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Fenced DES
Terry Ritter
April 17, 1994
Introduction
This article is one in a series which document my attempts to find
a fast, strong, acceptable extension to the U.S. Data Encryption
Standard (DES), which I believe is now dangerously insecure.
The intent is to find a relatively-simple and believable construct
which uses DES as a building block, thus avoiding the need to
certify a complete new cipher. I note that currently there is no
institution which could and also would provide such certification.
In this article I propose a new "fenced" ciphering construct which
may be a solution. The experimental 256-bit-block implementation
takes about 1.2 times the computation (per byte) of DES alone, and
may have the strength of four DES keys.
In this design, some important block-cipher properties seem to
follow logically from the widely-accepted existence of those
properties in DES itself.
Wide Blocks
All practical block ciphers attempt to emulate a large substitution
table algorithmically; DES employs substantial computation simply
to behave like a substitution table of 2**64 elements. Accepting
DES as a reasonable design means that we have implicitly accepted
the argument that a fast 8-bit-wide substitution is not secure (by
itself). Certainly, if a small-block substitution were secure,
we would all use that simple and fast alternative instead of DES.
Since we do not, we must have accepted the fact that block size is
a significant factor in block cipher strength.
DES is often used to encipher language text, which contains a
surprisingly small amount of information. Since data-compression
programs routinely compress language text by 60%, we can expect
that a 64-bit block of language text may contain perhaps 26 bits
of information. While it is not currently known how this could
be exploited, a 256-bit-wide block should contain four times that
much information, which should solve any related problem.
A large block size also addresses some aspects of cryptoanalytic
weakness: Some attacks on block ciphers make use of the "birthday
paradox" to find a matching pair from a large number of ciphertexts.
With a 64-bit block about 2**32 ciphertext blocks would be expected
to be needed; a large number, admittedly, but still possible. But
the same attack on a 256-bit block would require about 2**128
ciphertext blocks, which is completely out of the question. Thus,
a large block size eliminates one type of attack on the cipher.
A large-block 4x-wide cipher need not expand ciphertext beyond the
normal expansion for DES (CBC initialization vector and key-length
aside), provided that one trailing 2x and one trailing 1x block can
be used if needed. All the preceding blocks would be 4x blocks.
The Two Problems
This project has had to address two major problems:
1. Weaknesses of Multi-Layer Constructs: Many simple multi-
level ciphering structures based on DES can be attacked by
working simultaneously on both the input and output layers,
given "known plaintext" or "defined plaintext." In general,
this means that two-level constructs are much weaker than one
might expect. This leads to three-level construct like
"triple-DES" which tend to be very slow.
2) Weakness in Multi-Block Constructs: Similarly, simple
large-block structures based on DES can be attacked by
defining or "fixing" the input values of all but one DES
block, using "defined plaintext." Apparently, any composite
structure which does not have each bit affect the every DES
ciphering will have this weakness.
To expand the effective block size while using DES itself, Fenced-
DES uses the "block mixing transform" construct which I described
in the previous article. In this article I want to clarify how
those transforms can be used to create a cipher with a large block
size out of smaller blocks, despite the mixing having no strength
of its own.
The Block Mixing Transform
In a previous article I introduced the concept of a "block mixing
transform" (extended from work by Eli Biham) as a tool to mix the
information in two data blocks, and then recover that information.
This concept could be expressed as two pairs of expressions:
X := f1( A, B ); Y := f2( A, B );
A := f3( X, Y ); B := f4( X, B );
The term "transform" is taken from the ability to change the
data into a different data-space, and then recover the original
values, and also the similarity to the "fast Fourier transform"
"butterfly" operation. This "block mixing transform" should
be distinguished from the "mixing transformation" described by
Shannon [10: 711].
The particular form I suggested was:
X := 2A + 3B; Y := 3A + 2B;
A := 2X + 3Y; B := 3X + 2Y;
with operations mod-2 and mod-p, where p is some primitive mod-2
polynomial of appropriate degree for the data blocks X, Y, A and B.
