Toward Axiomatic Fenced DES (long!)

Ritter Software Engineering 2609 Choctaw Trail Austin, Texas 78745 (512) 892-0494, ritter@io.com Toward Axiomatic Fenced DES Terry Ritter May 26, 1994 Introduction This article continues the development of a block cipher which I have been calling "Fenced DES." This unique construct uses the U.S. Data Encryption Standard (DES) as a component in a strength- enhanced cipher. Even though DES is slow and is now becoming vulnerable to advancing attack technology, DES is also well-known and trusted, and industry would be grateful to continue to use it if only it were stronger. The time has come to replace ordinary DES. One alternative is the complete certification of a totally new cipher at tremendous cost in both treasure and time. Another alternative is "triple- DES," at three times the computation of ordinary DES. But if a strength-enhancing construction can be found which is sufficiently clear and elegant, we may hope for a "derivative certification," based only assumptions about the strength of DES itself. In this article I start the process of proving some things about the Fenced DES cipher. In particular, I prove that the resulting cipher is invertible and has the avalanche property, two admittedly modest characteristics, but ones we do associate with a good block cipher. I claim that the construct is certainly guaranteed to be no weaker than DES. I also argue--with some theoretical support-- that the construct should be expected to be much stronger, at least 120 bits. In other words, it should be "strong enough" for the next couple of decades. The system of definitions, proofs and arguments which takes up the major part of this article is by no means finished, and is known to be casual and inconsistent in places. (Some of these problems could be fixed by expanding the mathematical base, which I avoid for now.) In spite of this, I believe it to be an interesting approach, even if it is an approach to which others are probably far better suited than myself. Therefore, let us just agree to accept it for what it is, and see how close it gets to what we need. The definitions apply to this particular construction. Those generally familiar with combinatorics might start with section 7, "Block Mixing Transforms." Fenced DES Here is the current 4x Fenced DES 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 Each "S" represents a separately-shuffled and independent 8-bit substitution table (which also implies the presence of a keyed cryptographic RNG to shuffle the tables). We have 32 input substitutions and 32 output substitutions, for an overall block size of 256 bits. This is only 32 bytes, which should be much smaller than the typical message. Trailing 2x and 1x blocks would reduce data expansion to only that needed by DES itself. Each "---DES---" represents an ordinary 64-bit-block DES operation. Each "---mix---" represents the mixing of the covered data blocks using "block mixing transform" technology. There are two levels of mixing on each side of the DES operations: The innermost levels each have two mixings which combine two 64-bit blocks; the outermost levels each have just a single mixing which combines two 128-bit blocks, a substantial mixing operation. This entire construct requires about 4.8 times the computation to cipher 4 times the data. In contrast, triple-DES would of course need 12 times the computation to cipher 4 times the data. The Proofs 1. SETS ======= 1.1 DEFINITION: A SET is a collection of objects in which any object either is or is not a part of the set. A set S can be described by a list of the elements in the set, viz. S = { a1, a2, ..., an }. 1.2 DEFINITION: The SIZE OF SET S is the number of elements in S, and is denoted |S|. 2. CODES ======== 2.1 DEFINITION: A CODE is a string of symbols in which the symbol in each position is taken from some common set S. When S consists of numeric values, a code can be seen as a polynomial with coefficients in S. 2.2 DEFINITION: An N-POSITION code is a code which has n positions for symbols, and can be denoted by S**n. 2.3 DEFINITION: A BINARY code is a code in which the common set is the set {0,1}. 2.4 DEFINITION: An N-BIT binary code is a binary code with n positions and can be denoted by {0,1}**n or by S**n with S = {0,1}. 2.5 THEOREM: (Size of code.) There are |S|**n distinct code values in an n-position code. (Proof: Each position in a code string can be any possible symbol, there are |S| possible symbols and n positions in each code string, so there are |S|**n possible code values of length n.) 2.6 THEOREM: (No special positions.) Taken over all possible code values, each string position has exactly the same number of occurrences of each symbol. (Proof: Each position in a code string can be any possible symbol. For any particular combination of symbols in other positions, in the selected position each possible symbol occurs once. So for every possible combination of symbols in other positions, in the selected position each possible symbol occurs the same number of times.) 