METHOD AND SYSTEM FOR OBFUSCATING A CRYPTOGRAPHIC FUNCTION

Abstract

In a system 300, a server 310 provides a cryptographic function F to an execution device in an obfuscated form. The function F composes output of a plurality of the mapping tables Ti(0≦i≦n; n≧1) using an Abelian group operator . A processor 312 chooses tables O and C such that C[x]O[x]=0, ∀x±Di and creates tables T′i, 0≦i≦m; n≦m≦n+1, where for 0≦i≦n, each Table T′i represents the respective corresponding table T,j and at least one table T′o1, 0≦01, ≦n, being formed through an Abelian composition of To1 and O, and at least one table T′c10≦c1,≦m, c1#oj, being formed through an Abelian composition that includes C. Means 314 are used for providing the tables T′i, to the executing device. The executing device 320 includes means 324 for receiving the tables and a processor 322 for forming a function F′ that is functionally equivalent to the cryptographic function F by an Abelian composition of the tables T′i.

Description

FIELD OF THE INVENTION

The invention relates to a method of obfuscating a cascaded cryptographic function. The invention also relates to a system for supplying a cascaded cryptographic function to an execution device in an obfuscated way. The invention further relates to an execution device for executing a cascaded cryptographic function provided in an obfuscated way.

BACKGROUND OF THE INVENTION

The Internet provides users with convenient and ubiquitous access to digital content. Because of the potential of the Internet as a powerful distribution channel, many CE products strive to directly access the Internet or to interoperate with the PC platform—the predominant portal to the Internet. The CE products include, but are not limited to, digital set top boxes, digital TVs, game consoles, PCs and, increasingly, hand-held devices such as PDAs, mobile phones, and mobile storage and rendering devices, such as Apple's iPod. The use of the Internet as a distribution medium for copyrighted content creates the compelling challenge to secure the interests of the content provider. In particular it is required to warrant the copyrights and business models of the content providers. Increasingly, CE platforms are operated using a processor loaded with suitable software. Such software may include the main part of functionality for rendering (playback) of digital content, such as audio and/or video. Control of the playback software is one way to enforce the interests of the content owner including the terms and conditions under which the content may be used. Where traditionally many CE platforms (with the exception of a PC and PDA) used to be closed, nowadays more and more platforms at least partially are open. In particular for the PC platform, some users may be assumed to have complete control over the hardware and software that provides access to the content and a large amount of time and resources to attack and bypass any content protection mechanisms. As a consequence, content providers must deliver content to legitimate users across a hostile network to a community where not all users can be trusted. The general approach in digital rights management for protected content distributed to CE platforms is to encrypt the digital content (for instance using DES or AES) and to store the decryption key (or the “license”) in a so-called License database (for a PC, the License database is typically stored on the PC's hard disk). The largest vulnerability of digital rights management relying on encryption is the key distribution and handling. For playback, a software player has to retrieve a decryption key from the license database, it then has to store this decryption key somewhere in memory for the decryption of the encrypted content. This leaves an attacker two options for an attack of the key handling in a software player: firstly, reverse engineering of the license database access function could result in a black box software (i.e., the attacker does not have to understand the internal workings of the software function) capable of retrieving asset keys from all license databases. Secondly, by observation of the accesses to memory used during content decryption it is possible to retrieve the asset key.

Typically, digital rights management systems use an encryption technique based on block ciphers that process the data stream in blocks using a sequence of encryption/decryption steps, referred to as rounds. During each round, a round-specific function is performed. The round-specific function may be based on a same round function that is executed under control of a round-specific sub-key. For many encryption systems, the round function is specified using mapping tables for different parts of the domain of the function. Even if no explicit tables were used, nevertheless frequently tables are used for different parts of the domain of the function for efficient execution in software of encryption/decryption functions. The computer code accesses or combines table values into the range value of the function. Instead of distributing keys, that may be user-specific, it becomes more interesting to distribute user specific algorithms instead of keys for en- or decryption algorithms. These algorithms, most often functions (mappings), have to be obfuscated (hidden) in order to prevent redesign or prohibit the re-computation of elements that are key-like. On computers, tables accompanied with some computer code often represent these functions.

SUMMARY OF THE INVENTION

It is an object of the invention to provide a method, system and execution device of the type set forth with better protection of a cascaded cryptographic function such as a Feistel network.

To meet the object of the invention, a method of obfuscating a cryptographic function F that cryptographically converts an input x from a predetermined domain D to an output y using a plurality of non-linear mapping tables T_i(0≦i≦n; n≧1), each with key-like information and operating on a respective part D_i of the domain D, the function F composing output of the mapping tables using an Abelian group operator (for example, F(x)=T₀[x₀] . . . T_n[x_n], x=(x₀ , x₁ , . . . , x_n), x_iεD_i), includes:

choosing a table O and a table C such that C[x]O[x]=0, ∀xεD_i;

creating a plurality of tables T′_i, 0≦i≦m; n≦m≦n+1, where for 0≦i≦n, each table T′_i represents the respective corresponding table T_i and at least one of those tables T′_o1, 0≦o₁≦n, hereinafter referred to as obfuscated table, being formed through a composition, using the Abelian group operator, of the corresponding table T_o1 and the table O (for example, T′_o1[x]=O[x]T_o1[x], ∀xεD_o1), and at least one of the other tables T′_c1, 0≦c₁≦m, c₁≠o₁, hereinafter referred to as compensating table, being formed through a composition, using the Abelian group operator, that includes the table C (for example, T′_c1[x]=C[x]T_c1[x], ∀xεD_c1); and

forming an obfuscated function F′ that is functionally equivalent to the cryptographic function F by composing the plurality of tables T′_i, 0≦i ≦m, using the Abelian group operator, such that the tables O and C are compensated through the composition.

According to the invention at least one of the original tables is obfuscated through an Abelian composition with a chosen table O. In this way it is more difficult to obtain the content of the original table. The fact that the cryptographic function F can be implemented as an Abelian composition of tables is used to compensate for the disturbance caused by the table O. To this end, a table C that is the inverse of table O under the Abelian operator is also included in the new composition of the tables, canceling out the effect of O. Preferably all of the original tables are obfuscated. The compensation table(s) may be included in one or more of the tables T′_i, 0≦i≦n or in an additional table T′_n+1.