(Later work shows that p need not be primitive, but p must be
irreducible in cryptographic service.) This transform is a self-
inverse, has good mixing correlation properties, is statistically
balanced, and has a processing cost which is linear with block size.
Efficient implementation suggests a re-labeling as follows:
X := 3A + 2B; Y := 2A + 3B;
A := 3X + 2Y; B := 2X + 3Y;
Comments on the original "block mixing transform" article
have uncovered a few other references to fixed-size math
transforms, including Agarwal and Burrus [1], Pollard [6],
and Rader [7], but none related to cryptography. I would
be glad to hear of any other references of any sort.
The mixing transform need not be a cipher by itself. Indeed,
it need have no "strength" at all, but must provide at least a
minimal level of mixing and be cryptographically-balanced; it
should also be expandable and fast. Although speed is not an
issue in most individual ciphering, speed is a major issue for
industrial applications, including centralized network servers.
The application in this article mixes blocks of substantial size,
making many other forms of mixing completely impractical.
4x Fenced-DES
Consider the following construct:
S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S
--------------mix-------------- --------------mix--------------
------------------------------mix------------------------------
------DES------ ------DES------ ------DES------ ------DES------
------------------------------mix------------------------------
--------------mix-------------- --------------mix--------------
S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S
Here each "S" represents an 8-bit substitution table. Thus, we
have 32 input substitutions and 32 output substitutions, each a
separately-shuffled and independent table, and an overall block
size of 256 bits. We also have four DES operations, plus two
levels of input mixing and two levels of output mixing. Note
that the innermost mixing levels combine two 128-bit blocks, a
substantial operation which is nevertheless practical using the
selected block mixing transform.
The idea is to spread the effect of each input bit to each of the
four DES operations, and to produce a particular output bit from
a combination of all four DES results. If this works, The Opponent
would be forced to search all four DES keyspaces simultaneously to
match a particular known-plaintext pair.
An experimental implementation of the above construct performs all
64 substitutions and all 6 mixings in less time than a single DES
computation. Currently, it ciphers 4 times the data with about
4.8 times the computation, and has, perhaps, a keyspace of 224 bits
or so. (A much faster hybrid implementation might do the DES
computations in hardware.)
In the experimental implementation, table and key
initialization take about 200 times the computation of a
single 256-bit-block ciphering. (This is mainly a
consequence of shuffling 64 small substitution tables.)
Even so, it is probably faster to compute the 16K initial
state than to decipher 16K of saved state with software
DES or Fenced-DES: Construction is faster than ciphering.
The keyed construction of the substitution tables implies the
presence of a specific cryptographic RNG. This means that any
overall Fenced-DES specification will pin-down the key
processing which varies so widely in current DES applications.
The current implementation uses a fast 992-bit Additive RNG
and the nonlinear "jitterizer" [8] which I have discussed many
times with respect to my Penknife cipher and my other Dynamic
Substitution [9] ciphers.
In the experimental implementation, a User Key of arbitrary
length and content is hashed (CRC'd) by 32 separate degree-31
primitive mod-2 polynomials (11- through 19-nomials), producing
the 992-bit RNG state, which also eventually generates the DES
keys. Note that this approach eliminates the need for keys to
have a specific format unique to this particular cipher. This
enables the selection of an arbitrary cipher from among many
different ciphers, all of which can use the exact same key.
Deciphering simply uses inverse substitutions (the inverse of each
encipher output substitution is used for decipher input) and DES
in decipher mode. The selected block mixing transform is a self-
inverse and needs no changes.
Mixing Levels
The arrangement of the mixing levels deserves some comment.
First, note that a change in any one input data bit produces a
distribution of changes out of the associated input substitution,
depending on the particular substitution, original input value,
and change. Any possible byte input has a 50 percent probability
of affecting each of the eight output bits from that substitution.
