Subject: BLOWFISH - Paper from Dec 94 Conference
From: [email-masked] (Bruce Schneier)
Date: Fri, 8 Apr 1994 16:50:28 GMT
Newsgroups: sci.crypt

Quite a few people have been asking me about this paper, so I have
decided to post it here.  It will appear in the conference proceedings,
which will be published by Springer-Verlag sometime soon.  The three
figures are not here; sorry.

Bruce

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Description of a New Variable-Length Key, 64-Bit Block Cipher     
                       (Blowfish)

                      Bruce Schneier
Counterpane Systems, 730 Fair Oaks Ave, Oak Park, IL  60302
                   [email-masked]

                       Abstract:

Blowfish, a new secret-key block cipher, is proposed.  It is a
Feistel network, iterating a simple encryption function 16 times. 
The block size is 64 bits, and the key can be any length up to
448 bits.  Although there is a complex initialization phase
required before any encryption can take place, the actual
encryption of data is very efficient on large microprocessors.

The cryptographic community needs to provide the world with a new
encryption standard.  DES [16], the workhorse encryption
algorithm for the past fifteen years, is nearing the end of its
useful life.  Its 56-bit key size is vulnerable to a brute-force
attack [22], and recent advances in differential cryptanalysis
[1] and linear cryptanalysis [10] indicate that DES is vulnerable
to other attacks as well.

Many of the other unbroken algorithms in the literaturebKhufu
[11,12], REDOC II [2,23, 20], and IDEA [7,8,9]bare protected by
patents.  RC2 and RC4, approved for export with a small key size,
are proprietary [18].  GOST [6], a Soviet government algorithm,
is specified without the S-boxes.  The U.S. government is moving
towards secret algorithms, such as the Skipjack algorithm in the
Clipper and Capstone chips [17].

If the world is to have a secure, unpatented, and freely-
available encryption algorithm by the turn of the century, we
need to develop several candidate encryption algorithms now. 
These algorithms can then be subjected to years of public
scrutiny and cryptanalysis.  Then, the hope is that one or more
candidate algorithms will survive this process, and can
eventually become a new standard.

This paper discusses the requirements for a standard encryption
algorithm.  While it may not be possible to satisfy all
requirements with a single algorithm, it may be possible to
satisfy them with a family of algorithms based on the same
cryptographic principles.

AREAS OF APPLICATION

A standard encryption algorithm must be suitable for many
different applications:

         Bulk encryption.  The algorithm should be efficient in
                 encrypting data files or a continuous data stream.

         Random bit generation.  The algorithm should be efficient in
                 producing single random bits.

         Packet encryption.  The algorithm should be efficient in
                 encrypting packet-sized data.  (An ATM packet has a 48-
                 byte data field.)  It should implementable in an
                 application where successive packets may be encrypted
                 or decrypted with different keys.

         Hashing.  The algorithm should be efficient in being
                 converted to a one-way hash function.

PLATFORMS

A standard encryption algorithm must be implementable on a
variety of different platforms, each with their own requirements. 
These include:

         Special hardware.  The algorithm should be efficiently
                 implementable in custom VLSI hardware.

         Large processors.  While dedicated hardware will always be
                 used for the fastest applications, software
                 implementations are more common.  The algorithm should
                 be efficient on 32-bit microprocessors with 4 kbyte
                 program and data caches.

         Medium-size processors.  The algorithm should run on
                 microcontrollers and other medium-size processors, such
                 as the 68HC11.

         Small processors.  It should be possible to implement the
         algorithm on smart cards, even inefficiently.

The requirements for small processors are the most difficult. 
RAM and ROM limitations are severe for this platform.  Also,
efficiency is more important on these small machines. 
Workstations double their capacity almost annually.  Small
embedded systems are the same year after year, and there is
little capacity to spare.  If there is a choice, the extra
computation burden should be on large processors rather than
small processors.

