ELECTRONIC CALCULATING DEVICE FOR PERFORMING OBFUSCATED ARITHMETIC
20170324544 · 2017-11-09
Inventors
- Leandro Marin (Eindhoven, NL)
- Alphons Antonius Maria Lambertus Bruekers (Nederweert, NL)
- Paulus Mathias Hubertus Mechtildis Antonius Gorissen (Eindhoven, NL)
Cpc classification
International classification
Abstract
An electronic calculating device (100) for performing obfuscated arithmetic in a commutative ring (Z.sub.n; Z.sub.n[x]/f(x)) is presented. The calculating device comprising a storage (110) arranged to store an increment table (T) defined for an increment ring element (1; u.sup.t), the increment table mapping an input ring element (k=u.sup.k1−u.sup.k2) to an output integer-list (T((k.sub.1,k.sub.2))=(l.sub.1, l.sub.2)) encoding an output ring element (I=u.sup.l1−u.sup.l2), such that the output ring element equals the increment ring element ring-added to the input ring element (I=k+1). Using the increment table, a ring addition unit (130) adds a first addition-input integer-list ((a.sub.1, a.sub.2)) encoding a first addition-input ring element and a second addition-input integer list ((b.sub.1, b.sub.2)) encoding a second addition-input ring element. The device may comprise a ring multiplication unit (140) also using the increment table.
Claims
1. An electronic calculating device for performing obfuscated arithmetic in a commutative ring (.sub.n;
.sub.n[x]/f(x)), the ring having a finite number of ring elements, a ring-addition and a ring-multiplication being defined on the ring elements, the calculating device operating on integer-lists (a.sub.1/a.sub.2)) encoding ring elements (u.sup.a.sup.
2. A calculating device as in claim 1, wherein the size of the allowed-differences list is less than 50% the order, more preferably less than 20% the order, even more preferably, less than 10% the order, even more preferably less than 5% the order.
3. A calculating device as in claim 1, wherein the base ring element has an order, and the order (k) times the size of the allowed-differences list minus one (c−1) divided by the size of the ring minus one (N−1) is less than 2, more preferably equals 1
4. A calculating device as in claim 1, wherein the base ring element has an order, and the order (k) times the size of the allowed-differences list minus one (c−1) divided by the size of the ring minus one (N−1) is more or equal than two
5. A calculation device as in claim 1 wherein the increment table takes as input an input integer-lists (k.sub.1, k.sub.2) representing the input ring element, wherein the arithmetical difference between a first integer (k.sub.1) of the input integer-lists ((k.sub.1, k.sub.2)) and a second integer (k.sub.2) of the integer list is comprised in the allowed-differences list, or an integer (k.sub.1; k.sub.2) of the input integer-lists ((k.sub.1, k.sub.2)) is comprised in the allowed-differences list.
6. A calculating device as in claim 1, comprising a ring addition unit arranged to receive a first addition-input integer-list ((a.sub.1, a.sub.2)) encoding a first addition-input ring element and a second addition-input integer list (b.sub.1, b.sub.2)) encoding a second addition-input ring element, determining an addition-output integer-list encoding an addition-output ring element by applying the increment table to ring elements determined from the first and second addition-input integer-lists, the addition-output ring element being equal to the ring-addition of the first addition-input ring element and the second addition-input ring element.
7. A calculation device as in as in claim 6, wherein determining the addition-output integer-list comprises determining an intermediate addition integer-list ((c.sub.1,c.sub.2)) encoding an intermediate addition ring element by a first application of the increment table to a ring element (u.sup.a.sup.
8. A calculating device as in claim 1, comprising a ring multiplication unit arranged to receive a first multiplication-input integer-list ((r.sub.1, r.sub.2)) encoding a first multiplication-input ring element and a second multiplication-input integer list ((s.sub.1,s.sub.2)) encoding a second multiplication-input ring element, determining a multiplication-output integer-list encoding a multiplication-output ring element by applying the increment table to ring elements determined from the first and second multiplication-input integer-lists, the multiplication-output ring element being equal to the ring-multiplication of the first multiplication-input ring element and the second multiplication-input ring element.
9. A calculation device as in claim 1, wherein the commutative ring is a ring formed by the integers modulo an integer modulus (.sub.n), and/or the commutative ring is a ring formed by the integer polynomials modulo an integer polynomial modulus (
.sub.n[x]/f(x)).
10. A ring encoding device for encoding a ring element of a commutative ring (.sub.n;
.sub.n [x]/f(x)) as an integer list, for use with a calculation device as in claim 1, the ring encoding device comprising a storage arranged to store an encoding table defined for a base ring element (u), the encoding table mapping the ring element (x) to an integer-list ((a, b)) such that the ring element equals a linear combination of powers of the base ring element (x=u.sup.a−u.sup.b), wherein the powers have exponents determined by the integer-list, a difference between a first exponent (a.sub.1) and a second exponent (a.sub.2) is comprised in an allowed-differences list (A), and wherein the size of the allowed-differences list is less than the order (k).
11. A ring decoding device for decoding an integer-list ((a,b)) to a ring element (x) of a commutative ring (.sub.n;
.sub.n[x]/f(x)), for use with a calculation device as in claim 1, the ring decoding device being arranged to determine for one or more base ring element (u), the ring element (x) such that the ring element equals a linear combination of powers of one or more base ring element (x=u.sup.a−u.sup.b), wherein the powers have exponents determined by the integer-list, a difference between a first exponent (a.sub.1) and a second exponent (a.sub.2) is comprised in an allowed-differences list (A), and wherein the size of the allowed-differences list is less than the order (k).
12. A table computation device for computing an increment table for use in a calculating device for performing obfuscated arithmetic in a commutative ring (.sub.n;
.sub.n[x]/f(x)), the ring having a finite number of ring elements, a ring-addition and a ring-multiplication being defined on the ring elements, the calculating device operating on integer-lists ((a.sub.1, a.sub.2)) encoding ring elements (u.sup.a.sup.