2.7 THEOREM: (Position difference counts.) The number of n-position code values which differ in exactly m positions is (n) m (m) * (|S|-1) . (n) (Proof: There are (m) combinations of m positions out of n possible positions, and in any particular combination of m positions each position can take on |S|-1 other symbols producing (|S|-1)**m other code values for each combination.) 2.8 EXAMPLE: The number of 8-bit binary codes which differ in m bits is: distance count 0 1 1 8 2 28 3 56 4 70 5 56 6 28 7 8 8 1 --- 256 = 2**8 (Comment: There are 256 8-bit binary code values, and 255 values which differ in at least one position from any particular code value.) 2.9 THEOREM: (Average distance and distribution.) The expected number of elements which differ between two n-position code values is n * (1 - 1/|S|), and the distribution is binomial. (Proof: Assume the number of code differences is the binomial (n) m n-m probability of a difference B(m;n,p) = (m) p q , where where p = 1 - 1/|S| and q = 1-p, times the total number of n code values (1/q) : (n) m (n) m n-m -n (m) * (|S|-1) = (m) p q q (n) m = (m) (p / (1-p)) which is correct, so the expected number of different elements is the binomial expectation np.) 2.10 EXAMPLE: The expected number of elements which differ between two 8-bit binary code values is: 8 * (1 - 0.5) = 4. 2.11 EXAMPLE: The probability of having two 8-bit binary code values which differ in exactly two elements is: (8) 2 6 (2) (0.5) (0.5) = 0.109 = 28 / 256. 2.12 EXAMPLE: The expected number of elements which differ between two 64-bit binary code values is: 64 * (1 - 0.5) = 32. 2.13 EXAMPLE: The probability of getting a 64-bit binary code value which differs in exactly m bits from some other value is: difference probability 16 0.000026 28 0.061 29 0.075 30 0.088 31 0.096 32 0.099 (Comment: The 9 difference values 28..36 account for about 74 percent of all possible difference counts, even though they are only about 14 percent of all 65 possibilities.) 3. DISCRETE FUNCTIONS ===================== 3.1 DEFINITION: A DISCRETE FUNCTION takes an input code value to an output code value for a finite number of input code values. 3.2 DEFINITION: A RANDOM discrete function allows each output code value to be selected independently for each possible input condition. 3.3 THEOREM: (Number of random functions.) There are 2**2n possible random functions with an n-bit binary input code and an n-bit binary output code. (Proof: An n-bit binary code can make 2**n possible selections, each of which can be 2**n possible values, and (2**n)*(2**n) = 2**2n.) 4. SUBSTITUTION =============== 4.1 DEFINITION: A SUBSTITUTION is a mapping from input values or positions to output values. (Comment: A SUBSTITUTION can be seen as an indexable vector of substitute values. A SUBSTITUTION can also be seen as a "codebook" with an entry for every possible input code, and storage for each corresponding output code. A SUBSTITUTION can also be seen as an "arbitrary" discrete function, since any possible discrete function can be described by using a separate output code for each possible input condition. A SUBSTITUTION can also be seen as the relation joining substitute values with the position of each value.) 4.2 DEFINITION: SIMPLE substitution is the operation of using a substitution table or codebook to "encode" a string of input values by replacing each value in the string with its associated substitute value. (Comment: If the substitution is invertible, we can use an inverse substitution to "decode" the resulting encoded values and recover the original values.) 4.3 THEOREM: (Unique substitute values.) An invertible substitution can contain any particular output code at most once. (Proof: Suppose not: Then two different values into a substitution will produce the same output value. But that output value can inverse-substitute to only one inverse value, making the other input value unreachable, which contradicts invertibility, so this is false.) 4.4 THEOREM: (Number of invertible substitutions.) There are (2**n)! possible invertible substitutions for an n-bit binary input code. (Proof: The first substitution element can be any one of 2**n elements, the second element can be any except the first element, or (2**n)-1 elements, the third can be any except the first and second, for (2**n)-2 elements, and so on.) 4.5 THEOREM: (Guaranteed change propagation.) A change of even one input bit to an invertible substitution is guaranteed to produce a change in at least one output bit from the substitution. (Proof: Each input bit can select between two different input code values, which will select two different output code values, since an invertible substitution contains no duplicate values. Since any two different codes must be different in at least one bit, any input bit-change will produce at least one output bit-change.) 4.6 DEFINITION: A COMPLETE substitution contains every value of an n-position code, for some n. 4.7 THEOREM: (Probable change propagation.) Any change whatsoever to the input value to a complete invertible substitution is likely to change about half the bits in the output value. (Proof: Changing the input value selects among all remaining output code values. If the output is considered to be binary bits, we expect about half those bits to change (2.9).) 