A substitution table S is an indexable n-element vector of
output codes. An invertible substitution table S with
inverse table inv(S) has the property that for any input code
i in n, inv(S)[ S[i] ] = i. This implies that S contains
n different output codes.
An invertible substitution table S contains each output code
value exactly once. Since each possible index selects a
different element, any index change will select a different
output code. Since different code values must differ in at
least one bit, any input change must produce a change in at
least one output bit.
Given invertible substitution table S with shuffled contents,
define the output distribution for any input code change to be
an arbitrary selection from the output codes which differ from
the current output code. If the output codes are a complete
set of 2**m values (0..(2**m-1)) for some m, counting arguments
show that it is likely that about half of the output bits will
change for any input code change of any nature whatsoever.
Conversely, since each output bit is produced by an output
code, and the selected output code is completely dependent
upon every bit in the input code, each output bit is dependent
on every bit of the input. A network with this property is
normally called "complete" [5], and localized completeness is
also the basis for "avalanche" [3: 22] in an iterated block
cipher.
Next, note that we first mix two 64-bit blocks (twice), then two
128-bit blocks. Suppose we have a change in any one input data
bit: this produces an 8-bit substituted result which would normally
affect just a single DES block. But the 64-bit mixing extends
those changes to two DES blocks, and the 128-bit mixing extends the
changes to all four DES blocks. Thus, any change of even a single
input bit will affect all four DES operations.
Using the transformation X := 3A + 2B; Y := 2A + 3B; any
value change to A or B must be reflected in both X and Y:
Suppose some change C is added to A:
X := 3A + 2B (mod 2, mod p)
X' := 3(A+C) + 2B
X' := 3A + 3C + 2B
dX := X' - X = 3C
but 3C is non-zero (thus affecting the output) for any C which
is not zero, and if C is zero, there has been no change to A.
Suppose some change C added to B:
X := 3A + 2B (mod 2, mod p)
X' := 3A + 2(B+C)
X' := 3A + 2B + 2C
dX := X' - X = 2C
Similarly, 2C is also non-zero for any C which is not zero.
Suppose we try to make C half the value of p plus the highest
bit (2**(deg(p)-1)) so that p will be activated and 2C will
cancel the lower bits of p: Alas, p is irreducible so there
is no q S.T. 2q = p.
Similar arguments apply for Y := 2A + 3B.
The experimental implementation uses the degree-128 irreducible
0100004000000400200002000004000001 (hex), and the degree-64
irreducible 010002000000800201 as block mixing polynomials.
The output from each DES operation is, of course, random-like, so
one might think it could be used directly. However, a three-
level structure is still necessary to prevent, for example, "fix-
in-the-middle" attacks, so the output substitutions are important.
We also need the output mixing so that the result from a single
DES block cannot be isolated and worked on independently.
The guaranteed performance of the input substitution and the block
mixing transform imply that each DES input block collectively
depends upon each and every input bit. The expected performance
of the DES algorithm extends this, making every DES output bit
depend upon each and every input bit in the entire large input
block, thus making all DES outputs "complete" over the large input
block.
Cryptographic Strength
First let's review where modern cryptographic science stands with
respect to "strength":
1. There is no algorithmic test to "certify" or evaluate the
"strength" of a cipher.
2. Despite a half-century of intensive mathematical work, we
still have exactly one cipher which is commonly accepted as
having been proven "unbreakable," and that cipher is normally
impractical. Despite this immense effort, and the fact that
a "proof" of cipher strength is unfulfilled for any practical
cipher whatsoever, there are still calls for "proofs" of new
cipher designs.
3. While various cryptanalytic attack strategies are known,
each such attack is necessarily specific to the particular
cipher being attacked. Attack names represent strategies,
rather than generally-applicable algorithms. Simply knowing
the history of previous attacks does not necessarily provide
insight into applying those attacks to a new cipher.