ADDITIONAL REQUIREMENTS

These additional requirements should, if possible, be levied on a
standard encryption algorithm.

         The algorithm should be simple to code.  Experiences with
         DES [19] show that programmers will often make
         implementation mistakes if the algorithm is complicated.  If
         possible, the algorithm should be robust against these
         mistakes.

         The algorithm should have a flat keyspace, allowing any
         random bit string of the required length to be a possible
         key.  There should be no weak keys.

         The algorithm should facilitate easy key-management for
         software implementations.  Software implementations of DES
         generally use poor key management techniques.  In
         particular, the password that the user types in becomes the
         key.  This means that although DES has a theoretical
         keyspace of 2^56, the actual keyspace is limited to keys
         constructed with the 95 characters of printable ASCII. 
         Additionally, keys corresponding to words and near words are
         much more likely.

         The algorithm should be easily modifiable for different
         levels of security, both minimum and maximum requirements.

         All operations should manipulate data in byte-sized blocks. 
         Where possible, operations should manipulate data in 32-bit
         blocks.

DESIGN DECISIONS

Based on the above parameters, we have made these design
decisions.  The algorithm should:

         Manipulate data in large blocks, preferably 32 bits in size
         (and not in single bits, such as DES).

         Have either a 64-bit or a 128-bit block size.

         Have a scalable key, from 32 bits to at least 256 bits.

         Use simple operations that are efficient on microprocessors: 
         e.g., exclusive-or, addition, table lookup, modular-
         multiplication.  It should not use variable-length shifts or
         bit-wise permutations, or conditional jumps.

         Be implementable on an 8-bit processor with a minimum of 24
         bytes of RAM (in addition to the RAM required to store the
         key) and 1 kilobyte of ROM.

         Employ precomputable subkeys.  On large-memory systems,
         these subkeys can be precomputed for faster operation.  Not
         precomputing the subkeys will result in slower operation,
         but it should still be possible to encrypt data without any
         precomputations.

         Consist of a variable number of iterations.  For
         applications with a small key size, the trade-off between
         the complexity of a brute-force attack and a differential
         attack make a large number of iterations superfluous. 
         Hence, it should be possible to reduce the number of
         iterations with no loss of security (beyond that of the
         reduced key size).

         If possible, have no weak keys.  If not possible, the
         proportion of weak keys should be small enough to make it
         unlikely to choose one at random.  Also, any weak keys
         should be explicitly known so they can be weeded out during
         the key generation process.

         Use subkeys that are a one-way hash of the key.  This would
       of long passphrases for the key without
         compromising security.

         Have no linear structures (e.g., the complementation
         propertie of DES) that reduce the complexity of exhaustive
         search [4].

         Use a design that is simple to understand.  This will
         facilitate analysis and increase the confidence in the
         algorithm.  In practice, this means that the algorithm will
         be a Feistel iterated block cipher [21].

Most of these design decisions are not new.  Almost all block
ciphers since Lucifer [5,21] are Feistel ciphers, and all have a
flat keyspace (with the possible exception of a few weak keys). 
FEAL [13,14,15] and Khufu [11] use a variable number of
iterations.  Khufu [11] has a large number of subkeys that are a
one-way function of the key.  RC2 [18] has a variable-length key. 
GOST [6] uses a 32-bit word length and a 64-bit block size.  MMB
[2] uses a 32-bit word length and a 128-bit block size.

BUILDING BLOCKS

There are a number of building blocks that have been demonstrated
to produce strong ciphers.  Many of these can be efficiently
implemented on 32-bit microprocessors.

         Large S-boxes.  Larger S-boxes are more resistant to
         differential cryptanalysis.  An algorithm with a 32-bit word
         length can use 32-bit S-boxes.  Khufu and REDOC III both use
         a 256-entry, 32-bit wide S-box [11,20].