13. An electronic configuration device for configuring an electronic calculating device (100) for performing obfuscated arithmetic in a commutative ring (.sub.n;
.sub.n[x]/f(x)) as in claim 1, providing a ring having a finite number of ring elements, providing a base ring element having an order k, defining an equivalence relation on
.sub.k such that two elements r.sub.1 and r.sub.2 are equivalent if and only if (u.sup.r.sup.
.sub.k, the equivalence relation inducing equivalence classes on
.sub.k, determining an allowed-differences list by selecting at least one element from each equivalence class.
14. An electronic calculating method for performing obfuscated arithmetic in a commutative ring (.sub.n;
.sub.n[x]/f(x)), the ring having a finite number of ring elements, a ring-addition and a ring-multiplication being defined on the ring elements, the calculating method operating on integer-lists ((a.sub.1, a.sub.2)) encoding ring elements (u.sup.a.sup.
15. An electronic calculating method as in claim 14, comprising ring adding, the ring adding comprising receiving a first addition-input integer-list ((a.sub.1,a.sub.2)) encoding a first addition-input ring element and a second addition-input integer list ((b.sub.1,b.sub.2)) encoding a second addition-input ring element, determining an addition-output integer-list encoding an addition-output ring element by applying the increment table to ring elements determined from the first and second addition-input integer-lists, the addition-output ring element being equal to the ring-addition of the first addition-input ring element and the second addition-input ring element,
16. A computer program comprising computer program instructions arranged to perform the method of claim 14 when the computer program is run on a programmable apparatus.
17. A computer readable medium comprising the computer program according to claim 16, and/or an increment table for performing obfuscated arithmetic in a commutative ring (.sub.n;
.sub.n[x]/f(x)), the ring having a finite number of ring elements, a ring-addition and a ring-multiplication being defined on the ring elements, integer-lists ((a.sub.1, a.sub.2)) encoding ring elements (u.sup.a.sup.
Description
BRIEF DESCRIPTION OF THE DRAWINGS
[0018] These and other aspects of the invention are apparent from and will be elucidated with reference to the embodiments described hereinafter. In the drawings,
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[0029] Items which have the same reference numbers in different figures, have the same structural features and the same functions, or are the same signals. Where the function and/or structure of such an item has been explained, there is no necessity for repeated explanation thereof in the detailed description.
LIST OF REFERENCE NUMERALS IN FIG. 1
[0030] 100 a calculating device [0031] 110 a storage arranged to store an increment table [0032] 120 a ring negating unit [0033] 130 a ring addition unit [0034] 140 a ring multiplication unit [0035] 150 an operand store [0036] 160 a decoding unit [0037] 170 an encoding unit [0038] 172 a storage arranged to store an encoding table
DETAILED DESCRIPTION OF EMBODIMENTS
[0039] While this invention is susceptible of embodiment in many different forms, there is shown in the drawings and will herein be described in detail one or more specific embodiments, with the understanding that the present disclosure is to be considered as exemplary of the principles of the invention and not intended to limit the invention to the specific embodiments shown and described.
[0040] In the following, for sake of understanding, elements of embodiments are described in operation. However, it will be apparent that the respective elements are arranged to perform the functions being described as performed by them.
[0041] Embodiments allow multiplication and addition operations to be performed using the same table. Below, first a number of possible architectures of embodiments of calculating devices are discussed. Next, a number of alternative ways to perform obfuscated arithmetic are discussed.
[0042] .sub.n) and the polynomials modulo a number and a polynomial (
.sub.n[x]/f(x)). Other embodiment may use other commutative rings.
[0043] The elements of the ring are referred to as ring elements. On the ring elements an addition and a multiplication is defined, the latter are referred to as the ring-addition and the ring-multiplication.
[0044] Ring elements may be represented in any suitable form, should that be needed. For example, elements of .sub.n may be represented as integers; elements of
.sub.n[x]/f(x) as polynomials. However, in calculation device 100, ring elements are represented as integer-lists. For example, a ring element a may be represented in calculation device 100 by a list (a.sub.1, a.sub.2). The latter even holds for non-integer rings, say polynomial rings. An integer-list encodes a ring-element according to some mapping between ring elements and integers lists; given any ring-element there is at least one integer list that represents the ring-element, and given any integer list, there is exactly one ring element that it represents. In embodiments any ring element may be represented as an integer list.
[0045] The integer-lists comprising at least two integers, and encode a ring element, such that the ring element equals a linear combination of powers of base ring element wherein the powers have exponents determined by the integer-list. For example, the integer-list (a.sub.1, a.sub.2) may encode the ring element a such that the ring element equals a linear combination of powers, e.g., a=u.sup.a.sup.
[0046] The difference between the exponents is comprised in an allowed-differences list. If the exponents themselves are in the integer list then a first integer, e.g., a.sub.1, of the integer-list and a second integer, e.g., a.sub.2 of the integer list is comprised in an allowed-differences list. On the other hand this is not necessary. For example, the integer list (a.sub.1, a.sub.2) may also map to a=u.sup.a.sup.
The size of the allowed-differences list is less than the order of the base element. This means that the number of choices for an integer list is reduced. For example, given a single base ring element u with order k, the total number of different representations would be k.sup.2. However when the difference is required to be in an allowed-differences list of size c, the total number of representations becomes kc which is less than k.sup.2. We will use the letter A to denote the allowed-differences list.
[0047] The integer lists have at least two elements. As it turns out, the addition and multiplication operations require fewer steps if the integer list is shorter. Accordingly, in an embodiment the integer lists always have two elements. In the main description we will assume that the integer lists are integer pairs, however, examples of integer lists having more than two elements are provided.
[0048] As an example, the integer list (a.sub.1, a.sub.2) may map to the ring element (u.sup.a.sup.