4. Ordinarily we speak of the "strength" of a cipher as the
minimum effort needed to "break" the cipher. Unfortunately,
we are necessarily limited to discussing what we know now, and
not what can be known in the future. Any current minimum may
not last, and we may not be able to know whether it will last
or not.
With those points in mind, the current "strength" for 4x Fenced-DES
is ((2**56)**4)(256!**64) keys, a very big number. I would be
delighted to learn of a simpler attack.
It would of course be ridiculous to accept this sort of number as
a true indication of strength. Personally, I would be happy with
anything over 112 bits, since this should be sufficient for the
next couple of decades and then we may have a stronger basis for
cryptographic design.
Design Strength
Note that we need assume no "strength" for the mixing layers, but
simply mixing: Each mixed output block must be a function of each
and every bit in both input blocks. In this particular design we
need only two levels of mixing to make sure that every input bit has
propagated to all four DES blocks. And then we need two more to
make sure that all four DES blocks participate in every output bit.
The purpose of the small substitutions is to prevent the (weak and
known) mixing functions from being exploited to divide-and-conquer
the DES operations. Small substitutions appear to be sufficient
to isolate the mixing functions, because "known plaintext" is only
available across the entire cipher, and not across the internal
layers of the cipher. When known-plaintext is not available, and
substitutions cannot be separated for divide-and-conquer, little
substitutions can be surprisingly strong.
In the 4x construct, we might lay all the strength on the four DES
keys, which would imply a 224-bit value. On the other hand, an
attack which is able to isolate one of the DES keys (perhaps as a
consequence of 1x operation using the same state), would reduce
this to 168 bits. Note that the substitutions must be keyed even
if we discount their "strength."
Strength Arguments by Attack
Exhaustive Search: Try each key until the correct one is found.
Preventing this now requires a keyspace substantially larger than
56 bits (or, with a computationally-expensive setup phase, perhaps
a few bits less). It seems reasonable to claim that Fenced-DES has
at least a 224-bit keyspace. Note that this is not four times the
DES keyspace, but four times the key size, which is 2**168 times
the conventional DES keyspace.
Known-Plaintext/Defined Plaintext: Somehow "obtain" both the
plaintext and the corresponding ciphertext for some large number
of encipherings (under one key). This has many flavors:
Codebook: Try to obtain all possible ciphertexts and associated
plaintext; then, when a ciphertext occurs, look it up. This is
normally prevented by having a large number of transformations,
which implies both a large block size and a large keyspace.
Fenced-DES has both.
Codebook approaches can be combined with "divide-and-conquer" to
isolate and define parts of some ciphers. Fenced-DES tries to
avoid these attacks by not allowing the parts to be isolated and
worked on separately.
Meet-in-the-Middle: With a multi-layered structure, given known-
or defined-plaintext, search the top keyspace to find every
possible result, and search the bottom keyspace to find every
possible value. With a two-level construct, matches can be
verified with some subsequent known-plaintext/ciphertext pairs.
Fenced-DES avoids this by using a three-level construction, and
by using outer layers which have a huge "keyspace."
Differential Cryptanalysis: Given a S-P iteration cipher with
known tables, use any statistical unbalance in the tables to peer
back into previous steps. Fenced-DES avoids this by having no
fixed tables, by using only balanced full-substitution tables,
and by using a fully-balanced block mixing transform to avoid
"divide-and-conquer."
Important Aspects of the Design
First, the Fenced-DES construct is more like a Kam-Davida
substitution-permutation (S-P) design [5] than the common iterated
Feistel design [3] represented by DES itself. The block mixing
transform is specifically intended to avoid the sort of weakness
exploited by the recent Heys-Tavares attack [4] on S-P designs.
Next, it seems that there is a fundamental weakness in any two-
layer construct for some form of "meet in the middle" attack when
we assume "defined-plaintext" capabilities. Fenced-DES has three
independent layers to avoid such attacks.
Conventional block-cipher designs generally use unkeyed static
substitution tables which are selected for "optimum" performance.