         Key-dependent S-boxes.  While fixed S-boxes must be designed
         to be resistant to differential and linear cryptanalysis,
         key-dependent S-boxes are much more resistant to these
         attacks.  They are used in the Khufu algorithm [11]. 
         Variable S-boxes, which could possibly be key dependent, are
         used in GOST [6].

         Combining operations from different algebraic groups.  The
         IDEA cipher introduced this concept, combining XOR mod 2^16,
         addition mod 2^16, and multiplication mod 2^16+1 [7].  The
         MMB cipher uses a 32-bit word, and combines XOR mod 232 with
         multiplication mod 2^32-1 [2].

         Key-dependent permutations.  The fixed initial and final
         permutations of DES have been long regarded as
         cryptographically worthless.  Khufu XORs the text block with
         key material at the beginning and the end of the algorithm
         [11].

BLOWFISH

Blowfish is a variable-length key block cipher.  It does not meet
all the requirements for a new cryptographic standard discussed
above: it is only suitable for applications where the key does
not change often, like a communications link or an automatic file
encryptor.  It is significantly faster than DES when implemented
on 32-bit microprocessors with large data caches, such as the
Pentium and the PowerPC.

DESCRIPTION OF THE ALGORITHM

Blowfish is a variable-length key, 64-bit block cipher.  The
algorithm consists of two parts: a key-expansion part and a data-
encryption part.  Key expansion converts a key of at most 448
bits into several subkey arrays totaling 4168 bytes. 

Data encryption occurs via a 16-round Feistel network.  Each
round consists of a key-dependent permutation, and a key- and
data-dependent substitution.  All operations are XORs and
additions on 32-bit words.  The only additional operations are
four indexed array data lookups per round.

Subkeys:

Blowfish uses a large number of subkeys.  These keys must be
precomputed before any data encryption or decryption.

         1.  The P-array consists of 18 32-bit subkeys:
                 P[1], P[2],..., P[18].

         2.  There are four 32-bit S-boxes with 256 entries each:
                 S[1,0], S[1,1],..., S[1,255]; 
                 S[2,0], S[2,1],..,, S[2,255];
                 S[3,0], S[3,1],..., S[3,255];
                 S[4,0], S[4,1],..,, S[4,255].

The exact method used to calculate these subkeys will be
described later.

Encryption:

Blowfish is a Feistel network consisting of 16 rounds (see Figure
1).  The input is a 64-bit data element, x.

         Divide x into two 32-bit halves:  x[L], x[R]
         For i = 1 to 16:
                 x[L] = x[L] XOR P[i]
                 x[R] = F(x[L]) XOR x[R]
                 Swap x[L] and x[R]
         Swap x[L] and x[R]  (Undo the last swap.)
         x[R] = x[R] XOR P[17]
         x[L] = x[L] XOR P[18]
         Recombine x[L] and x[R]

         Function F (see Figure 2):
                 Divide x[L] into four eight-bit quarters:  a, b, c, and
                 d
                 F(x[L]) = ((S[1,a] + S[2,b] mod 2^32) XOR S[3,c]) 
                           + S[4,d] mod 2^32

Decryption is exactly the same as encryption, except that P[1],
P[2],..., P[18] are used in the reverse order.

Implementations of Blowfish that require the fastest speeds
should unroll the loop and ensure that all subkeys are stored in
cache.

Generating the Subkeys:

The subkeys are calculated using the Blowfish algorithm.  The
exact method is as follows:

         1.  Initialize first the P-array and then the four S-boxes,
         in order, with a fixed string.  This string consists of the
         hexadecimal digits of pi (less the initial 3).  For example:

                 P[1] = 0x243f6a88
                 P[2] = 0x85a308d3 
                 P[3] = 0x13198a2e
                 P[4] = 0x03707344 

         2.  XOR P[1] with the first 32 bits of the key, XOR P[2]
         with the second 32-bits of the key, and so on for all bits
         of the key (possibly up to P[14]).  Repeatedly cycle through
         the key bits until the entire P-array has been XORed with
         key bits.  (For every short key, there is at least one
         equivalent longer key; for example, if A is a 64-bit key,
         then AA, AAA, etc., are equivalent keys.)