4.8 DEFINITION: AVALANCHE is a statistical property of a discrete function in which any change whatsoever on the input is expected to produce a change in about half the bits in the output value. 4.9 THEOREM: (Avalanche is automatic.) Avalanche is an inherent property of complete invertible substitution. (Proof: See 4.5, 4.7, and 2.9.) 4.10 THEOREM: (No special input bits.) Each input bit to an invertible substitution has exactly the same power to produce the same expected change in output bits. (Proof: Consider any possible change to any possible input value: from all possible input values any particular bit-change will produce all possible input values. Thus, any possible bit-change must produce the same overall expectation.) 4.11 THEOREM: (No special output bits.) Each output bit from a complete invertible substitution has exactly the same change expectation as any other output bit. (Proof: See 2.6.) 4.12 THEOREM: (Not a random function.) An invertible substitution cannot be a random function. (Proof: Suppose a value is selected for placement somewhere in a substitution. Since an invertible substitution cannot allow another occurrence of that same value, other values cannot be selected independently.) 4.13 DEFINITION: In a KEYED substitution the substitute element values have been permuted or re-arranged as a function of some key value or function. 4.14 THEOREM: (Reconstruction requires information linking output values to input values.) An unknown invertible substitution cannot be resolved without simultaneous information about both the input value or position and the output value. (Proof: To the extent that a particular substitution can be said to have an identity, that identity is the relation between substitute values and their position. This relation is both necessary and sufficient to define the substitution.) 5. BIT MIXERS ============= 5.1 DEFINITION: A BIT-MIXER combines multiple input bits such that each output value is defined by each and every input bit. 5.2 THEOREM: An invertible substitution is a bit-mixer. (Proof: Each and every input bit can select between two different input code values. Any input value change into an invertible substitution must necessarily select a different output value. Thus, the output value, and every bit in the output value, inherently depends upon each and every bit of the input value.) 6. BLOCK CIPHERS ================ 6.1 DEFINITION: A CIPHER is a keyed invertible translation from a plaintext element to a ciphertext element. 6.2 THEOREM: A CIPHER is a keyed invertible substitution. (Proof: For "translation" read "substitution.") 6.3 DEFINITION: A BLOCK cipher is a cipher in which the size of the code element is prohibitively large to be exhaustively explored. 6.4 THEOREM: (Not a random function.) No static block cipher can be a random function. (Proof: A cipher must be an invertible function, and no invertible function can have elements which are independent.) 6.5 ASSERTION: (Just a large substitution.) There is no property of a block cipher which is not ideally modelled by a substitution table of appropriate size containing a key-selected permutation of the possible output values. (Invertibility argument: A permutation of the possible output values is just a re-arrangement of values, without duplication. As long as there are no duplicate output values, the substitution is invertible.) (Avalanche argument: Avalanche is an expected property of an invertible substitution (4.9).) 7. BLOCK MIXING TRANSFORMS ========================= 7.1 DEFINITION: A BLOCK MIXING TRANSFORM is a mapping from multiple input code values to the same number of output code values, in which: 1. (Invertible.) The mapping is invertible. (Every possible input will imply a different output, and every possible output will imply a different input.) 2. (Each Output a Function of All Inputs.) Every output code value is a function of all input code values. 3. (Changes Propagate to All Outputs.) Any change to any one of the input code values will change all of the output code values. 4. (Balance and Input Independence.) Stepping any input through all possible values (with the other inputs held fixed) will step every output through all possible values. 7.2 ASSERTION: (We have a finite field.) Mod-2 polynomials modulo some irreducible polynomial p generate a finite field. (Comment: Proofs can use algebra.) 7.3 THEOREM: (Example block mixing transform.) The equations X = 3A + 2B = A + 2(A + B) Y = 2A + 3B = B + 2(A + B) and the inverse A = X + 2(X + Y) B = Y + 2(X + Y) mod 2 and mod p, where p is some mod 2 irreducible polynomial, represent a block mixing transform. (Inverse Proof: assume true, thus A = A + 2(A + B) + 2(A + 2(A + B) + B + 2(A + B)) = A + 2(A + B) + 2(A + B) = A and B = B + 2(A + B) + 2(A + 2(A + B) + B + 2(A + B)) = B + 2(A + B) + 2(A + B) = B which are both correct, so the inverse does exist for any polynomials X and Y.) (Function Proof: the equations for output code X includes both input code values A and B, so X is a function of both input codes. Y reasons similarly.) (Change Propagation Proof: First consider one term of one output block equation: 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 So, for any non-zero change, X has changed. Similar reasoning covers the other term, and the other equation.) (Balance Proof: Suppose not. Assuming A is fixed, then there must be two different values, B and B', which produce the same X: X = 3A + 2B = 3A + 2B' so X + 3A = 2B = 2B' which implies that B = B' a contradiction. Fixing B or working on the other block reason similarly.) 