[0049] The allowed-differences list A may be regarded as a subset of .sub.k. The difference allowed-differences list may be represented as a list, say an array, or positive integers. Verifying if a difference is in the list may require performing an modulo k operation. However, in practical implementation the use of the allowed-differences list will be implicit; encoding operations produce integer lists satisfying the condition, operations on the integer lists produce as output integer lists that satisfy the condition.
[0050] In an embodiment, the integers in an integers list are non-negative. This simplifies the computation, but is not necessary. Furthermore, in an embodiment, the integers in the integer list are taken modulo the order of a base element. The order of a base element u is the smallest integer k such that u.sup.k=1. It is convenient to keep the values in an integer list in the range [0, k−1], e.g., by performing modulo k operations.
[0051] Calculation device 100 may comprise an operand store 150. Operands are stored as integer lists in operand store 150. Arithmetic may be performed on operands stored in operand store 150. Results of said arithmetic may be stored in operand store 150, where they may be used in new operations, or may be output to a different device, etc.
[0052] An integer lists, e.g., the operands in operand store 150 may be represented in several different ways. An integer list may be represented as a list of integers, say an array. For example, the arithmetical difference between a first integer of the integer-lists and a second integer of the integer list is comprised in the allowed-differences list. For example, taking the preferred case of integer lists having two elements, the integer list may be represented as the ordered list (k.sub.1, k.sub.2), such that k.sub.1−k.sub.2εA. In this example, both parameters k.sub.1 and k.sub.2 may be chosen mod k, i.e., modulo the order of the base ring element. This is not strictly necessary, the parameters may be any integer, in this case, the requirement may be implemented, e.g., as (k.sub.1-k.sub.2) mod kεA.
[0053] As an another example, an increment list may be represented by choosing one of the integers in the integer list from the allowed-differences list. For example, this representation may take an integer-list (k.sub.1, k.sub.2) wherein say k.sub.2 is comprised in the allowed-differences list, the other integer or integers may be taken modulo the order k of the base ring element u. The allowed-differences list may be significantly smaller than the order, in this representation, fewer bits may be required to represent elements of A, thus reducing space requirements. In an embodiment, the integer in the integer list that is required to be a member of the allowed-differences list A is at a fixed position, say the first, or second position. But the latter is not necessary; the position may change as a further obfuscation measure.
[0054] Calculation device 100 comprises a storage 110 arranged to store an increment table T defined for an increment ring element. The increment table maps an input ring element to an output integer-list encoding an output ring element, such that the output ring element equals the increment ring element ring-added to the input ring element. In an embodiment, the input ring element is represented as an integer list. In this way, table T maps integer lists to integer lists; both according to the same encoding, e.g. the same mapping. However, there are embodiments in which the input ring element is represented as an integer list in an alternative encoding. In any case, the input ring element is represented in digital form, allowing the table to map the input ring element to an output ring element.
[0055] The output integer-list of the increment table satisfies the requirement that the output ring element which the output integer-list represents equals a linear combination of powers of the base ring element (u), wherein the powers have exponents determined by the output integer-list, a difference between a first exponent (l.sub.1) and a second exponent (l.sub.2) is comprised in an allowed-differences list (A). This will be referred to as the limited-differences requirement. In embodiments, the limited-differences requirements translates to a requirement on the integers in the integer list, e.g., that a function of the integers in the integer list is comprised in the allowed differences list.
[0056] The table may list input ring elements, in some format, together with the associated output integer list. The table may also be represented in storage by omitting the input ring and only listing the output integer lists. For example, this may be done if the input ring is represented in a canonical format.
[0057] For example, assuming the example encoding, the input ring element k=u.sup.k.sup.
[0058] Not all possible combinations of input integer list need to be represented in the table, as many integer lists do not represent ring elements according to the limited differences requirement.
[0059] The increment table is much smaller than the tables described in the background. The latter tables take two inputs, e.g., two encoded numbers to produce an encoded output. However, the table T takes only one encoded input to produce one encoded output; the increment ring element is fixed. Assuming the encodings take a similar amount of space, the input space of T is reduced to about the square root. This is a significant size improvement.
[0060] With only the increment table, the calculation device can represent ring elements in an obfuscated fashion, and can add the increment elements to such an encoded ring element. This may be used, e.g., to implement a counter.
[0061] In an embodiment, calculation device 100 comprises a ring addition unit 130 and a ring multiplication unit 140. Calculation device 100 may also comprise a ring negation unit 120. In an embodiment, the ring multiplication unit 140 may use addition unit 130 to perform additions; addition unit 130 may use negating unit 120. This has been indicated in
[0062] Negation unit 120 may receive a negation-input integer list (a.sub.1, a.sub.2) encoding a negation-input ring element a. Negation unit 120 is arranged to determine an negation output integer list (b.sub.1, b.sub.2) encoding a negation-output ring element b. The negation output ring element is the negative of the negation input ring element, e.g., the negation output ring element equals the neutral ring element for the addition (0) minus the negation input ring element. Thus b=−a.
[0063] In an embodiment, the negation unit may compute the output integer list by permuting the negation input integer list. Using the example encoding, a=u.sup.a.sup.
[0064] In an embodiment, the negation unit may compute the output integer list by adding a constant to each integer of the integer list. For example, in the example encoding using an integer m such that u.sup.m=−1; for example the output integer list may be (a.sub.1+m, a.sub.2+m).
[0065] In an embodiment, the negation unit comprises a negation table. The negation table is configured to receive a negation-input integer-list ((a.sub.1, a.sub.2)) encoding a negation-input ring element a and to produce a negation-output integer list ((b.sub.1, b.sub.2)) encoding a negation-output ring element b, such that b=−a. The negation table is not very large as it is unary, moreover the negation table needs only to contain the negation-output integer list for input that satisfy the limited-differences requirement on the integer lists. The negation table is used in the same manner during addition and multiplication operations.