         3.  Encrypt the all-zero string with the Blowfish algorithm,
         using the subkeys described in steps (1) and (2).

         4.  Replace P[1] and P[2] with the output of step (3).

         5.  Encrypt the output of step (3) using the Blowfish
         algorithm with the modified subkeys.

         6.  Replace P[3] and P[4] with the output of step (5).

         7.  Continue the process, replacing all entries of the P-
         array, and then all four S-boxes in order, with the output
         of the continuously-changing Blowfish algorithm.

In total, 521 iterations are required to generate all required
subkeys.  Applications can store the subkeys rather than execute
this derivation process multiple times.

MINI-BLOWFISH

The following mini versions of Blowfish are defined solely for
cryptanalysis.  They are not suggested for actual implementation. 
Blowfish-32 has a 32-bit block size and subkey arrays of 16-bit
entries (each S-box has 16 entries).  Blowfish-16 has a 16-bit
block size and subkey arrays of 8-bit entries (each S-box has 4
entries).

DESIGN DECISIONS

The underlying philosophy behind Blowfish is that simplicity of
design yields an algorithm that is both easier to understand and
easier to implement.  Through the use of a streamlined Feistel
networkba simple S-box substitution and a simple P-box
substitutionbI hope that the design will not contain any flaws.

A 64-bit block size yields a 32-bit word size, and maintains
block-size compatibility with existing algorithms.  Blowfish is
easy to scale up to a 128-bit block, and down to smaller block
sizes.  Cryptanalysis of the mini-Blowfish variants may be
significantly easier than cryptanalysis of the full version.

The fundamental operations were chosen with speed in mind.  XOR,
ADD, and MOV from a cache are efficient on both Intel and
Motorola architectures.  All subkeys fit in the cache of a 80486,
68040, Pentium, and PowerPC.

The Feistel network that makes up the body of Blowfish is
designed to be as simple as possible, while still retaining the
desirable cryptographic properties of the structure.  Figure 3 is
round i of a general Feistel network:  R[n,i] are reversible
functions of text and key, and N[i] is a non-reversible function
of text and key.  For speed and simplicity, I chose XOR as my
reversible function.  This let me collapse the four XORs into a
single XOR, since:

1] = R[1,i+1] XOR R[2,i-1] XOR R[3,i] XOR R[4,i]

This is the P-array substitution in Blowfish.  The XOR can also
be considered to be part of the non-reversible function, N[i],
occurring at the end of the function.  (Although equivalent, I
chose not to illustrate them in this way because it simplifies
description of the subkey-generation process.)  There are two
XORs that remain after this reduction: R[1] in the first round
and R[2] in the last round.  I chose not to eliminate these in
order to hide the input to the first non-reversible function.

I considered a more complicated reversible function, one with
modular multiplications and rotations.  However, these operations
would greatly increase the algorithm's execution time.  Since
function F is the primary source of the algorithm's security, I
decided to save time-consuming complications for that function.

Function F, the non-reversible function, gives Blowfish the best
possible avalanche effect for a Feistel network:  every text bit
on the left half of the round affects every text bit on the right
half.  Additionally, since every subkey bit is affected by every
key bit, the function also has a perfect avalanche effect between
the key and the right half of the text after every round.  Hence,
the algorithm exhibits a perfect avalanche effect after three
rounds and again every two rounds after that.

I considered adding a reversible mixing function, more
complicated than XOR, before the first and after the last round. 
This would further confuse the entry values into the Feistel
network and ensure a complete avalanche effect after the first
two rounds.  I eventually discarded the addition as a time-
consuming complication with no clear cryptographic benefits.

