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*To*: [email protected]*Subject*: Block Mixing Transforms*From*: [email protected] (Terry Ritter)*Date*: Sat, 12 Mar 1994 23:16:36 -0600 (CST)*Sender*: [email protected]

Ritter Software Engineering 2609 Choctaw Trail Austin, Texas 78745 (512) 892-0494, [email protected] Keyed Balanced Size-Preserving Block Mixing Transforms Terry Ritter March 12, 1994 Introduction Modern block ciphers seek to emulate extremely large substitution tables algorithmically, using complex combinations of various simple internal mechanisms. These internal mechanisms include small substitutions and trivial combinings, but the art and mystery of block cipher design is how to couple these simple and weak operations in ways which produce a strong overall cipher. One apparently new type of mechanism which might be useful in block cipher design would take two blocks in, share data between them, and then produce two generally-different blocks as a result. In particular, this mechanism might be used to mix data to (and from) a pair of substitutions, thus hopefully producing a stronger result than the two substitutions operating separately and independently. In most cases, it would be necessary for the mechanism to have an inverse, and to produce output blocks of the same size as the input. The result would be a mechanism which could be inserted anywhere in the internal data paths common in block-cipher designs. Block Mixing Transforms Consider constructs like this: A B | | v v Mixing Transform | | v v X Y X Y | | v v Inverse Transform | | v v A B Capital letters represent data blocks. Alternately, we can describe the transform, in general, as: X := f1( A, B ); Y := f2( A, B ); A := f3( X, Y ); B := f4( X, Y ); The intent of such a system is to mix two input blocks in a complex yet reversible way. This could provide two advantages: 1) It should make each output bit a function of all the input bits (on average), thus providing a way to expand block size while using smaller block-cipher functions. Hopefully the construct would also defeat attempts to "divide-and-conquer" the smaller functions separately. 2) It could provide a way to connect block-cipher functions in sequence, while eliminating any fixed direct connection between the blocks, such connections being vulnerable to "fix-in-the-middle" attack. A mixing transform is not unlike a "butterfly" section in a fast Fourier transform (FFT) [3]. But the usual FFT operates on complex values which are normally represented in floating-point. When implemented in fixed-point (as needed for mixing data blocks), the normal FFT butterfly expands the range of the input values, thus requiring a larger amount of storage (a larger block size) for the result. Fast Hadamard / Walsh transforms [2] behave similarly. For cryptography, we need transforms which are "size preserving" so that we can perform fixed-size block operations (such as DES) either on the input data or on the transformed results. It was not clear to me that this was going to be possible (at least with equations of practical complexity) until Eli Biham provided some examples of size-preserving mixing transforms: X := A - B; Y := 2A - B; A := Y - X; B := Y - 2X; for n-bit blocks, A, B, X, and Y, and arithmetic mod 2^n. There are actually many such transforms, and Biham has found a generalized form: (-1 1 ) (-w w-1) and (w-1 -1) (w -1) where w is some constant. For example, when w = 2: X := -1*A + 1*B = B - A Y := -2*A + (2-1)*B = B - 2A A := (2-1)*X + -1*Y = X - Y B := 2*X + -1*Y = 2X - Y with the arithmetic mod 2^n. To see inverse, note that A = X - Y = (B - A) - (B - 2A) = A B = 2X - Y = 2(B - A) - (B - 2A) = B These are fixed, linear transformations. If we know the input values, and the transformation, we will also know the output values. Even when the full equation is unknown, the simplicity and linearity of these transforms means that they require special protection in cryptographic applications. Mixing transforms can only be used when both the input and the output values cannot be exposed simultaneously. Alas, the transform mentioned above has a problem: Specifically, the least-significant-bit (lsb); that is, lsb(Y) = lsb(B). This is because the expression B - 2A has shifted A left one bit, leaving the bottom bit of B exposed. This provides a bit of direct correlation between an input value and an output value. This is probably sufficient to support a practical "fix-in-the-middle" attack if the transform is used to isolate two DES operations. Consider these correlation experiments on the above transform with 4-bit blocks: x3 x2 x1 x0 y3 y2 y1 y0 b0 