According to a measure of the dependent claim 2, the tables O and C represent a same non-trivial constant b₁; the step of creating the obfuscated table T′_o1, 0≦o₁≦n, c₁≠o₁ includes using the Abelian group operator to combine the constant b_i with each element of the corresponding table T_o1; and the step of creating the compensating table T′_c1, 0≦c₁≦m, c₁≠o₁ includes using the Abelian group operator to combine the constant b_i with each element of the corresponding table T_c1. Using a constant is a simple way to hide the table T_o1. The constant is combined with the table values and thus has the same range (for example, a typical range of 0 . . . 2⁶⁴⁻¹ can be represented by 64-bits, giving a 64-bit constant). The constant can be seen as a key for obfuscating the table. By obfuscating several of the tables, preferably each using a respective constant, an even more secure system can be built.

According to a measure of the dependent claim 3, m=n. In this approach the number of tables used for the obfuscated cryptographic function F′ is the same as used for F. No additional compensation table(s) are required.

According to a measure of the dependent claim 4, a further table O₂ is chosen and a further table C₂ such that C₂[x]O₂[x]=0, ∀xεD₁; O₂ being distinct from O; and at least one further obfuscated table T′_o2, 0≦o₂≦n, o₁≠o₂ is formed through a composition using the Abelian group operator of the corresponding table T_o1 and the table O₂ (for example, T′_o2[x]=O₂[x]T_o2[x], ∀xεD_o2) and the compensating table T′_c1 is formed through a composition that also includes the table C₂ (for example,

T′_c1[x]=C₂[x]C[x]T_c1[x], ∀xεD_c1). Using constants makes it simple to compensate for multiple distinct obfuscations in only one compensating table. By including more than one compensation in this compensating table, the individual compensations are hidden better, improving security.

According to the measure of the dependent claim 5, m=n+1 and c₁=m implying that the compensation table is an additional table. This additional table may represent one or more of the obfuscation tables. Using an additional table gives additional freedom, resulting in further possibilities for the obfuscation that can now be compensated for.

According to the measure of the dependent claim 6, the obfuscated table T′_o1 is applied the domain part D_o1; O being a mapping table applied to D_o1, C[x]O[X]=0, ∀xεD_o1. The method includes extending the domain D to D′=(D′_o, . . . , D′_n, D′_n+1), where D_i=D′_i for i<=n and D_n+1 includes a representation of D_o1 such that the composition that forms the compensating table T′_c1 includes C[x], xεD_o1. By extending the domain and ensuring that the domain extension includes the obfuscated part, it is now also possible to obfuscate the range of the tables since that can now be compensated due to the fact that the undisturbed domain part is still available and can be used as input for the compensation. In principle, each disturbed domain part could be included as a separate extended domain part.

According to the measure of the dependent claim 7, advantageously O is a homomorphism and the method includes forming a plurality of obfuscated tables T′_i through a composition, using the Abelian group operator, of the corresponding table T_i and the table O (for example, T′_i[x]=O[x])T_i[x], xεD_i) and forming the domain extension D′_n+1 through a composition using the Abelian operator of the corresponding subdomains of the involved tables T_i (for example, D′_n+1[x]=D_o1[x]D_o2[x]D_o3[x], ∀x). The fact that O is a homomorphism enables the combined input represented in only one domain extension part D′_n+1 to have a compensating effect on the respective obfuscated tables. Thus even if several original tables are obfuscated using the homomorphism, nevertheless only one extended domain part is required and not several. This increases security and reduces the amount of data involved.

These and other aspects of the invention are apparent from and will be elucidated with reference to the embodiments described hereinafter.

BRIEF DESCRIPTION OF THE DRAWINGS

In the drawings:

FIG. 1 shows a block diagram of a system in which the invention may be used;

FIG. 2 shows an exemplary structure of a prior art execution device;

FIG. 3 shows a block diagram of a system according to the invention; and

FIG. 4 shows a further embodiment of a system according to the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

Digital content on the PC is typically rendered using media players, such as Microsoft's Media Player, Real's RealOne Player, Apple's QuickTime or iTunes player. Such players can load for a specific content format a respective plug-in for performing the format-specific decoding. Those content formats may include AVI, DV, Motion JPEG, MPEG-1, MPEG-2, MPEG-4 (AAC), WMV, Audio CD, MP3, WMA, WAV, AIFF/AIFC, AU, etc. The player and plug-in structure is illustrated in FIG. 1, where a media player 100 includes a core player 100 and several format-specific plug-ins (shown are plug-ins 120, 122 and 124). The core player 100 may, for example, provide the user interface for controlling the player. Each plug-in includes a respective decoder. It may send the decoded content directly to rendering HW/SW, such as a sound-card, or pass it on to the core player 100 for further processing. For secure rendering, a secure plug-in is used that not only decodes the content in the specific format but also decrypts the content. This is illustrated in FIG. 2, where the encrypted content is first fed through a decryptor 230 and next the decrypted content is fed through the format-specific decoder 220. The decryptor 230 may receive a decryption key/license from a license database 210.

The largest vulnerability of digital rights management relying on encryption is the key distribution and handling. For playback, a software player has to retrieve a decryption key from the license database, it then has to store this decryption key somewhere in memory for the decryption of the encrypted content. Typically digital rights management systems use an encryption technique based on block ciphers that process the data stream in blocks using a sequence of encryption/decryption steps, referred to as rounds. The output of i-1^th round is the input of the i^th round. Thus, for a system with N rounds the algorithm can be described as a function cascade ƒ_N o . . . o ƒ₁(x), where function ƒ_i represents the functionality of round i. Most block algorithms are Feistel networks. In such networks, the input data block x of even length n is divided in two halves of length n/2, usually referred to as L and R. So, the input x fed to the first round is given as x=L₀, R₀. The i^th round (i>0) performs the function ƒ_i, where ƒ_i is defined as

ƒ_i(L_i-1, R_i-1)=R_i-1, (L_i-1⊕F(R_i-1,K_i)),

K_i is a subkey used in the i^th round and F is an arbitrary round function.

The input and output for an encryption/decryption algorithm usually consists of a sequence of a predetermined number of N bits (digits with values of 0 or 1). These sequences will be referred to as blocks and the number of bits they contain will be referred to as their length. The bits within such sequences may be numbered starting at zero and ending at one less than the sequence length (block length or key length). For example, the AES/Rijndael system uses 128-bit blocks (N=128). The number i attached to a bit is known as its index and will be in the ranges 0≦i<N for an N-bit block.