The non-reversible function is designed for strength, speed, and
simplicity.  Ideally, I wanted a single S-box with 2^32 32-bit
words, but that was impractical.  My eventual choice of 256-entry
S-boxes was a compromise between my three design goals.  The
small-number of bits to large-number of bits may have weaknesses
with respect to linear cryptanalysis, but these weaknesses are
hidden both by combining the output of four S-boxes and making
them dependent on the key.

I used four different S-boxes instead of one S-box primarily to
avoid symmetries when different bytes of the input are equal, or
when the 32-bit input to function F is a bytewise permutation of
another 32-bit input.  I could have used one S-box and made each
of the four different outputs a non-trivial permutation of the
single output, but the four S-box design is faster, easier to
program, and seems more secure.

The function that combines the four S-box outputs is as fast as
possible.  A simpler function would be to XOR the four values,
but mixing addition mod 2^32 and XOR combines two different
algebraic groups with no additional instructions.  The
alternation of addition and XOR ends with an addition operation
because an XOR combines the final result with x[R].

If the four indexes chose values out of the same S-box, a more
complex combining function would be required to eliminate
symmetries.  I considered using a more complex combining function
in Blowfish (using modular multiplications, rotations, etc.), but
chose not to because the added complication seemed unnecessary.

The key-dependent S-boxes protect against differential and linear
cryptanalysis.  Since the structure of the S-boxes is completely
hidden from the cryptanalyst, these attacks have a more difficult
time exploiting that structure.  While it would be possible to
replace these variable S-boxes with four fixed S-boxes that were
designed to be resistant to these attacks, key-dependent S-boxes
are easier to implement and less susceptible to arguments of
"hidden" properties.  Additionally, these S-boxes can be created
on demand, reducing the need for large data structures stored
with the algorithm.

Each bit of x[L] is only used as the input to one S-box.  In DES
many bits are used as inputs to two S-boxes, which strengthens
the algorithm considerably against differential attacks.  I feel
that this added complication is not as necessary with key-
dependent S-boxes.  Additionally, larger S-boxes would take up
considerably more memory space.

Function F does not depend on the iteration.  I considered adding
this dependency, but did not feel that it had any cryptographic
merit.  The P-array substitution can be considered to be part of
this function, and that is already iteration-dependent.

The number of rounds is set at 16 primarily out of desire to be
conservative.  However, this number affects the size of the P-
array and therefore the subkey-generation process; 16 iterations
permits key lengths up to 448 bits.  I expect to be able to
reduce this number, and greatly speed up the algorithm in the
process, as I accumulate more cryptanalysis data.

In algorithm design, there are two basic ways to ensure that the
key is long enough to ensure a particular security level.  One is
to carefully design the algorithm so that the entire entropy of
the key is preserved, so there is no better way to cryptanalyze
the algorithm other than brute force.  The other is to design the
algorithm with so many key bits that attacks that reduce the
effective key length by several bits are irrelevant.  Since
Blowfish is designed for large microprocessors with large amounts
of memory, I chose the latter.

The subkey generation process is designed to preserve the entire
entropy of the key and to distribute that entropy uniformly
throughout the subkeys.  It is also designed to distribute the
set of allowed subkeys randomly throughout the domain of possible
subkeys.  I chose the digits of pi as the initial subkey table
for two reasons: because it is a random sequence not related to
the algorithm, and because it could either be stored as part of
the algorithm or derived when needed.  There is nothing sacred
about pi; any string of random bitsbdigits of e, RAND tables,
output of a random number generatorbwill suffice.  However, if
the initial string is non-random in any way (for example, ASCII
text with the high bit of every byte a 0), this non-randomness
will propagate throughout the algorithm.

In the subkey generation process, the subkeys change slightly
with every pair of subkeys generated.  This is primarily to
protect against any attacked of the subkey generation process
that exploit the fixed and known subkeys.  It also reduces
storage requirements.  The 448 limit on the key size ensures that
the every bit of every subkey depends on every bit of the key. 
(Note that every bit of P[15], P[16], P[17], and P[18] does not
affect every bit of the ciphertext, and that any S-box entry only
has a .06 probability of affecting any single ciphertext block.)