64 64 64 64 64 64 64 128 b1 64 64 64 64 64 64 64 64 b2 64 64 64 64 64 64 64 64 b3 64 64 64 64 64 64 64 64 a0 64 64 64 64 64 64 64 64 a1 64 64 64 64 64 64 64 64 a2 64 64 64 64 64 64 64 64 a3 64 64 64 64 64 64 64 64 This is a 0 -> 0 correlation count. For each possible input value (over both A and B), for each input bit which is zero (somewhere in A and B) and each output bit which is zero (somewhere in X and Y), a count is recorded. The count of 128 means that y0, the lsb of Y, occurs twice as often as expected when the lsb of B is zero. Similarly, 64 64 64 64 64 64 64 0 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 a 0 -> 1 correlation count, shows that no cases exist where the lsb of B is a one and the lsb of Y is a zero. Cryptographic Mixing In [8] I introduced a new type of reversible stream-cipher combiner (the first stream-cipher combiner, which we now call "exclusive-OR" or "mod-2 addition" was described by Vernam [12]). "Combiner" is the traditional cryptographic name for a mixing function. [11,5,1] (Non-reversible combiners are also used, typically to make confusion sequences difficult to penetrate. [e.g., 6]) Combiners and mixing transforms have much in common. Basically, a combiner will look like any other two-input one-output function: A B | | v v Mixing Function | v C C B | | v v Inverse Function | v A The capital letters represent the block size; in a typical stream cipher these are byte values. A is the plaintext, B the confusion stream, C the ciphertext. Note that exactly the same confusion stream is needed to recover the original data; this is the heart of stream-cipher security. There are many two-input functions, but most are not useful as cryptographic data combiners, which must be reversible and must have no correlation between either input and the output. Combiners which do have correlation [e.g., 4] fall to statistical attacks [e.g., 10]. If we see mixing transforms as a matched-set of cryptographic combiners, we can see that correlation is a problem with the example transform. (Biham did have an example of one balanced but non-keyed transform based on rotation and subtraction mod 2^n.) Mixing in Mod-2 Polynomials Since the "weak" exclusive-OR form of combiner has long been available, modern combiner designs are normally intended to be "stronger" and, thus, are more complex. But it is not at all clear that "stronger" is what we need in a mixing transform. Presumably, "strength" can be provided more efficiently by some other function, like DES, or a substitution table. Thus, we may really want a modest-strength extremely-fast mixing solution, and one approach is to consider the well-known field of mod-2 polynomials. In mod-2 arithmetic, addition is the same as subtraction X + Y = X - Y and any value added to itself is zero X + X = 0 so, in general, multiplication cannot be achieved by addition X + X <> 2X (assuming X is non-zero) but is instead achieved by shifting. Then 2X + X = 3X so multiplication is not restricted to binary powers. Of course 3X + X = 2X which just shows that mod-2 arithmetic can be surprising. It is interesting to see just how unusual good mixing transforms are. Consider a first approach X := A + B; Y := A - B; (mod-2, mod-p, where p is some primitive mod-2 polynomial of appropriate degree for the size of the data blocks). While this is a reasonable approach in the integers, in mod-2 polys, A + B = A - B. This means that X = Y, and the two resulting identical blocks cannot possibly carry enough information to provide an inverse transform for two arbitrary input blocks. It does not work. Next consider X := A + B; Y := A + 2B; with inverse operations A := (2X + Y) / 3; B := (X + Y) / 3; (mod-2, mod-p), and the division done by multiplying by the inverse of 3, mod p. (Appropriate inverse equations may not always exist; finding the inverse equations is interesting in itself.) This works. But here X is never affected by p at all, thus producing an extremely regular (and un-keyed) transformation. And the inverse multiplication is, in general, far more expensive than multiplication by a small integer. Finally, consider X := 2A + 3B; Y := 3A + 2B; A := 2X + 3Y; B := 3X + 2Y; Again, operations are 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. This works, and the transform is a self-inverse. The primitive affects the result in both data blocks. And the multiplications are simple. Correlation experiments conducted as before show a nice, balanced, uncorrelated system: 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 These functions are extremely fast. Addition is a simple exclusive-OR. Multiplication by two is simply a left-shift