7.4 THEOREM: It is easy to manipulate both input blocks to a block mixing transform so as to fix one of the output blocks at a constant value. (Proof: Just inverse-transform the desired output blocks.) 7.5 ASSERTION: A block cipher can be used as a block mixing transform. (Method: Just divide the input block and output block into smaller "sub-blocks.") (Inverse Proof: A block cipher is invertible (6.1) and (6.3).) (Function Proof: To the extent that the block cipher can be considered an invertible substitution, each output bit is a function of each input bit (4.5), so each sub-block result is certainly a function of all sub-block input values.) (Change Propagation Argument: In a statistical sense, assuming substantial sub-blocks, each sub-block is extremely likely to change for any input change whatsoever (2.9).) (Balance Argument: In a statistical sense, over all possible inputs and all possible keys, any output value is equally likely, so any set of input changes is likely to produce a statistically-balanced result.) 8. 1X FENCED DES STRUCTURES ========================= 8.1 DEFINITION: A 1X INPUT-FENCED DES STRUCTURE is a 64-bit- wide construct consisting of eight keyed invertible byte- substitutions feeding a single DES ciphering: S S S S S S S S ------DES------ 8.2 THEOREM: Any data change whatsoever into a 1x input-fenced DES structure will produce a different result, and is expected to change about half of the output bits. (Proof: Every bit in the input block enters some small substitution which selects a keyed or arbitrary value from its set of output codes. Any input-change into an invertible substitution is is guaranteed to produce a change to at least one output bit (4.5). We model the DES ciphering as a large invertible substitution (6.5), and so expect that any change to the input will select a different output code value, which is likely to change about half of the output bits (4.7).) 8.3 DEFINITION: A 1X OUTPUT-FENCED DES STRUCTURE is a 64-bit-wide construct consisting of a single DES ciphering and eight keyed invertible byte-substitutions on the output: ------DES------ S S S S S S S S 8.4 THEOREM: Any data change whatsoever into a 1x output-fenced DES structure is expected to change about half of the output bits. (Proof: We model the DES ciphering as a large invertible substitution (6.5) and expect that any change to the input will change about half the bits in the output value (4.7). Since every possible DES result may occur, there are no special bits or bit subsets (2.6). Each of the output substitutions samples a bit subset in which about half of the bits are expected to change. Any change into an output substitution will select a different output code value, thus changing about half of the output bits (4.7) in every output substitution, and, thus, the overall output.) (Comment: One time in 255 there is no change to an output substitution, which is exactly what is required for an even output distribution. ) 8.5 DEFINITION: A 1X FENCED DES CIPHER is a 64-bit-wide construct consisting of eight keyed invertible byte-substitutions on the input, a single DES ciphering, and eight keyed invertible byte-substitutions on the output: S S S S S S S S ------DES------ S S S S S S S S 8.6 THEOREM: (Avalanche.) In 1x Fenced DES, any change of even a single bit in the large input block can be expected to change about half the bits in the large output block. (Proof: See 8.2 and 8.4.) 8.7 THEOREM: (Invertibility.) A 1x Fenced DES cipher is invertible. (Proof: From the construction of 1x Fenced DES, the small input substitutions are invertible, as are the small output substitutions. DES is assumed to be invertible. Since all elements in sequence from input to output are separately invertible, the sequential combination of these elements must also be invertible.) 9. 2X FENCED DES STRUCTURES ============================ 9.1 DEFINITION: A 2X INPUT-FENCED DES STRUCTURE is a 128-bit- wide construct consisting of 16 keyed invertible byte-substitutions feeding a block mixing transform, which feeds two DES cipherings: S S S S S S S S S S S S S S S S --------------mix-------------- ------DES------ ------DES------ 9.2 THEOREM: Any data change whatsoever into a 2x input-fenced DES structure will produce a different result, and is expected to change about half of the output bits. (Proof: Any change into an invertible substitution is guaranteed to produce a change to at least one output bit (4.5). Any change to either input block of a two-block block mixing transform is guaranteed to produce a change to both output blocks (7.1.3). We model the DES cipherings as large invertible substitutions (6.5) and so expect that any change to the input will select a different output code value, which is likely to change about half of the output bits (4.7).) 