Typically, the basic unit for processing in en encryption/decryption algorithm is a smaller than the block length. Commonly used unit lengths are 4-bit (a 4-bit unit is normally referred to as a nibble) or 8-bit (normally referred to as a byte). Those unit lengths can be processed more easily on conventional processors. If so desired also other unit lengths may be used. A unit (a contiguous sequence of a plurality of n-bits) is treated as a single entity. The input to the algorithm is processed as an array of such units that is formed by dividing these sequences into k groups of n contiguous bits to form arrays of units, where

k*n=N.

For an input denoted by x, the units in the resulting array will be referenced as x_i, where i will be in the following range: 0≦i<k. The i-th nibble can be described as:

x _i=nibble(i,x)=(x/b 2⁴ⁱ)mod 2⁴

Similarly, the i-th byte can be described as:

x _i=byte(i,x)=(x/2⁸ⁱ)mod 2⁸

All unit values will be presented as the concatenation of its individual bit values (0 or 1) between braces in the order {b_n-1, b_n-2, . . . , b₁, b₀}. For most ciphers, these units are interpreted as finite field elements using a polynomial representation:

$\sum _{j=0}^{n-1}b_jx^j.$

In this notation, the factor x is used only to comply with a customary notation of such polynomial representation and does not represent the input. Frequently, encryption/decryption algorithms use Galois Fields as finite fields (for example GF(2ⁿ)). In the polynomial representation, for an 8-bit unit {01100011} identifies the specific finite field element x⁶+x⁵+x+1. Usually, the unit value is also indicated in a hexadecimal notation, for example the element {01100011} can be represented as {63}. Since the encryption/decryption algorithm itself is not subject of the invention, this will not be described here in more detail.

In many existing en- and decryption algorithms, like AES (Rijndael) and Feistel ciphers, a round function F is typically constructed using tables (T_i[x]) for the different units of the input (i.e. different parts of the domain of the function). A role of such a table may be to perform a non-linear substitution; such tables are also referred to as S-boxes. In addition to such an S-box role, the table may also represent pre-processing and/or post-processing with respect to such an S-box. In general, the table maps n input bits to m output bits. Such a table is referred to as an n*m table. DES uses eight different 6*4 S-boxes. Some systems (e.g. Khufu and Khafre) use a single 8*32 bit S-box, IDEA effectively has a 16*16 S-box and AES/Rijndael uses an 8*8 S-box.

The composition of the output of these often-small tables is done with an Abelian group operator on the ranges of these functions. Any Abelian group operator may be chosen. A frequently used operator is the bit-wise XOR operator. The system according to the invention uses this Abelian composition to hide the actual values of the tables without changing the functional behavior of F. According to the invention, at least one of the tables is obfuscated. This table will be referred to as obfuscated table. The other tables may also be obfuscated or kept unmodified. Thus, a plurality of tables T′_i, 0≦i≦ n are created, where for each table T′_i represents the respective corresponding table T_i in a modified or unmodified form. As described at least one of those created tables is an obfuscated version of a corresponding original table. First a description is given wherein only one table is directly obfuscated. This table is referred to as T′_o1, 0≦o₁≦n. According to the invention this obfuscated table is formed through a composition, using the Abelian group operator, of the corresponding table T′_i and a table O. A straightforward composition is T′_o1=0T_o1. More complex compositions (e.g. using additional tables that are also composed using the Abelian operator) are also possible. Please note that the notation T′_o1=OT_o1 is a short notation for the purpose of simplicity. A mathematically more accurate notation is T′_o1[x]=O[x]T_o1[x] for x in the relevant domain to which it is applied. The short notation will be used in several places in the description and a skilled person will with above explanation understand the meaning. The table O may be chosen in any suitable way. Some preferred choices will be described below. The system and method according to the invention use the Abelian composition in such a way that an obfuscation of one or more of the tables is compensated for in the Abelian composition of the tables created tables T′_i. At least one of those tables T+_i thus compensates (in the Abelian composition) for the obfuscating caused by table O. In this way the Abelian composition of the tables T′_i thus still performs the function F, although some or all of the tables differ from the original tables T_i. The table that compensates for O will be indicated as T′_c1, 0≦c₁≦m, c₁≠o₁, and referred to as compensating table. This table is formed through a composition, using the Abelian group operator, that includes a table C. A straightforward composition is T′_c1=CT_ci. The table C is chosen such that CO=0 (thus C is the inverse of O under the Abelian operator, also denoted as C=O⁻¹). Depending on the choice for the disturbance O, the compensating table may be one of the tables T′_i described above, or an additional table may be used. Thus, either n or n+1 tables are created: T′_i, 0≦i≦m; n≦m≦n+1. It will be appreciated that preferably all original tables are changed. In the system and method according to the invention, an obfuscated function F′ is formed that is functionally equivalent to the cryptographic function F by composing the plurality of tables T′₁, 0≦i≦m, using the Abelian group operator, where the composition is such that the tables O and C compensate each other through the composition. It will be understood that instead of regarding table O as the table causing the obfuscating and C causing the compensation, these roles can equally well be seen as reverse.

The invention will be illustrated for a function F: F:0 . . . 2⁶⁴⁻¹→0 . . . 2⁶⁴⁻¹ The function F may be a round function of a block cipher, such as a Feistel cipher. In the example, function F can be represented as an Abelian composition of sixteen table-implemented functions, each with a respective nibble as input:

F(x)=T₀[nibble(0,x)] . . . T₁₅[nibble(15,x)]

This can also be written as:

$F\left(x\right)=\prod _{i=0}^{15}\otimes T_i\left[\mathrm{nibble}\left(i,x\right)\right]=\prod _{i=0}^{15}\otimes T_i\left[x_i\right],$

and

$T_i\left[x_i\right]\text{:}0\dots 15\to 0\dots 2^{64-1},\forall i\text{:}i\in 0\dots 15$

It will be appreciated that the range (0 . . . 2⁶⁴⁻¹) and the unit length (in this example a 4-bit nibble) are a mere choice for the purpose of the example. Other choices may also be made.

Two preferred approaches for the obfuscating of the tables according to the invention are described in more detail below. In the first approach the domain on which the obfuscated tables work is the same as used for the original tables. In a preferred embodiment, constants are used for the obfuscating. In the second approach the domain is extended. In preferred approach a homomorphism that depends on the input x is used for the obfuscation and compensation.