and a conditional add of the primitive. Multiplication by three is a multiplication by two plus an addition. Keyed Mixing Transforms The mod-2 polynomial transforms depend on having some primitive of the appropriate degree. Different primitives produce different mixing functions, with similar overall performance. This leads to the possibility of keying the transforms by selecting arbitrary primitives. (Some references to primitive-finding algorithms are given in [9].) Rabin gives the number of degree-n primitives as about p^n / n [7]. Thus, for degree 64, we have about 2^64 / 2^6 or about 2^58 primitives. This means that each randomly-selected degree-64 primitive carries about 58 bits of key. Of course, this key can only be effective to the extent that the linear transformation cannot be attacked and the primitive thus deduced. Some Consequences If a single input bit changes on one of the mixing transform input blocks, we can be sure that at least one bit will change in both output blocks. If two input bits change, we can be sure that these bits will not "cancel" each other; changes will still occur in the output blocks. If many input bits are changed, and the transform primitive is known, it is possible to engineer a no-change in one output block (although this is unlikely to happen by chance). Should this be undesirable, it might be made impossible by design (such as ciphering the input blocks before mixing), or by keying the transform (so the necessary bit patterns are unknown). If it becomes possible to define the input to, and what the output must be from a ciphering element, it will be possible to key-search that element independent of other elements, and this is what we hope to avoid. To prevent this it may be necessary to use keyed input and output transforms, or even multiple ciphering levels between transforms. Applications It is crucial to remember that these simple, high-speed, but linear mixing transforms can be said to have "strength" only if the input and output values are never both available. That is, these structures do not by themselves handle "known-plaintext" attack. (Of course, the same could be said for many other simple internal mechanisms used in block cipher construction.) Simple constructs like A B | | v v MixTrans | | v v C D are not likely to be very useful as ciphers by themselves, even if the mixing transformation is keyed and the blocks are large. On the other hand, constructs like A B | p1 | v v v MixTrans | | v v DES1 DES2 | | | p2 | v v v MixTrans | | v v C D are considerably more interesting. Note that this construct ciphers a double-size DES block at single-DES rates. It seems to require keyed mixing transforms. Similarly, A B | | v v DES1 DES2 | | | p | v v v MixTrans | | v v DES3 DES4 | | v v C D will cipher a double-size DES block at double-DES rates, and at least superficially avoids all weakness in the mixing transform by placing strength in each input and output port. This may avoid the need to key the mixing transform. Alternately, A B | k1 | v v | XOR <- DES1-----| | | | k2 | | v v |---- DES2 -> XOR | | | p | v v v Mixing Transform | | | k3 | v v | XOR <- DES3 ----| | | | k4 | | v v |---- DES4 -> XOR | | v v C D also ciphers at double-DES rates. Of course, larger external blocks mean an increase in the number of internal data paths, making various sorts of interconnection configurations possible. Thus A B C D | p1 | | p2 | v v v v v v MixTrans1 MixTrans2 p3 | | p4 | | v v v v v v -Trans3 MixTrans4 Mix- | | | | v v v v DES1 DES2 DES3 DES4 | | | | | p5 | | p6 | v v v v v v MixTrans5 MixTrans6 p7 | | p8 | | v v v v v v -Trans7 MixTrans8 Mix- | | | | v v v v E F G H will cipher quadruple-size DES blocks at single-DES rates, A B C D | | | | v v v v DES1 DES2 DES3 DES4 | | | | | p1 | | p2 | v v v v v v MixTrans1 MixTrans2 p3 | | p4 | | v v v v v v -Trans3 MixTrans4 Mix- | | | | v v v v DES5 DES6 DES7 DES8 | | | | v v v v E F G H will cipher quadruple-size DES blocks at double-DES rates, and A B C D | k1 | | k2 | v v | v v | XOR <- DES1 ----| XOR <- DES2 ----| | | | | | k3 | | k4 | | v v | v v |---- DES3 -> XOR |---- DES4 -> XOR | | | | | | | | | p1 | | p2 | v v v v v v MixingTransform1 MixingTransform2 p3 | | p4 | | v v v v v v -Transform3 MixingTransform4 Mixing- | | | | | k5 | | k6 | v v | | v | XOR <- DES5 ----| XOR <- DES6 ----| | | | | | k7 | | k8 | | v v | v v |---- DES7 -> XOR |---- DES8 -> XOR | | | | v v v v E F G H will also cipher quad-size blocks at double-DES rates. But in each case, four double-level mixing transforms could be replaced by a single double-size mixing transform: A B C D | | p1 | | v v v v v ---------mix1--------- | | | | v v v v DES1 DES2 DES3 DES4 p2 | | | | v v v v v ix2--------- --------m | | | | v v v v E F G H A B C D | | | | v v v v DES1 DES2 DES3 DES4 | | | | | | p | | v v v v v ---------mix---------- | | | | v v v v DES5 DES6 DES7 DES8 | | | | v v v v E F G H A B C D | k1 | | k2 | v v | v v | XOR <- DES1 ----| XOR <- DES2 ----| | | | | | k3 | | k4 | | v v | v v |---- DES3 -> XOR |---- DES4 -> XOR | | | | | | p | | v v v v v ---------------------mix---------------------- | | | | | k5 | | k6 | v v | | v | XOR <- DES5 ----| XOR <- DES6 ----| | | | | | k7 | | k8 | | v v | v v |---- DES7 -> XOR |---- DES8 -> XOR | | | | v v v v E F G H These are new ciphering architectures. Clearly, it is not known how strong these constructs would be. However, this situation can hardly be considered unusual. Other opportunities exist when constructing completely new block ciphers. These might, for example, be based on byte-wide key- permuted substitutions, thus avoiding differential attacks on fixed "optimal" tables. Thus ------------------------------mix------------------------------ --------------mix-------------- --------------mix-------------- ------mix------ ------mix------ ------mix------ ------mix------ --mix-- --mix-- --mix-- --mix-- --mix-- --mix-- --mix-- --mix-- mix mix mix mix mix mix mix mix mix mix mix mix mix 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 mix mix mix mix mix mix mix mix mix mix mix mix mix mix mix mix --mix-- --mix-- --mix-- --mix-- --mix-- --mix-- --mix-- --mix-- ------mix------ ------mix------ ------mix------ ------mix------ --------------mix-------------- --------------mix-------------- ------------------------------mix------------------------------ enciphers 256-bit blocks through 32 keyed 8-bit substitutions by using five levels of input keyed mixing transform and five levels of output keyed mixing transforms of varying size. Clearly, there are a plethora of alternate interconnection possibilities here. For example, the mixing rows could be permuted, different sizes of mixing combined in some rows, the mixing not arranged on 2^n boundaries, etc., etc. Since the mixing transforms are extremely fast, we would expect this 256-bit system to be much faster than 64-bit single-DES. And, 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 mix mix mix mix mix mix mix mix mix mix mix mix mix --mix-- --mix-- --mix-- --mix-- --mix-- --mix-- --mix-- --mix-- ------mix------ ------mix------ ------mix------ ------mix------ --------------mix-------------- --------------mix-------------- ------------------------------mix------------------------------ --------------mix-------------- --------------mix-------------- ------mix------ ------mix------ ------mix------ ------mix------ --mix-- --mix-- --mix-- --mix-- --mix-- --mix-- --mix-- --mix-- mix mix mix mix mix mix mix mix mix mix mix mix mix 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 enciphers 256-bit blocks through 64 keyed 8-bit substitutions by using nine levels of mixing transforms of varying size. With the substitutions all keyed, we can probably avoid keying the mixing transforms. Again, there are a plethora of alternate interconnection possibilities. Summary Practical, high-speed, keyed, balanced, and size-preserving block mixing transforms are introduced for cryptographic service. References [1] Arko, R. 1961. Mechanical Signal Combiner. U.S. Patent 3,159,712. [2] Beauchamp, K. 1984. Applications of Walsh and Related Functions. Academic Press. [3] Brigham, E. 1974. The Fast Fourier Transform. Prentice-Hall. [4] Geffe, P. 1973. How to protect data with ciphers that are really hard to break. Electronics. January 4. 99-101. [5] Kohler, H. 1951. Combining Circuits. U.S. Patent 2,567,214. [6] Massey, J., and R. Rueppel. 1989. Method of, and Apparatus for, Transforming a Digital Data Sequence into an Encoded Form. U.S. Patent 4,797,922. [7] Rabin, M. 1980. Probabilistic Algorithms in Finite Fields. SIAM Journal on Computing. 9(2): 273-280. [8] Ritter, T. 1990. Substitution Cipher with Pseudo-Random Shuffling: The Dynamic Substitution Combiner. Cryptologia. 14(4): 289-303. [9] Ritter, T. 1991. The Efficient Generation of Cryptographic Confusion Sequences. Cryptologia. 15(2): 81-139. [10] Siegenthaler, T. 1985. Decrypting a Class of Stream Ciphers Using Ciphertext Only. IEEE Transactions on Computers. C-34: 81-85. [11] Smith, H. 1950. Combining Circuit. U.S. Patent 2,496,317. [12] Vernam, G. 1919. Secret Signaling System. U.S. Patent 1,310,719. --- Terry Ritter [email protected] (alas, cactus.org dies March 18) [email protected] (perhaps temporarily)

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