In contrast, Fenced-DES uses only key-generated tables, in which
any table permutation is as good as any other, making selection
unnecessary. (A shuffled substitution is very unlikely to be
linear [2], but linearity is itself unimportant when it cannot
be detected externally. The mid-level substitution--here
DES--acts to hide any S-box linearity.)
Conventional block-cipher designs are also very economical with
state, using either small tables (e.g., the 256 bytes in eight
6-bit to 4-bit tables in DES), or no tables at all (e.g., IDEA).
But 4x Fenced-DES uses 16K (bytes) of tables, all keyed.
More conventional S-P designs tend to use the same block size at
each substitution level, thus becoming vulnerable to Heys-Tavares
attacks [4]. Fenced-DES differs from this approach by having a
middle layer with a block size which is much larger than the outer
layers (this is similar to a Kam-Davida "partition" [5: 749] but
differs in that it is a single block). This should prevent those
small substitutions associated with a single internal block from
being separated and attacked individually.
Other contemporary block-cipher designs generally use a 64-bit
block size. This is much weaker than it was 20 years ago, when
that size was selected for DES. To avoid birthday attacks on
ciphertext, as well as unknown information-based attacks, 4x
Fenced-DES has a nominal block size of 256 bits, although 8x or
even 16x versions are both possible and practical. 2x and 1x
versions can be used to cipher the last part of a message, thus
reducing data expansion to that expected with DES alone.
A fundamental difference is that conventional S-P designs perform
only a bit-permutation (or "transposition") between substitution
layers; this is a weakness in that an input bit to one layer is
exactly the same as some output bit in the previous layer.
Fenced-DES differs from other block-cipher designs in the use of a
block mixing transform to make the input code to a middle-layer
substitution (in this case, DES) a function of every substitution
in the previous layer. This allows the external block size to be
expanded while preventing substitutions in the middle layer from
being separated and attacked individually.
An interesting aspect of the Fenced-DES design is the possibility
that assumed properties of DES--a cipher which has been studied
and evaluated for almost 20 years--can be provably expanded into
properties of the larger cipher.
Summary
A new type of cryptographic ciphering construct has been introduced
which uses DES as a building block. The result seems to provide
a larger block size and more strength than triple-DES (the leading
alternative), while operating almost three times as fast.
References
[1] Agarwal, R. and C. Burrus. 1974. Fast Convolution Using
Fermat Number Transforms with Applications to Digital
Filtering. IEEE Transactions on Acoustics, Speech, and
Signal Processing. ASSP-22(2): 87-97.
[2] Ayob, F. 1982. Probabilistic completeness of substitution-
permutation encryption. IEE Proceedings, Pt. E. 129(5):
195-199.
[3] Feistel, H. 1973. Cryptography and Computer Privacy.
Scientific American. 228(5): 15-23.
[4] Heys, H. and S. Tavares. 1993. Cryptanalysis of Tree-
Structured Substitution-Permutation Networks. Electronics
Letters. 29(1): 40-41.
[5] Kam, J. and G. Davida. 1979. Structured Design of
Substitution-Permutation Encryption Networks. IEEE
Transactions on Computers. C-28(10): 747-753.
[6] Pollard, J. 1971. The Fast Fourier Transform in a Finite
Field. Mathematics of Computation. 25(114): 365-374.
[7] Rader, C. 1972. Discrete Convolutions via Mersenne
Transforms. IEEE Transactions on Computers. C-21(12):
1269-1273.
[8] Ritter, T. 1991. The Efficient Generation of Cryptographic
Confusion Sequences. Cryptologia. 15(2): 81-139.
[9] Ritter, T. 1990. Substitution Cipher with Pseudo-Random
Shuffling: The Dynamic Substitution Combiner. Cryptologia.
14(4): 289-303.
[10] Shannon, C. 1949. Communication Theory of Secrecy Systems.
Bell System Technical Journal. 28: 656-715.
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Terry Ritter ritter(a)io.com