[0066] Ring addition unit 130 is arranged to receive a first addition-input integer-list (a.sub.1, a.sub.2) encoding a first addition-input ring element and a second addition-input integer list (b.sub.1, b.sub.2) encoding a second addition-input ring element. For example, ring addition unit 130 may receive the two operands from operand store 150. Ring addition unit 130 is arranged to determine an addition-output integer-list encoding an addition-output ring element by applying the increment table to ring elements determined from the first and second addition-input integer-lists, the addition-output ring element being equal to the ring-addition of the first addition-input ring element and the second addition-input ring element.
[0067] In an embodiment, the mapping of an integer list to a particular ring element comprises multiple sub-mapping, each sub-mapping being associated with an integer of the integer list, a sub-mapping mapping an integer to a ring element. The mapping is a linear combination, e.g., a sum, of the sub-mappings applied to the associated integers. A sub-mapping may be raising a base element to a power determined by the associated integer. For example, in the example encoding, (a.sub.1, a.sub.2) may be said to be a sum of the sub-mappings u.sup.a.sup.
[0068]
[0069] Ring multiplication unit 140 is arranged to receive a first multiplication-input integer-list (r.sub.1, r.sub.2) encoding a first multiplication-input ring element and a second multiplication-input integer list (s.sub.1, s.sub.2) encoding a second multiplication-input ring element. For example, multiplication unit 140 may receive two operands from operand store 150. Ring multiplication unit 140 is arranged to determine a multiplication-output integer-list encoding a multiplication-output ring element by applying the increment table to ring elements determined from the first and second multiplication-input integer-lists, the multiplication-output ring element being equal to the ring-multiplication of the first multiplication-input ring element and the second multiplication-input ring element.
[0070]
[0071] Calculation device 100 optionally comprise a ring encoding unit 170 for encoding a ring element of the commutative ring as an integer list, and a ring decoding unit 160 for decoding an integer-list (a, b) to a ring element (x) of the commutative ring. Encoding unit 170 and/or decoding unit 160 may be absent, e.g., when calculating device 100 receives encoded input and/or reports in encoded output. Encoding unit 170 and/or decoding unit 160 may be implemented as a standalone unit, e.g., as encoding device and/or decoding device 160.
[0072] The ring encoding unit 170 may comprise a storage 172 arranged to store an encoding table defined for a base ring element (u), the encoding table mapping the ring element (x) to an integer-list ((a, b)) such that the ring element equals a linear combination of powers of the base ring element (x=u.sup.a−u.sup.b), wherein the powers have exponents determined by the integer-list. Encoding unit 170 may store the encoded ring element in operator store 150. Encoding unit 170 allows the system to work with plain information.
[0073] The encoding table mapping the ring element (x) to an integer-list ((a, b)) such that the ring element equals a linear combination of powers of the base ring element (x=u.sup.a−u.sup.b), wherein the powers have exponents determined by the integer-list, a difference between a first exponent (a.sub.1) and a second exponent (a.sub.2) is comprised in an allowed-differences list (A), and wherein the size of the allowed-differences list is less than the order (k).
[0074] The ring decoding unit 160 is arranged to determine for one or more base ring element (u), the ring element (x) such that the ring element equals a linear combination of powers of one or more base ring element (x=u.sup.a−u.sup.b), wherein the powers have exponents determined by the integer-list. For example, decoding unit 160 may comprise a store storing a decoding table mapping integer lists to ring elements. For example, decoding unit 160 may comprise a computation unit to compute the powers and the linear combination thereof.
[0075] The decoding table need only map integer list that satisfy the limited-differences requirement, so that the decoding table has a smaller size than a decoding table that needs to maps all possible differences.
[0076] Many interesting embodiments omit one or both of the encoding and decoding units 160 and 170. For example, calculation device 100 may be configured to receive encoded information over a computer network, say the Internet. The proprietor of the system on which obfuscated calculation device 100 runs, e.g. a computer executing obfuscated calculation software may not know the encoding used for the input information, nor for information outputted by the system 100, e.g., transmitted back over the computer network. Accordingly, even though the calculations are performed in the cloud, the owner of the information has some assurance that his information is safe. Operating on information in encoded form is typically not possible using cryptography, say encryption. Even if a table system is used as outlined in the background this requires dual tables.
[0077] Typically, calculation device 100 comprises a microprocessor (not shown) which executes appropriate software stored at device 100; for example, that software may have been downloaded and/or stored in a corresponding memory, e.g., a volatile memory such as RAM or a non-volatile memory such as Flash (not shown). Alternatively, device 100 may, in whole or in part, be implemented in programmable logic, e.g., as field-programmable gate array (FPGA). Device 100 may be implemented, in whole or in part, as a so-called application-specific integrated circuit (ASIC), i.e. an integrated circuit (IC) customized for their particular use.
[0078] In an embodiment, the electronic calculating device comprises a ring addition circuit and a ring multiplication circuit arranged to execute the function of the corresponding unit. The calculation device may also comprise a negation circuit. The circuit may be integrated circuits, such as CMOS, e.g. obtained by describing the functions in a hardware description language, such as Verilog and VHDL. The circuits may be a processor circuit and storage circuit, the processor circuit executing instructions represented electronically in the storage circuits. The circuits may also be FPGA, ASIC or the like.
[0079] The table store 110 and operand store 150 may be implemented as an electronic storage, e.g. a memory. Both stores may be part of the same memory, but they may be distinct memories. Table store 110 may be a non-volatile, non-writable, say a ROM, or Write once read many (WORM) memory. Operand store 150 may be a volatile or non-volatile writable memory, say Flash or RAM.