9.3 DEFINITION: A 2X OUTPUT-FENCED DES STRUCTURE is a 128-bit- wide construct consisting of two DES cipherings which feed a two- block block mixing transform, which feeds 16 keyed invertible byte- substitutions. ------DES------ ------DES------ --------------mix-------------- S S S S S S S S S S S S S S S S 9.4 THEOREM: Any data change whatsoever into a 2x output-fenced DES structure is expected to change about half of the output bits. (Proof: We model the DES cipherings as large invertible substitutions (6.5) and expect that any change to their inputs will select a different output value from all possible output values (4.5). Since any DES result is possible, any value is possible from both block mixing transform outputs (7.1.4), so we expect about half of the output bits to change (4.7). Since any block mixing result value is possible, there are no special bits (2.6), and each of the output substitutions samples a bit subset in which about half of the bits are expected to change. Any change into an output substitution will select a different output code value, thus changing about half of the output bits (4.7) in every output substitution, and, thus, the overall output.) 9.5 DEFINITION: A 2X FENCED DES STRUCTURE is a 128-bit-wide construct consisting of 16 keyed invertible byte-substitutions which feed a block mixing transform which feeds two DES cipherings which feed another two-block block mixing transform, which feeds another 16 keyed invertible byte-substitutions: S S S S S S S S S S S S S S S S --------------mix-------------- ------DES------ ------DES------ --------------mix-------------- S S S S S S S S S S S S S S S S 9.6 THEOREM: (Avalanche.) In a 2x Fenced DES cipher, any change of even a single bit in the large input block can be expected to change about half the bits in the large output block. (Proof: See 9.2 and 9.4.) 9.7 THEOREM: (Invertibility.) A 2x Fenced DES cipher is invertible. (Proof: From the construction of 2x Fenced DES, the small input substitutions are invertible, as are the small output substitutions. The block mixing transform is invertible (7.1.1). DES is assumed to be invertible. Since all elements in sequence from input to output are separately invertible, the sequential combination of these elements must also be invertible.) 10. 4X FENCED DES STRUCTURES ============================ 10.1 DEFINITION: A 4X FENCED DES CIPHER is a 256-bit-wide construct consisting of 32 keyed invertible byte-substitutions feeding a block mixing transform with two 128-bit blocks, which then feeds two block mixing transforms each with two 64-bit blocks, which feed four DES cipherings. The DES results feed two block mixing transforms each with two 64-bit blocks, which feed a block mixing transform with 128-bit blocks, which feeds 32 more keyed invertible byte-substitutions. 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 10.2 THEOREM: (Every input bit affects every DES ciphering.) In 4x Fenced DES, every bit in the large input block will affect at least one bit of the input to each of the DES cipherings. (Proof: Every bit in the large block enters some small substitution. Any input-change into a substitution is guaranteed to produce a change to at least one output bit (4.5). Any change into either side of the first-level block mixing transform is guaranteed to change both sides of the output (7.1.3), so some change is guaranteed to be present in the input of both next-level block mixing transforms. Again, any change anywhere on those inputs is guaranteed to be present in both sides of both outputs, which are the inputs to each DES ciphering.) 10.3 THEOREM: (Each output bit is affected by every DES ciphering.) In 4x Fenced DES, any data change whatsoever into any of the four DES cipherings is expected to change about half of the output bits. (Proof: We model the DES cipherings as large invertible substitutions (6.5) and expect that any change to their inputs will select a different output value from all possible output values (4.5). Since any DES result is possible, any value is possible on both outputs of the first-level output block mixing transform (7.1.4). Any possible block mixing transform result can be produced by some BMT input, so any possible value can occur as the input to the second-level output block mixing transform. With any possible BMT input, every output will occur, so there are no special bits (2.6), and each of the output substitutions samples a bit subset in which about half of the bits are expected to change. Any change into an output substitution will select a different output code value, thus changing about half of the output bits (4.7) in every substitution, and, thus, the overall output.) 10.4 THEOREM: (Avalanche.) In 4x Fenced DES, any change of even a single bit in the large input block can be expected to change about half the bits in the large output block. (Proof: See 10.2 and 10.3.) 10.5 THEOREM: (Invertibility.) 