBS 303-772-1062 |
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