[0080]
[0081] The operations of multiple arithmetic units, e.g., addition, multiplication, negation, may be order in any order allowed by their data dependencies. For example, operation 140.3 may be inserted in the ordering 140.1, 140.2, 130.1, and 130.2 at any point before 130.1. Moreover the ordering of subsequent multiplications or additions may be reversed. Thus a diagram like diagram 2 may be translated in a linear ordering for a software program in many ways. It is not needed that the units are strictly separated; instructions for a first unit may be interspersed with instructions for another unit.
[0082]
For example, table computation device 200 may comprise an encoding unit like encoding unit 170. [0086] add an entry to increment table mapping the input ring element to the output integer-list satisfying the limited-difference requirement.
[0087] These steps may be performed until all ring elements have been mapped to an integer list. In some embodiments, elements may be skipped, building a partial increment table; e.g., it may be known from the context that certain ring elements will not occur.
[0088] Given a ring R, a potential base ring element u, an encoding, say the example encoding, and an integer list length, say 2, a decoding table may be generated as given below. Let k be the order of u. [0089] generate all integer lists satisfying the limited-difference requirement, e.g., by generating all integer lists of the integer list length, having integers mod k, and a first integer of the integer-list and a integer of the output integer-list being comprised in the allowed-differences list A.
[0090] For example, for an integer list of length two the following example algorithm may be used: for k1 from 0 to (k−1); for all d in A; generate (k1, k1+d mod k). For example, for a difference list A={0, 1, 4, 5, 902, 903, 906}, one may generate, generate: (0,0), (0,1), (0, 4), . . . , (1, 1), (1, 5), . . . etc. This example, differences list may be used with the ring Z.sub.907, U=10, k=151. [0091] for each generated integer list compute the ring element encoded by the integer list, and add an entry to the decoding table associating the integer list to the decoding. For example, continuing the above example, one may compute 151.sup.1−151.sup.5 mod 907=76. Thus the decoding table may receive the entry that (1,5) decodes to 76.
[0092] Although decoding may or may not use a decoding table such a table is also useful since an encoding table may be generated from a decoding table, e.g., by sorting the table for the ring elements. It may happen that a ring element has multiple encodings. For example the ring element 0 (neutral element for addition) may be represented as (a, a) in the example encoding for any a. Such multiple encodings may be removed from the table, e.g., by deleting all but 1 of the multiple entries for a given ring element; or by leaving the multiple encodings in the table and using the encoding table to encode to a random one of the multiple entries.
[0093] Constructing a decoding or encoding table may also be used to find out if a ring element u is a base ring element. If constructing an encoding table fails, because it turns out that some ring elements do not have an encoding, then u is not a base ring element.
[0094] Below a number of embodiments of encodings, increment tables, ring addition methods and ring multiplication methods are presented. The negation, addition and multiplication units of calculation device 100 may be configured for any of these embodiments. All examples apply to any commutative ring, in particular .sub.n and
.sub.n[x]/f(x). Herein is n a positive integer. Furthermore, it is greatly preferred that any element of the commutative ring may be represented in the chosen encoding. Not all commutative rings allow all elements to be represented in a given encoding, e.g., as a given type of integer list representation. Given a commutative ring R we will say that it allows full homogenous obfuscation if any element in R may be represented as an integer list using a given encoding type. The person skilled in the art can verify if a given commutative ring allows full homogenous obfuscation given an encoding, e.g., by generating all allowable encodings and verifying that together they represent all elements of a given ring. For some applications it may be allowed that the encoding has some gaps. This could have as a consequence that arithmetic cannot be performed on those gaps, at least not using the obfuscated integer list encoding. Specific examples of commutative rings allowing specific types of encodings are presented further below.
[0095] Below first a description is given of the example encoding in which the parameters are restricted to lie in the allowed-differences list. There are many types of encodings, which have in common that ring elements may be represented as lists of integers. These integers are not ring elements, e.g., even if the ring is not an integer ring say a polynomial ring, then nevertheless elements may be represented as integer lists. The encoding used, how a given integer list maps to a ring element is referred to as the encoding. Typically, the integer lists will always be of the same length, however this is not necessary. Generally, as the encoding allows more types of integer lists, e.g., longer lists, it becomes more likely that a given ring element may be encoded as an integer list in different ways. Given a commutative ring R, that has a base ring element u, such that any element a of R may be written as u.sup.a.sup.
[0096] An increment table T plays a central role both in the addition and multiplication operation. The increment table maps an input ring element, in this case an input ring element may be represented as an integer list. For example, given an input integer list (k.sub.1, k.sub.2) representing the input ring element k=u.sup.k.sup.
[0097] Below the operations, negation, addition, and multiplication are described.
Negation.
[0098] Given a negation-input integer-list (a.sub.1, a.sub.2) representing the negation input ring element a=u.sup.a.sup.
Addition.
[0099] To add received first addition-input integer-list (a.sub.1, a.sub.2) encoding a first addition-input ring element a=u.sup.a.sup.
[0100] The ring element c may be the first addition-input ring element a plus the base element u to a power determined from the second addition input integer list, in particular a first integer of the second addition input integer list. In this example, we may have c=u.sup.a.sup.
[0101] Through a first application of the increment table to the ring element u.sup.a.sup.
[0102] Thus the addition unit has obtained intermediate addition ring element c=u.sup.a.sup.
[0103] In this example, the addition-output integer-list may be determined through a second application of the increment table to ring elements determined from the intermediate addition integer-list and the second addition-input integer-list. This may comprise computing the sum of intermediate addition ring element c and minus the base element raised to a power determined from the second addition input integer list, e.g. the second integer of the second addition input integer list b.sub.2: c−u.sup.b.sup.
[0104] Subtracting −u.sup.b.sup.
Multiplication.
[0105] To multiply the received first multiplication-input integer-list (r.sub.1, r.sub.2) encoding a first multiplication-input ring element r=u.sup.r.sup.