4x Fenced DES is invertible. (Proof: From the construction of 4x Fenced DES, the small input substitutions are invertible, as are the small output substitutions. The block mixing transform is invertible (7.3.1). DES is assumed to be invertible. Since all elements in sequence from input to output are separately invertible, the sequential combination of these elements must also be invertible.) 11. 4X FENCED DES STRENGTH CHARACTERISTICS ========================================== 11.1 ASSERTION: (DES cipherings cannot be separated.) In 4x Fenced DES, it is not possible to isolate and work on a single DES ciphering unless the small input substitutions have first been resolved. (Argument: In order to key-search a single DES ciphering, it is necessary to develop the input and output value for that particular ciphering. The large input and output blocks are known, but the values sent to the internal cipherings are hidden by the input and output substitutions.) 11.2 ASSERTION: (Input substitutions cannot be separated.) In 4x Fenced DES, it is not possible to isolate and work on any one small input substitution unless all four of the DES keys and at least one element in each of the 32 small output substitutions have first been resolved. (Argument: Even though their input values are known, resolving the content of the small input substitutions requires some information about their output values. Since these values flow through the internal DES cipherings, if DES is effective, these values cannot be known without the DES keys. Further, each of the DES keys is required, since all of the DES cipherings combined produce the known output. There can be no statistical effects which identify particular values from the input substitutions, because any change of any number of bits whatsoever affects the large output block similarly. There can be no statistical effects which isolate individual input substitutions because each input substitution has the same effect on the large output block. Any change whatsoever from any input substitution changes about half the bits in the output block, making statistical issues about the content of the substitutions completely irrelevant.) 11.3 ASSERTION: (Output substitutions cannot be separated.) In 4x Fenced DES, it is not possible to isolate and work on any one small output substitution unless all four of the DES keys and at least one element in each of the 32 small input substitutions have first been resolved. (Argument: Even though their output values are known, resolving the content of the small output substitutions requires some information about their input values. Since the input values flow from the internal DES cipherings, if DES is effective, these values cannot be known without the DES keys. Further, each of the DES keys is required, since each DES ciphering affects all of the output substitutions. There can be no statistical effects which identify particular input values to the output substitutions, because any change of any number of bits whatsoever affects the output from the substitution similarly. There can be no statistical effects which isolate individual output substitutions, because each of their input values come from the the output of the DES cipherings, and these values are "random like." So there can be no statistic to use for attack.) 12. FENCED DES EXPECTED STRENGTH ================================ 12.1 THEOREM: (Absolute minimum strength of 1x Fenced DES.) Assuming a known-plaintext attack, further assuming that all the input and output substitutions are known, if DES has a strength of 56 bits, the 1x Fenced DES construct has a keyspace of 56 bits. (Proof: All data flows through each layer; if the input and output substitutions are known, they do not confuse the data, but they also do not undo whatever confusion DES provides.) 12.2 ASSERTION: (Expected strength of the substitution layers in 1x Fenced DES.) Assuming a known-plaintext attack, and further assuming that the DES key is known, the 1x Fenced DES construct has a keyspace exceeding 64 bits. (Argument: The overall input is known, so the small input substitution _positions_ are all known; the uncertainty lies wholly in the _values_ at those positions. There are 256 possible values at the known position for each of eight input substitutions, for 256**8, or 2**64 possibilities. (A 63-bit expectation.) The uncertainty in the output substitution positions is the same, but the input and output substitutions are not independent: Since the DES key is known, defining the input substitutions implies what the output substitutions must be (or vise versa), so only one substitution level contributes to the keyspace. When working on the small input substitutions, the individual substitutions are independent: If even one of the input substitute values is wrong, we expect that half of the DES result bits will be wrong, which will imply wrong positions for most output substitutions. The process is similar if we choose to work on the output substitutions instead. A 64-bit keysearch is guaranteed to identify one element in each of the eight small input substitutions (for