No Domain Extension, Using Constants

In this approach, the tables O and C represent a same non-trivial constant b₀. The obfuscated table T′_o1, 0≦o₁≦n, is created by using the Abelian group operator to combine the constant b_i with each element of the corresponding table T_o1. Similarly, the compensating table T′_c1, 0≦c₁≦m, c₁≠o₁ is created by using the Abelian group operator to combine the constant b₀ with each element of the corresponding table T_c1. It can be observed that in this scenario, two identical tables are used. As will also be clear from the example given below, this can be generalized to the situation wherein an even number of the original tables T_i are combined with a same constant using the Abelian composition. The constant is combined with the table values and thus has the same range (in the example, 0 . . . 2⁶⁴⁻¹) and is represented by the same number of bits (in the example, 64 bits). By combining an even number of tables with the same constant, the Abelian composition results in an automatic elimination of the disturbance when the obfuscated tables are combined. This is illustrated for a simple example wherein only the first table (o₁=0) and second table (c₁₌1) are modified through combination with the same constant b₀ and the other tables are not modified, giving:

T′₀[x₀]=b₀T₀[x₀],

T′₁[x₁]=b₀T₁[x],

T′_i[x_i]=T_i[x_i], ∀i: iε2 . . . 15

The created tables T′_i ∀i: iε0 . . . 15 of which some are modified (obfuscated) can be used in the new function F′ (obfuscated F):

$F^{\prime }\left(x\right)=\prod _{i=0}^{15}\otimes T_i^{\prime }\left[x_i\right]$

It can be observed that the tables O and C indeed are chosen such that CO=0, based on the fact that C=O and that for an Abelian composition holds that OO=0. As a consequence, also the functions F′ and F are functionally equal.

$\mathrm{Proof}\text{:}F^{\prime }\left(x\right)=\prod _{i=0}^{15}\otimes T_i\left[x_i\right]\left[\mathrm{by}\mathrm{definition}\mathrm{of}T_i\right]$ $F^{\prime }\left(x\right)=b_0\otimes T_0\left[x_0\right]\otimes b_0\otimes T_1\left[x_1\right]\otimes \prod _{i=2}^{15}\otimes T_i\left[x_i\right]\left[a\otimes b=b\otimes a,a\otimes \left(b\otimes c\right)=\left(a\otimes b\right)\otimes c\right]$ $F^{\prime }\left(x\right)=b_0\otimes b_0\otimes T_0\left[x_0\right]\otimes T_1\left[x_1\right]\otimes \prod _{i=2}^{15}\otimes T_i\left[x_i\right]\left[a\otimes a=0\right]$ $F^{\prime }\left(x\right)=T_0\left[x_0\right]\otimes T_1\left[x_1\right]\otimes \prod _{i=2}^{15}\otimes T_i\left[x_i\right]=\prod _{i=0}^{15}\otimes T_i\left[x_i\right]=F\left(x\right)$

It will be clear that this holds indeed for any obfuscation where the same constant is combined with an even number of tables using the Abelian group operator. So, the constant may be combined with 2, 4, 6, etc. number of tables. In the example, a total of 16 tables are used. So, it is possible to obfuscate all tables with just one constant. The constant in the example has 64 bits; this constant can be seen as a 64 bit key.

This embodiment will be illustrated further with two examples that for the purpose of the explanation have been simplified to the extent that no domain parts are used, instead each table operates on the entire domain. Only two original tables are used.

T₀={01,12,20}

T₁={00,12,21}

In these definitions of the tables 01 means that an input value 0 is mapped to an output value 1, 12 that 1 is mapped to 2, etc. In the examples, the tables O and C depend on the input x. The following obfuscating and compensating tables have been chosen:

$O\left[x\right]=\left\{0\mapsto 2,1\mapsto 1,2\mapsto 2\right\}\Rightarrow C\left[x\right]=\left\{0\mapsto \frac{1}{2},1\mapsto 1,2\mapsto \frac{1}{2}\right\}$

The created tables are then T′₀[x]=O[x]T₀[x], T′₁[x]=C[x]T₁[x]

EXAMPLE 1

F(x)=T₀[x]·T₁[x] (in this example, the Abelian operator is a multiplication·) giving

$T_1^{″}=\left\{0\mapsto 2,1\mapsto 2,2\mapsto 0\right\}$ $T_1^{″}=\left\{0\mapsto 0,1\mapsto 2,2\mapsto \frac{1}{2}\right\}$ $F^{\prime }\left(x\right)=T_0^{\prime }\left[x\right]·T_1^{\prime }\left[x\right]$

EXAMPLE 2

F(x)=T₀[x]T₁[x] (the Abelian operator is the XOR operator)

T′₀={03,10,22}

T′₁={02,10,23}

F′(x)=T′₀[x]T′₁[x]

In a further embodiment, security is improved further by using at least one more key (i.e. more constants may be used). In the simplest form two distinct tables are combined with the constant b₀ as shown above and two different tables are combined with the constant b₁. In this way, a further table O₂ and a further table C₂ are chosen such that C₂O₂=0, where O₂ is distinct from O and the tables O₂ and C₂ both represent the same non-trivial constant b₁. It will be appreciated that also more tables may be protected using the constant b₁. For example, it is then also possible to combine half of the tables with b₀ and half of the tables b₁. In this way, all tables are obfuscated and each constant is used an even number of times, resulting in the constant being eliminated in the composition. If so desired, some tables may be used in plain form (i.e. not obfuscated through an Abelian group operator).

In a further embodiment, the same tables O₂ and C₂ may be used. Table O₂ may be used as described above to obfuscate at least one further table T_o2, 0≦o₂≦n, o₁≠o₂ forming at least one further obfuscated table T′_o2 through a composition using the Abelian group operator of the corresponding table T_o1 and the table O₂ (for example, T′_o2=O₂T_o1). Instead of using C₂ to form a further compensating table, instead the existing compensating table T′_c1 is used to also compensate for O₂. This is achieved by including in the composition for T′_c1 also the further compensating table C₂ (for example, T′_c1=C₂CT_c1). This latter embodiment can be extended by including more than two compensations in T′_c1. In a preferred embodiment, all, but one, of the original tables T_i are obfuscated with a respective constant and the one remaining table is obfuscated with the Abelian composition of all constants. In this embodiment, all constants are used twice (i.e. an even number) and all tables are obfuscated. In the following example, the first fifteen tables (i=0 . . . 14) are combined with a respective constant b_i and the last table is combined with all constants, giving:

T′_i[x_i]=b_iT_i[x_i] ∀i: iε0 . . . 14 ∀x_i: x_iε0 . . . 15

and

T′₁₅[x₁₅ ]=b₀ . . . b₁₄T₁₅[x₁₅] ∀x_i: x_iε0 . . . 15

The obfuscated tables T′_i ∀i: iε0 . . . 15 are used in a new function F′ (obfuscated F):

$F^{\prime }\left(x\right)=\prod _{i=0}^{15}\otimes T_i^{\prime }\left[x_i\right]$

As before, the functions F′ and F are functionally equal.