The key bits are repeatedly XORed with the digits of pi in the
initial P-array to prevent the following potential attack: 
Assume that the key bits are not repeated, but instead padded
with zeros to extend it to the length of the P-array.  An
attacker might find two keys that differ only in the 64-bit value
XORed with P[1] and P[2] that, using the initial known subkeys,
produce the same encrypted value.  If so, he can find two keys
that produce all the same subkeys.  This is a highly tempting
attack for a malicious key generator.

To prevent this same type of attack, I fixed the initial
plaintext value in the subkey-generation process.  There is
nothing special about the all-zeros string, but it is important
that this value be fixed.

The subkey-generation algorithm does not assume that the key bits
are random.  Even highly correlated key bits, such as an
alphanumeric ASCII string with the bit of every byte set to 0,
will produce random subkeys.  However, to produce subkeys with
the same entropy, a longer alphanumeric key is required.

The time-consuming subkey-generation process adds considerable
complexity for a brute-force attack.  The subkeys are too long to
be stored on a massive tape, so they would have to be generated
by a brute-force cracking machine as required.  A total of 522
iterations of the encryption algorithm are required to test a
s9 steps to any brute-force
attack.

POSSIBLE SIMPLIFICATIONS

I am exploring several possible simplifications, aimed at
decreasing memory requirements and execution time.  These are
outlined below:

         Fewer and smaller S-boxes.  It may be possible to reduce the
         number of S-boxes from four to one.  Additionally, it may be
         possible to overlap entries in a single S-box: entry 0 would
         consist of bytes 0 through 3, entry 1 would consist of bytes
         1 through 4, etc.  The former simplification would reduce
         the memory requirements for the four S-boxes from 4096 bytes
         to 1024 bytes, the latter would reduce the requirements for
         a single S-box from 1024 bytes to 259 bytes.  Additional
         steps may be required to eliminate the symmetries that these
         simplifications would introduce.  Additionally, four
         different 10- or 12-bit indexes into a single large S-box
         could be used instead of the current series of S-boxes.

         Fewer iterations.  It is probably safe to reduce the number
         of iterations from 16 to 8 without compromising security. 
         The number of iterations required for security may be
         dependent on the length of the key.  Note that with the
         current subkey generation procedure, an 8-iteration
         algorithm cannot accept a key longer than 192 bits.

         On-the-fly subkey calculation.  The current method of subkey
         calculation requires all subkeys to be calculated advance of
         any data encryption.  In fact, it is impossible to calculate
         the last subkey of the last S-box without calculating every
         subkey that comes before.  An alternate method of subkey
         calculation would be preferable:  one where every subkey can
         be calculated independently of any other.  High-end
         implementations could still precompute the subkeys for
         increased speed, but low-end applications could only compute
         the required subkeys when needed.

CONCLUSIONS

I conjecture that the most efficient way to break Blowfish is
through exhaustive search of the keyspace.  I encourage all
cryptanalytic attacks, modifications, and improvements to the
algorithm.  Attacks on mini versions of Blowfish, those with a
32- or even a 16-bit block size, are also encouraged.  Source
code in C and test data can be provided to anyone wishing to
implement the algorithm, in accordance with U.S. export laws.

The software magazine Dr. Dobbs Journal is sponsoring $1000
contest for the best cryptanalysis of Blowfish received before
April 1995.  Please contact me for details.

Blowfish is unpatented, and will remain so in all countries.  The
algorithm is hereby placed in the public domain, and can be
freely used by anyone.

ACKNOWLEDGEMENTS

Much of the motivation for this algorithm, as well as the design
criteria, was developed with Niels Fergusen.  I would also like
to thank Eli Biham, Agnes Chan, Peter Gutmann, Angel Johnston,
Lars Kundsen, and Matt Robshaw for their helpful suggestions.

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         Patent 5,003,596, 26 Mar 1991.