[0106] Thus to multiply two ring elements represented as integer lists they may be transformed into two new integer lists that can be added to obtain the answer to the multiplication. The addition may be done as described above. For example, the multiplication unit may compute the intermediate integer lists and send them to the multiplication unit. For example, a first integer t.sub.1 of the first intermediate multiplication integer-list may comprise a first integer r.sub.1 of the first multiplication-input integer-list plus a first integer s.sub.1 of the second multiplication-input integer-list, and a second integer t.sub.2 of the first intermediate multiplication integer-list may comprise a first integer r.sub.1 of the first multiplication-input integer-list plus a second integer s.sub.2 of the second multiplication-input integer-list t.sub.1=r.sub.1+s.sub.1, t.sub.2=r.sub.1+s.sub.2; A first integer u.sub.1 of the second intermediate multiplication integer-list may comprise a second integer r.sub.2 of the first multiplication-input integer-list plus a second integer s.sub.2 of the second multiplication-input integer-list, and a second integer u.sub.2 of the second intermediate multiplication integer-list may comprise a second integer r.sub.2 of the first multiplication-input integer-list plus a first integer s.sub.1 of the second multiplication-input integer-list u.sub.1=r.sub.2+s.sub.2, u.sub.2=r.sub.2+s.sub.1,
[0107] In an embodiment, e.g., in the example just disclosed, the arithmetic is performed on integer lists, the ring elements do not need to be calculated as ring elements in some natural representation. Now a number of the variants are discussed. Many of the variants are independent, e.g., a variant encoding may be combined with a variant to perform addition.
[0108] Through the obfuscated arithmetic when calculations are performed in the integer list, corresponding e.g. to u.sup.a.sup.
The Increment Value.
[0109] The increment value need not be 1. There are at least two ways to use a different increment value. First, the equation c=u.sup.a.sup.
[0110] The principles illustrated for the example encoding may be applied to number of alternative encodings. A first alternative encoding is to encode a ring element a as integer list (a.sub.1, a.sub.2) using the encoding a=u.sup.a.sup.
[0111] Positive reduction may be applied twice, both for u.sup.b.sup.
The encodings given so far may be optionally multiplied by a constant ring element w=u.sup.v for some v. Thus an integer list (a.sub.1, a.sub.2) may represent the ring element a=w(u.sup.a.sup.
The Number of Integers in an Integer List.
[0112] In the example discussed so far the number of elements in the integer list was always two. This number has advantages, i.e., it reduces the number of computation steps. On the other hand allowing more elements in the integer list extends the number of rings that allow the obfuscation. The example, below considers three integers per list, but more is possible and works analogously.
[0113] Consider the first integer list (a.sub.1, a.sub.2, a.sub.3) and second integer list (b.sub.1, b.sub.2, b.sub.3) encoding the elements u.sup.a.sup.
[0114] If the integer list (a.sub.1, a.sub.2, a.sub.3) satisfies that two particular numbers of the list have a difference in the allowed differences list, e.g., a.sub.1−a.sub.2εA.sub.1 then so does (u.sup.a.sup.
[0115] Below examples for rings allowing negative and/or positive obfuscation are given, and how to find allowed-differences list. The ring R may be integer ring .sub.n, for a modulus n. The ring R may also be a polynomial ring
.sub.n[x] if (x) for some modulus n and reducing polynomial f(x).
[0116] Let R be a ring with obfuscatable arithmetic and base u, and let k=order (u). Two elements r.sub.1 and r.sub.2 both in .sub.k will be called equivalent if there exists tΣ
.sub.k such that (u.sup.r.sup.
.sub.151, the base element u=2, having order k=15. The elements 2 and 8 are equivalent since (2.sup.2−1)2.sup.11=2.sup.8−1. Likewise, the elements 4 and 10 are equivalent since (2.sup.4−1)2.sup.13=2.sup.10−1.
[0117] One may proof mathematically that if R is a ring such that u.sup.m=−1, then r and −r are always equivalent.
[0118] This equivalence relation induces equivalence classes on .sub.k, e.g., sets in which all elements are equivalent to each other. An allowed-differences list can now be determined by selecting the same number of elements of each equivalence class except from the equivalence class comprising 0. The element 0 will also be included in the allowed differences set, if it is needed to represent the ring element 0. This method for configuring an electronic calculating device (100) for performing obfuscated arithmetic in a commutative ring, may be executed by electronic configuration device, e.g., comprising a microprocessor and software for executing the software.
[0119] We provide the following examples:
Let R=F.sub.2.sub..sub.2 [x]/f(x), wherein f(x)=x.sup.8+x.sup.4+x.sup.3+x+1, and with the base ring element u=x.sup.5+x.sup.4+x+1, having order 51 and which is a base for obfuscated arithmetic in the field R. Note that since R has characteristic 2, positive and negative obfuscation are the same. The equivalence classes are:
[0]
[1; 3; 11; 16; 21; 23; 28; 30; 35; 40; 48; 50]
[2; 5; 6; 9; 19; 22; 29; 32; 42; 45; 46; 49]
[4; 7; 10; 12; 13; 18; 33; 38; 39; 41; 44; 47]
[8; 14; 15; 20; 24; 25; 26; 27; 31; 36; 37; 43]
[17; 34]
[0120] An allowed differences set may be obtained by selecting at least one element from each class, e.g., A={0, 1, 2, 4, 8, 17}.