$\mathrm{Proof}\text{:}$ $F^{\prime }\left(x\right)=\prod _{i=0}^{15}\otimes T_i\left[x_i\right]\left[\mathrm{by}\mathrm{definition}\mathrm{of}T_{15}\right]$ $F^{\prime }\left(x\right)=\left(b_0\otimes \dots \otimes b_{14}\right)\otimes T_{15}\left[x_{15}\right]\otimes \prod _{i=0}^{14}\otimes b_i\otimes T_i\left[x_i\right]\left[a\otimes b=b\otimes a,a\otimes \left(b\otimes c\right)=\left(a\otimes b\right)\otimes c\right]$ $F^{\prime }\left(x\right)=\left(b_0\otimes \dots \otimes b_{14}\right)\otimes \left(b_0\otimes \dots \otimes b_{14}\right)\otimes \prod _{i=0}^{15}\otimes T_i\left[x_i\right]\left[a\otimes a=0\right]$ $F^{\prime }\left(x\right)=0\otimes \prod _{i=0}^{15}\otimes T_i\left[x_i\right]\left[a\otimes 0=0\otimes a=a\right]$ $F^{\prime }\left(x\right)=\prod _{i=0}^{15}\otimes T_i\left[x_i\right]\left[\mathrm{by}\mathrm{definition}\mathrm{of}F\right]$ $F^{\prime }\left(x\right)=F\left(x\right)$

From above examples it will be clear that the skilled person has freedom in the level of obfuscating to be achieved. For a system where F is an Abelian composition of n tables, one to n constants may be used. Each constant is combined with an even number of tables. It will be appreciated that in principle any number of constants may be used. However, by associating every table with its own constant, using more than n constants will not increase security any further. Preferably, the constants are chosen ‘randomly’ using a suitable random or pseudo-random generator.

In the examples given above, the compensation C is combined with one or more of the original tables. Thus normally the number m of created tables T′ will be the same as the number n of original tables T. An alternative is to use one additional table for the compensation, thus m=n+1. In this embodiment, again all constants are used twice (i.e. an even number) giving an automatic compensation through the Abelian composition. In the following example, all sixteen original tables (i=0 . . . 15) are combined with a respective constant b_i and one additional table is created that is the combination of all constants, giving:

T′_i[x_i]=b_iT_i[x_i] ∀i:iε0 . . . 15 ∀x_i:x_iε0 . . . 15

and

T′₁₆=b₀ . . . b₁₄b₁₅ ∀x_i: x_iε0 . . . 15

Domain Extension

In above approach, combining an original table T with a constant b means that every entry in the table T is combined with the same value b using the Abelian group operator

In the second approach, the obfuscation depends at least partly on the input to the table. In order to be able to compensate for this, the compensation table needs access to the input. To this end, the domain is extended. As described above, the original tables T_i(0≦i≦n; n≧1), each operate on a respective part D_i of the domain D. At least one of the created T′_o1, 0≦o₁≦n, the obfuscated table, is formed through a composition, using the Abelian group operator, of the corresponding table T_i and the table O (for example, T′_o1=OT_o1). As a consequence, the obfuscated table T′_o1 is applied to the domain part D_o1 and O is a mapping table that is applied to D_o1. As described above, at least one of the other tables T′_c1, 0≦c₁≦m, c₁≠o₁, the compensating table, is formed through a composition, using the Abelian group operator, that includes the table C (for example, T′_c1=CT_c1). For this to work it is required that C[x]O[x]=0, ∀xεD_o1. To this end, the domain D is extended to D′=(D′₀, . . . , D′_n, D′_n+1), where D_i=D′_i for i<=n and D_n+1 includes a representation of D_o1 such that the composition that forms the compensating table T′_c1 includes C[x],xεD_o1.

According to the invention at least one table is obfuscated through a combination with a table O that at least partly depends on the same input; the combination being performed by the Abelian group operator. For example, the first table T₀ is obfuscated:

T″₀[x₀]=O(x₀)T₀[x₀]

Keeping all other tables unmodified:

T″_i[x_i]=Ti[x], ∀i:iε1 . . . 15

would not give a compensation for O(x₀). According to the invention, the compensation is achieved by using at least one additional table that covers the terms that need to be compensated with respect to the Abelian group operator. In the example given above, the additional table could simply be:

T″₁₆[x]=O(x₀)

A new function F″ (obfuscated F) could then be defined as:

$F^{″}\left(x^{″}\right)=\prod _{i=0}^{16}\otimes T_i^{″}\left[x_i^{″}\right]$

It can be observed that the domain of F″ is extended compared to the domain of F to supply input for the additional table. In the example, the domain of the F″ is now 0 . . . 2^(64+4)−1. For this example, the extended domain, indicated as E(x), can be defined as:

E(x)=(x″₀,x″₁, . . . , x″₁₄,x″₁₅,x″₁₆)=(x₀,x₁, . . . , x₁₄,x₁₅,x₀)

Using these definitions, F″ is functionally equivalent to F:

$F^{″}\left(x^{″}\right)=\prod _{i=0}^{16}\otimes T_i^{″}\left[x_i^{″}\right]=O\left(x_0\right)\otimes T_0\left[x_0\right]\otimes \prod _{i=1}^{15}\otimes T_i\left[x_i\right]\otimes O\left(x_0\right)=\prod _{i=0}^{15}\otimes T_i\left[x_i\right]=F\left(x\right)$

In this simple example, where only table T_o is obfuscated, it is relatively simple for an attacker to break the obfuscation, since the table O(x₀) that is used for the obfuscation is available in plain text form. One way to overcome this would be to use the first approach and to combine O(x₀) with a constant and use the constant also to obfuscate an odd number of other tables. A preferred way to hide O(x₀) is to hide several of the original tables T_i by a respective table O_i(x_i) and use the plurality of O-tables to hide one another. It should be noted that the respective mapping used for O_i may but need not be the same.