example). Then, assuming infinite known-plaintext, we just look for data blocks which are the same as the solved block in seven of the eight bytes. For each possible value of the eighth byte we can easily try each of the 254, 253,..., 2 remaining values (which will implicitly define many of the output substitutions) at almost no cost beyond holding and finding appropriate messages. With only a limited amount of known-plaintext there will be fewer if any messages which differ in just one byte, few if any quick byte searches, and many more-substantial searches until the input substitutions are filled in.) (Comment: DES with a known key is an example of a block mixing transform with absolutely no strength at all by itself, which nevertheless adds strength through bit mixing.) 12.3 ASSERTION: (Expected strength of 1x Fenced DES.) Assuming a known-plaintext attack, the 1x Fenced DES construct has a keyspace exceeding 120 bits. (Argument: When the DES key is known, the strength is 64 bits; the unknown DES key adds 56 bits more, for a total of 120 bits. (This is 2**64 times the complexity of DES.) It is not possible to separate the substitution layers from the cipher layer and so work on either independently, because the data flows through both. In addition, each DES operation is a function of every input bit (8.2) and each output bit is a function of every DES output (8.4), so individual DES operations cannot be isolated by particular input or output bits. A 120-bit keysearch will identify the DES key and one element in each of the eight small substitutions, and then we need to fill out the rest of each substitution as above.) 12.4 THEOREM: (Absolute minimum strength of 4x Fenced DES.) Assuming a known-plaintext attack, further assuming that all the input and output substitutions are known, if DES has a strength of 56 bits, the 4x Fenced DES construct has a keyspace exceeding 56 bits. (Proof: All data flows through each layer. The information content of the data is 256 bits; to recover that data, all four DES operations must be solved. Even if we assume that some aspect of the construction allows the DES operations to be solved separately, the resulting strength is still somewhat more than a single DES cipher.) 12.5 ASSERTION: (Expected strength of separated 4x Fenced DES.) Assuming a known-plaintext attack, and assuming that the internal ciphers _can_ be isolated and worked on separately, the 4x Fenced DES construct has an overall keyspace of not less than 120 bits. (Argument: The substitution and ciphering occur in series, consequently, at least one eight-byte substitution (input or output) and one DES ciphering must be solved simultaneously, even if the block mixing transform fails.) 12.6 ASSERTION: (Expected strength of 4x Fenced DES.) Assuming a known-plaintext attack, and assuming that the internal ciphers _cannot_ be isolated and worked on separately, the 4x Fenced DES construct has an overall keyspace exceeding 480 bits. (Argument: The small substitutions (input or output) jointly contribute 256 bits, and the four DES keys contribute 224 bits for a total of 480 bits. That is, searching a 480-bit keyspace will solve the system for a particular input (or output) block. This identifies the DES keys, but only solves 1/256th of each of 32 substitutions. Once the system is solved for a particular block, the 255 other entries in each of 32 substitutions must be filled in to completely solve the cipher.) Results It appears that Fenced DES can reasonably be proven to be an invertible block cipher which has the avalanche property (provided, of course, that DES has that property) with a strength at least that of DES itself. Reasonable-sounding arguments suggest that the internal ciphers cannot be separated and worked on independently, and that the resulting cipher has substantial strength. It would be nice to tighten this up; any and all suggestions are welcome. Appendix Some Fenced DES constructions: 1x Fenced DES S S S S S S S S ------DES------ S S S S S S S S 2x Fenced DES S S S S S S S S S S S S S S S S --------------mix-------------- ------DES------ ------DES------ --------------mix-------------- S S S S S S S S S S S S S S S S 4x Construct with 1x Strength 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 ------DES------ ------DES------ ------DES------ ------DES------ 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 Original 4x Fenced DES 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 Current 4x Fenced DES 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 4x Fenced DES with Less Storage and Strength (A..H and S..Z represent 16 keyed byte-substitutions, each used four times.) A B C D E F G H A B C D E F G H A B C D E F G H A B C D E F G H ------------------------------mix------------------------------ --------------mix-------------- --------------mix-------------- ------DES------ ------DES------ ------DES------ ------DES------ --------------mix-------------- --------------mix-------------- ------------------------------mix------------------------------ S T U V W X Y Z S T U V W X Y Z S T U V W X Y Z S T U V W X Y Z --- Terry Ritter ritter@io.com