[0121] Let R=F.sub.2.sub..sub.2 [x] if (x), wherein f(x)=x.sup.12+x.sup.7+x.sup.6+x.sup.5+x.sup.3+x+1, and with the base ring element u=x.sup.11+x.sup.10+x.sup.7+x.sup.5+x.sup.3+x.sup.2+x+1, having order 195 and which is a base for obfuscated arithmetic in the field R. The equivalence classes are: [0122] [0] [0123] [13; 26; 39; 52; 65; 78; 91; 104; 117; 130; 143; 156; 169; 182] [0124] [1; 8; 64; 73; 84; 87; 108; 111; 122; 131; 187; 194] [0125] [2; 16; 21; 27; 49; 67; 128; 146; 168; 174; 179; 193] [0126] [4; 32; 42; 54; 61; 97; 98; 134; 141; 153; 163; 191] [0127] [3; 47; 72; 83; 112; 123; 148; 192] [0128] [6; 29; 51; 94; 101; 144; 166; 189] [0129] [7; 12; 58; 93; 102; 137; 183; 188] [0130] [9; 14; 24; 79; 116; 171; 181; 186] [0131] [18; 28; 37; 48; 147; 158; 167; 177] [0132] [36; 56; 74; 96; 99; 121; 139; 159] [0133] [5; 25; 40; 70; 125; 155; 170; 190] [0134] [10; 50; 55; 80; 115; 140; 145; 185] [0135] [20; 35; 85; 95; 100; 110; 160; 175] [0136] [11; 69; 76; 90; 105; 119; 126; 184] [0137] [15; 22; 43; 57; 138; 152; 173; 180] [0138] [30; 44; 81; 86; 109; 114; 151; 165] [0139] [23; 33; 60; 88; 107; 135; 162; 172] [0140] [19; 46; 66; 75; 120; 129; 149; 176] [0141] [38; 45; 63; 92; 103; 132; 150; 157] [0142] [17; 53; 62; 68; 77; 82; 113; 118; 127; 133; 142; 178] [0143] [31; 34; 41; 59; 71; 89; 106; 124; 136; 154; 161; 164]
[0144] Again an allowed differences set may be obtained by selecting at least one element from each equivalence class. The selection may be random.
[0145] Let R=.sub.907, with the base ring element u=10 having order 151 and which is a base for obfuscated arithmetic in the field R. The seven equivalence classes are: [0146] [0] [0147] ±[1; 3; 8; 16; 18; 30; 31; 32; 33; 43; 53; 56; 62; 75; 78; 79; 101; 102; 113; 116; 124; 126; 128; 140; 149] [0148] ±[4; 9; 12; 15; 28; 44; 45; 47; 55; 57; 66; 67; 69; 71; 81; 83; 92; 99; 112; 114; 115; 127; 137; 138] [0149] ±[5; 6; 7; 21; 26; 29; 34; 40; 41; 51; 54; 61; 63; 64; 74; 86; 91; 93; 103; 105; 109; 129; 131; 132; 134; 141]
[0150] Note that the equivalency classes have a symmetry, which is used to denote the classes in shortened form.
[0151] Let R=.sub.1255, with the base ring element u=2 having order 100. Note that this ring is not a field. The 17 equivalence classes are: [0152] [0] [0153] ±[1; 23; 25; 27; 42; 49; 94] [0154] ±[2; 34; 54; 55; 57; 67; 74; 83; 93; 95] [0155] ±[3; 11; 39; 47; 82; 90] [0156] [4; 16; 24; 48; 52; 76; 84; 96] [0157] [8; 44; 56; 92] [0158] +[9; 19; 31; 38; 41; 70; 78] [0159] [12; 20; 28; 72; 80; 88] [0160] ±[13; 14; 21; 29; 37; 65; 85] [0161] [32; 40; 60; 68] [0162] [36; 64] [0163] [50]
[0164] One can proof that in a field, given an r we have that (u.sup.r−1)u.sup.s is different for all values of sΣ.sub.k. Therefore each class can represent exactly k elements. We have therefore the formula k(c−1)=n−1 for fields. Numerical experiments show that for commutative rings that are not fields the ratio k(c−1)/(n−1) is close to 1, especially as n grows.
[0165] This also implies that given a field, and an allowed differences list which was obtained by selecting exactly one element from each equivalence class, and integer lists of length 2, gives a unique representation of each field element. By selecting more elements from each equivalence class, ring elements obtain multiple representations.
[0166] Below a number of examples are summarized and additional examples are given. For polynomial rings the details of base ring element and generator are given above. Note that φ(1255)=1000 (Euler's totient function), which is why the ring given .sub.1255 has only 1000 elements.
TABLE-US-00001 Number of Base ring equivalence c/k * Ring R Size = n element u Order(u) = k classes = c 100 k(c − 1)/(n − 1) F2{circumflex over ( )}8 256 51 6 11,76471 1 F2{circumflex over ( )}12 4096 195 22 11,28205 1 Z907 907 10 151 7 4,635762 1 Z1511 1511 3 151 11 7,284768 1 Z2417 2417 15 151 17 11,25828 1 Z2719 2719 23 151 19 12,58278 1 Z3323 3323 30 151 23 15,23179 1 Z4229 4229 20 151 29 19,2053 1 Z1255 1000 2 100 17 17 1,601602
[0167] As can also be seen from the table above, rings with various sizes of allowed differences list may be obtained by generating rings with obfuscating arithmetic and computing the equivalence classes.
[0168] Having a smaller allowed differences lists has the advantage that this reduces the size of the increment table. In an embodiment, the increment table does not map an input ring element represented by the input integer list as a linear combination of powers (k=u.sup.k.sup.
[0169] By searching more rings, the size of the allowed-differences list may be chosen small with respect to the to order k. An allowed-differences list may be reduced in size by selecting only one element of each equivalence class; an allowed-differences list may be extended in size by selecting more than one element of each equivalence class. It is preferred, if the size of the allowed-differences list is less than 50% the order, more preferably less than 20% the order, even more preferably, less than 10% the order, even more preferably less than 5% the order. In case, of unique representation in fields, it is desired that c I k is less than 0.5, 0.2, 0.1 and 0.05.