This approach will be illustrated using the following example. In this example, the first table T₀ is obfuscated as follows:

T″₀[x₀]=O_o(x₀)T₀[x₀]

The second table is obfuscated by:

T″₁[x]=O₁(x₁)T₁[x₁]

Keeping all the other tables unmodified:

T″_i[x_i]=T_i[x_i], ∀i:iε2 . . . 15

The new table can now be defined as:

T″₁₆[x₁₆]=O₀(x₀)O₁(x₁)

Now this seventeenth table no longer reveals one of the tables used for the hiding. It will be observed that this last table now needs two domain parts as input, giving a double extension of the domain. It will be understood that this increases the data rate and reveals part of the operation of the system.

In a preferred embodiment, O is a non-trivial homomorphism and the method includes forming a plurality of obfuscated tables T′_i through a composition, using the Abelian group operator, of the corresponding table T_i and the table O example, T′_o1[x]=O[x]T₀₁[x], xεD_o1) and forming the domain extension D′_n+1 through a composition using the Abelian operator of the corresponding subdomains of the involved tables T_i (for example, D′_n+1=D_o1D_o2D_o3). In the example below, the non-trivial homomorphism will be referred to as H(x). Any such homomorphism may be chosen. The mathematical definition of group homomorphism is as follows. Let G₁ and G₂ be groups with the operations O₁ and O₂, respectively. A mapping M from G₁to G₂ is a homomorphism if M(O₁(x,y))=O₂(M(x),M(y)) for all x,y in G₁. In the example given below, H(x) is a linear mapping on GF(2⁴) to GF(2⁶⁴ ). Applying this approach to the same example as given above, the following is obtained:

T″₀[x₀]=H(x₀)T₀[x₀]

The second table is obfuscated by:

T″₁[x₁]=H(x₁)T₁[x₁]

Keeping all the other tables unmodified:

T″_i[x_i]=T_i[x_i], ∀i: iε2 . . . 15

The new table can now be defined as:

T″₁₆[x₁₆]=H(x₀)H(x₁)

Using the homomorphism, the extended domain can now be E(x)=(x₀, x₁, . . . , x₁₄, x₁₅, x₀x₁). Only one additional domain part is used, representing multiple parts. Compared to the previous approach this reduces the data rate and increases security.

It will be appreciated that in such a way each of the tables T[x_i] can be obfuscated by a respective H(x_i). How many tables are obfuscated is a choice that can be made by the skilled person depending on the security requirements.

Preferably all tables are hidden. For the exemplary system, this would give:

T″_i[x_i]=H(x_i)T_i[x_i] ∀i:iε0 . . . 15 ∀x_i: x_iε0 . . . 15

and

T″₁₆[x₁₆]=H(x₁₆) ∀x₁₆: x₁₆ε0 . . . 15

Let

$F^{″}\left(x^{″}\right)=\prod _{i=0}^{16}\otimes T_i^{″}\left[x_i^{″}\right]$

The extended domain E(x) is given by:

E(x)=(x″₀, x″₁, . . . , x″₁₄, x″₁₅, x″₁₆), where

x″_i=x_i, for i=0 . . . 15

and

$x_{16}^{″}=\prod _{i=0}^{15}\otimes x_i$

The functions F″o E, and F are functionally equal.

$\mathrm{Proof}\text{:}$ $\left(F^{″}\circ E\right)\left(x\right)=\prod _{i=0}^{16}\otimes T_i^{″}\left[\mathrm{nibble}\left(i,E\left(x\right)\right)\right]\left[\mathrm{by}\mathrm{definition}\mathrm{of}T^{″}\right]$ $\left(F^{″}\circ E\right)\left(x\right)=H\left(\mathrm{nibble}\left(16,E\left(x\right)\right)\right)\otimes \prod _{i=0}^{15}\otimes \left(H\left(\mathrm{nibble}\left(i,E\left(x\right)\right)\right)\otimes T_i\left[\mathrm{nibble}\left(i,E\left(x\right)\right)\right]\right)\left[a\otimes b=b\otimes a,a\otimes \left(b\otimes c\right)=\left(a\otimes b\right)\otimes c\right]$ $\left(F^{″}\circ E\right)\left(x\right)=H\left(\mathrm{nibble}\left(16,E\left(x\right)\right)\right)\otimes \prod _{i=0}^{15}\otimes H\left(\mathrm{nibble}\left(i,E\left(x\right)\right)\right)\otimes \prod _{i=0}^{15}\otimes T_i\left[\mathrm{nibble}\left(i,E\left(x\right)\right)\right]\left[\mathrm{by}\mathrm{the}\mathrm{definition}\mathrm{of}E\right]$ $\left(F^{″}\circ E\right)\left(x\right)=H\left(\prod _{i=0}^{15}\otimes x_i\right)\otimes \left(\prod _{i=0}^{15}\otimes H\left(x_i\right)\right)\otimes \prod _{i=0}^{15}\otimes T_i\left[x_i\right]\left[\mathrm{by}\mathrm{the}\mathrm{definition}\mathrm{of}F\right]$ $\left(F^{″}\circ E\right)\left(x\right)=F\left(x\right)\otimes H\left(\prod _{i=0}^{15}\otimes x_i\right)\otimes \left(\prod _{i=0}^{15}\otimes H\left(x_i\right)\right)\left[\mathrm{homomorphism}\right]$ $\left(F^{″}\circ E\right)\left(x\right)=F\left(x\right)\otimes H\left(\prod _{i=0}^{15}\otimes x_i\right)\otimes H\left(\prod _{i=0}^{15}\otimes x_i\right)\left[a\otimes a=0\right]$ $\left(F^{″}\circ E\right)\left(x\right)=F\left(x\right)\otimes 0\left[a\otimes 0=0\otimes a=a\right]$ $\left(F^{″}\circ E\right)\left(x\right)=F\left(x\right)\left[\right]$

It will be appreciated that, although all examples show for the purpose of explanation that the extension is the last part of the extended domain, in fact the extension may be anywhere, i.e. the tables and the respective domain parts may be mixed.

In itself, extension of the domain might be observed and used to attack the obfuscated tables. Any suitable technique may be used to hide details of the extension. Many ciphers, like Rijndael/AES or Feistel ciphers, use a sequence of round functions (usually the same round function made specific using a round-specific key) can be modeled as function cascade: FC=F_No . . . o F₁o F₀, where F_i is the round function of round i.

In such a system, the domain extension (i.e. the part that has been extended) can be hidden in the following way:Consider a function cascade of F_j ∀_j: jε1 . . . N like:

FC=F_N o . . . o F₁o F₀

FC=(F″_N o E_n) o . . . o(F″₁ o E₁)o F₀

FC=F″_N o(E_N o F″_N−1) o . . . o(E₁o F₀)

The E_i o F″_i-1 ∀i: 1 . . . N−1 can be computed upfront, turning every T_j,i[x] of F_j in a mapping 0 . . . 15→0 . . . 2^(=+4)−1 (hiding the domain extension function by composition).

In a preferred embodiment, approach 1 and 2 are combined. Thus some, and preferably all, tables T″ are combined with a respective constant. In this way, also the table with the domain extension (in the example T₁₆) can be hidden. An advantage of combining T₁₆ with a hiding constant is that the fact that H(x) is a homomorphism is hidden, since the property H(0)=0, which hold for every homomorphism with is lost.

FIG. 3 illustrates a system in which the invention may be employed. The system 300 includes a server 310 and at least one executing device 320. The server 310 provides a cryptographic function F to the execution device 320 in an obfuscated form. The function F cryptographically converts an input x from a predetermined domain D to an output y using a plurality of non-linear mapping tables T_i(0≦i≦n; n≧1), each with key-like information and operating on a respective part D_i of the domain D. The function F composes output of the mapping tables using an Abelian group operator (for example, F(x)=T₀[x₀] . . . T_n[x_n], x=(x₀, x₁ , . . . , x_n), x_iεDi). The server may be implemented on a conventional computer platform, for example on a platform used as a server, such as a web server, or file server. The server includes a processor 312. The processor 312 is operated under control of a program. The program may be permanently embedded in the processor in an embedded storage, like embedded ROM, but may also be loaded from a background storage, such as a hard disk (not shown). Under control of the program, the processor 312:

chooses a table O and a table C such that CO=0;

creates a plurality of tables T′_i0≦i≦m; n≦m≦n+1, where for 0≦i≦n, each table T′_i represents the respective corresponding table T_i and at least one of those tables T′_o1, 0≦o₁≦n, the obfuscated table, being formed through a composition, using the Abelian group operator, of the corresponding table T_i and the table O (for example, T′_o1=OT_o1), and at least one of the other tables T′_c1, 0≦c₁≦m, c₁≠o₁, the compensating table, being formed through a composition, using the Abelian group operator, that includes the table C (for example, T′_c1=CT_c1).

Additionally, the server includes means 314 for providing the plurality of tables T′ⁱ, 0≦i≦m; n≦m≦n+1, to the executing device. The server may do this in any suitable form. For example, in a factory the tables may be stored in a storage module of the executing device during the manufacturing of the executing device 320. FIG. 3 shows that the tables are downloaded through the Internet 330 directly to the executing device 320.

The executing device 320 includes means 324 for receiving the plurality of T′_i, 0≦i≦m; n≦m≦n+1, from the server 310. These means 326 cooperate with the means 314 of the server and will not be described further. The executing device 320 further includes a processor 322. The processor may be of any suitable type, such as a processor known from personal computers or an embedded microcontroller. The processor 322 is operated under control of a program. The program may be permanently embedded in the processor 322 using an embedded storage, like embedded ROM, but may also be loaded from a background storage, such as a hard disk (not shown). Under control of the program, the processor 322 forms a function F′ that is functionally equivalent to the cryptographic function F by composing the plurality of tables T″_i, 0≦i≦m, using the Abelian group operator, such that the tables O and C are compensated through the composition. In the example of FIG. 3, the executing device 320 includes means 324 for retrieving the tables, for example, through the Internet 330 or from a storage medium 350. Similarly, the executing device 320 may retrieve encrypted content 352 from the medium 350, and decrypt this using the processor 322. The processor may also decode the decrypted content. The server 310 may have stored the encrypted content 342 in the medium 350 possibly together with the tables 344.

FIG. 4 shows a preferred embodiment wherein the tables T′ are provided to the executing device 320 embedded in a software program 410 for execution by the processor 322. Same numbers in FIG. 4 refers to the same items as used in FIG. 3. The software program 410 may be a plug-in for a program like a media player. Thus, the means 314 of FIG. 4 may supply this plug-in 410 via the Internet (e.g. item 330 of FIG. 4) or embed it directly into the executing device 320 during manufacturing.

It will be appreciated that the invention also extends to computer programs, particularly computer programs on or in a carrier, adapted for putting the invention into practice. The program may be in the form of source code, object code, a code intermediate source and object code such as partially compiled form, or in any other form suitable for use in the implementation of the method according to the invention. The carrier be any entity or device capable of carrying the program. For example, the carrier may include a storage medium, such as a ROM, for example a CD ROM or a semiconductor ROM, or a magnetic recording medium, for example a floppy disc or hard disk. Further the carrier may be a transmissible carrier such as an electrical or optical signal that may be conveyed via electrical or optical cable or by radio or other means. When the program is embodied in such a signal, the carrier may be constituted by such cable or other device or means. Alternatively, the carrier may be an integrated circuit in which the program is embedded, the integrated circuit being adapted for performing, or for use in the performance of, the relevant method.

It should be noted that the above-mentioned embodiments illustrate rather than limit the invention, and that those skilled in the art will be able to design many alternative embodiments without departing from the scope of the appended claims. In the claims, any reference signs placed between parentheses shall not be construed as limiting the claim. Use of the verb “comprise” and its conjugations does not exclude the presence of elements or steps other than those stated in a claim. The article “a” or “an” preceding an element does not exclude the presence of a plurality of such elements. The invention may be implemented by means of hardware comprising several distinct elements, and by means of a suitably programmed computer. In the device claim enumerating several means, several of these means may be embodied by one and the same item of hardware. The mere fact that certain measures are recited in mutually different dependent claims does not indicate that a combination of these measures cannot be used to advantage.

Claims

1-13. (canceled)

14. A method of providing a cryptographic function F to an execution device in obfuscated form; the function F cryptographically converting an input x from a predetermined domain D to an output y using a plurality of non-linear mapping tables Ti(0≦i≦n; n≧1), each with key-like information and operating on a respective part Di of the domain D, the function F composing output of the mapping tables using an Abelian group operator (for example, F(x)=T0[x0] . . . Tn[xn], (x0, x1 , . . . , xn), xiεDi); the method including: choosing a table O and a table C such that C[x]O[x]=0, ∀xεDi;creating a plurality of tables T′i, 0≦i≦m; n ≦m≦n+1, where for 0≦i≦n, each table T′i represents the respective corresponding table Ti and at least one of those tables T′o1, 0≦o1≦n, hereinafter referred to as obfuscated table, being formed through a composition, using the Abelian group operator, of the corresponding table To1 and the table O (for example, T′o1[x]=O[x]To1[x], ∀xεDo1), and at least one of the other tables T′c1, 0≦c1≦m, c1≠o1, hereinafter referred to as compensating table, being formed through a composition, using the Abelian group operator, that includes the table C (for example, T′c1[x]C[x]Tc1[x], ∀xεDc1);providing the plurality of tables T′i, 0≦i≦m; n≦m≦n+1 to the executing device; andat the executing device, forming a function F′ that is functionally equivalent to the cryptographic function F by composing the plurality of tables T′i, 0≦i≦m, using the Abelian group operator, such that the tables O and C are compensated through the composition.

15. A method as claimed in claim 14, wherein the tables O and C represent a same non-trivial constant b1; the step of creating the obfuscated table T′o1, 0≦o1≦n, includes using the Abelian group operator to combine the constant b1 with each element of the corresponding table To1; and the step of creating the compensating table T′c1, 0≦c1≦m, c1≠o1 includes using the Abelian group operator to combine the constant bi with each element of the corresponding table Tc1.

16. A method as claimed in claim 15, wherein m=n.

17. A method as claimed in claim 15, including: choosing a further table O2 and a further table C2 such that C2[x]O2[x]=0, ∀xεDi; O2 being distinct from O, the tables O2 and C2 representing a same non-trivial constant b2; andforming at least one further obfuscated table T′o2, 0≦o2≦n, o1≠o2 through a composition using the Abelian group operator of the corresponding table To1 and the table O2 (for example, T′o2[x]=O2[x]To2[x], ∀xεDo2) and forming the compensating table T′c1 through a composition that also includes the table C2 (for example, T′c1[x]=C2[x]C[x]Tc1[x], ∀xεDc1).

18. A method as claimed in claim 14, wherein m=n+1and c1=m.

19. A method as claimed in claim 18, wherein the obfuscated table T′ois applied the domain part Do1; O being a mapping table being applied to Do1, C[x]O[x]=0, ∀xεDo1; the method including extending the domain D to D′=(D′0, . . . , D′n, D′n+1), where Di=D′i for i<=n and Dn+1 including a representation of Do1 such that the composition that forms the compensating table T′c1 includes C[x], xεDo1.

20. A method as claimed in claim 19, wherein O is a homomorphism and the method includes forming a plurality of obfuscated tables T′i through a composition, using the Abelian group operator, of the corresponding table Ti and the table O (for example, T′i[x]=O[x]Ti[x], xεDi) and forming the domain extension D′n+1 through a composition using the Abelian operator of the corresponding subdomains of the involved tables Ti (for example,

21. A system (300) for providing a cryptographic function F to an execution device (320) in an obfuscated form; the system including a server (310) and an executing device (320); the function F cryptographically converting an input x from a predetermined domain D to an output y using a plurality of non-linear mapping tables Ti(0≦i≦n; n≧1), each with key-like information and operating on a respective part Di of the domain D, the function F composing output of the mapping tables using an Abelian group operator (for example,

22. A system (300) as claimed in claim 21, wherein the program in the server (310) is operative to cause the processor (312) to form the program for the execution device (320) where the program for the execution device includes the plurality of tables T′i, 0≦i≦m; n≦m≦n+1, and includes instructions for causing the processor of the execution device to form the obfuscated function F′.

23. A server (320) for use in the system (300) as claimed in claim 21, wherein a cryptographic function F is provided to an execution device in an obfuscated form; the function F cryptographically converting an input x from a predetermined domain D to an output y using a plurality of non-linear mapping tables Ti(0≦i≦n; n≧1), each with key-like information and operating on a respective part Di of the domain D, the function F composing output of the mapping tables using an Abelian group operator (for example,

24. A computer program product for controlling a processor (312) in a server (310) as claimed in claim 23, wherein a cryptographic function F is provided to an execution device (320) in an obfuscated form; the function F cryptographically converting an input x from a predetermined domain D to an output y using a plurality of non-linear mapping tables Ti(0≦i≦n; n≧1), each with key-like information and operating on a respective part Di of the domain D, the function F composing output of the mapping tables using an Abelian group operator (for example, F(x)=T0[x0] . . . Tn[xn], x=(x0, x1 , . . . , xn), xiεDi); the computer program product including program instructions to cause the processor to choosing a table O and a table C such that C[x]O[x]=0, ∀xεDi; and to create a plurality of tables T′i, 0≦i≦m; n≦m≦n+1, where for 0≦i≦n, each table T′i represents the respective corresponding table Ti and at least one of those tables T′o1, 0≦o1≦n, hereinafter referred to as obfuscated table, being formed through a composition, using the Abelian group operator, of the corresponding table To1 and the table O (for example, T′o1[x]=O[x]To1[x], ∀xεDo1), and at least one of the other tables T′c1, 0≦c1≦m, c1≠o1, hereinafter referred to as compensating table, being formed through a composition, using the Abelian group operator, that includes the table C (for example, T′c1[x]=C[x]Tc1[x], ∀xεDc1).

25. An execution device (320) for use in the system (300) as claimed in claim 21, wherein a cryptographic function F is provided to the execution device in an obfuscated form; the function F cryptographically converting an input x from a predetermined domain D to an output y using a plurality of non-linear mapping tables Ti(0≦i≦n; n≧1), each with key-like information and operating on a respective part Di of the domain D, the function F composing output of the mapping tables using an Abelian group operator (for example,

26. A computer program product for controlling a processor (322) in an execution device (320), wherein a cryptographic function F is provided to the execution device in an obfuscated form; the function F cryptographically converting an input x from a predetermined domain D to an output y using a plurality of non-linear mapping tables Ti(0≦i≦n; n≧1), each with key-like information and operating on a respective part Di of the domain D, the function F composing output of the mapping tables using an Abelian group operator (for example,