[0170] By choosing only one element of each equivalence class it is obtained that
this provides a larger reduction for the table. (Here c represents the number of elements in the allowed differences list.) For rings that are not fields, the bound may be somewhat larger than 1, say less than 2. In the latter case, the representation is sometimes unique but not always. On the other hand, it may be desirable to have multiple different representations for ring elements. This may be achieved by growing the allowed differences list so that
[0171] .sub.n;
.sub.n[x]/f(x)), the ring having a finite number of ring elements, a ring-addition and a ring-multiplication being defined on the ring elements, the calculating method operating on integer-lists ((a.sub.1, a.sub.2)) encoding ring elements (u.sup.a.sup.
The method may further comprise ring adding: [0173] receiving 310 a first addition-input integer-list ((a.sub.1, a.sub.2)) encoding a first addition-input ring element and a second addition-input integer list ((b.sub.1, b.sub.2)) encoding a second addition-input ring element, [0174] determining 320 an addition-output integer-list encoding an addition-output ring element by applying the increment table to ring elements determined from the first and second addition-input integer-lists, the addition-output ring element being equal to the ring-addition of the first addition-input ring element and the second addition-input ring element.
The method may further comprise ring multiplication: [0175] receiving 330 a first multiplication-input integer-list ((r.sub.1, r.sub.2)) encoding a first multiplication-input ring element and a second multiplication-input integer list ((s.sub.1, s.sub.2)) encoding a second multiplication-input ring element, [0176] determining 340 a multiplication-output integer-list encoding a multiplication-output ring element by applying the increment table to ring elements determined from the first and second multiplication-input integer-lists, the multiplication-output ring element being equal to the ring-multiplication of the first multiplication-input ring element and the second multiplication-input ring element.
[0177]
[0178] Method 400 comprises receiving addition operands 410. This may comprise receiving 410 a first addition input integer list, e.g., (a.sub.1, a.sub.2) and receiving 420 a second addition input integer list, e.g., (b.sub.1, b.sub.2).
[0179] The addition input integer lists each encode a ring element such that the ring element equals a linear combination of powers of a base ring element, the base ring element having an order in the ring, wherein the powers have exponents determined by the integer-list. The exponents satisfy the requirement that a difference between a first exponent and a second exponent is comprised in an allowed-differences list.
[0180] Method 400 further comprises determining 420 an intermediate addition integer-list, e.g. (c.sub.1, c.sub.2). For example, this may comprise applying the increment table to a ring element determined from the first and second addition-input integer-lists. In particular, the increment table may be applied to an integer list, the elements in the integer being derived from the elements in the input integer lists. The output of the table satisfies the requirement. For example, determining 420 may comprise applying 422 the increment table to (a.sub.1−b.sub.1, a.sub.2−b.sub.1), e.g. obtaining (d.sub.1, d.sub.2); and adding 424 the integer b.sub.1 determined from second addition-input integer-lists to the integers in the integer-list resulting from first application, e.g., (c.sub.1, c.sub.2) (d.sub.1+b.sub.1, d.sub.2+b.sub.1).
[0181] Method 400 further comprises determining 430 the addition-output integer-list through a second application of the increment table to a ring element determined from the intermediate addition integer-list and the second addition-input integer-list. For longer integer lists this may involve additional increment table applications. For example, this may comprise negating 431 the intermediate addition integer-list, e.g., permuting to (c.sub.2, c.sub.1). Applying the 432 the increment table, and adding 434 are the same as applying 422 and adding 424 except that the addition input integer lists (a.sub.1, a.sub.2) is replaced by the intermediate integer list (c.sub.2, c.sub.1) and b.sub.1 by b.sub.2. Finally the result of 434 is negated 453 to obtain the result of the obfuscated addition.
[0182] If instead of negative obfuscation, as here, positive obfuscation is used, then negating 431, 435 may be omitted.
[0183]
[0184] Method 500 comprises receiving multiplication operands 510. This may comprise receiving 510 a first multiplication-input integer-list, e.g. (r.sub.1, r.sub.2), and receiving 514 a second multiplication-input integer list (s.sub.1, s.sub.2).
[0185] Method 500 further comprises determining 520 a first and second intermediate multiplication integer-list. For example, 520 may comprise determining 522 a first intermediate multiplication integer-list and determining 524 a second intermediate multiplication integer-list. These may for example, be chosen as (s.sub.1+r.sub.1, s.sub.1+r.sub.2) and (s.sub.2+r.sub.2, s.sub.2+r.sub.1), respectively, although there are other choices. The multiplication continues by adding these numbers in the addition method 400.
[0186] Note that the table is only used in applying 422 and applying 432 and nowhere else in methods 400 and 500. Both addition and multiplication use the same table, and both use the table the same number of times (2). The other operations comprise small arithmetic operations on the integers in the integer list, e.g., modulo the order of the base ring element Many different ways of executing the methods are possible, as will be apparent to a person skilled in the art. For example, the order of the steps can be varied or some steps may be executed in parallel. Moreover, in between steps other method steps may be inserted. The inserted steps may represent refinements of the method such as described herein, or may be unrelated to the method. Moreover, a given step may not have finished completely before a next step is started.
[0187] A method according to an embodiment may be executed using software, which comprises instructions for causing a processor system to perform any one of methods 300, 400 and 500. Software may only include those steps taken by a particular sub-entity of the system. The software may be stored in a suitable storage medium, such as a hard disk, a floppy, a memory etc. The software may be sent as a signal along a wire, or wireless, or using a data network, e.g., the Internet. The software may be made available for download and/or for remote usage on a server. A method may be executed using a bitstream arranged to configure programmable logic, e.g., a field-programmable gate array (FPGA), to perform the method.
[0188] It will be appreciated that an embodiment also extends to computer programs, particularly computer programs on or in a carrier, adapted for putting an embodiment 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 an embodiment. An embodiment relating to a computer program product comprises computer executable instructions corresponding to each of the processing steps of at least one of the methods set forth. These instructions may be subdivided into subroutines and/or be stored in one or more files that may be linked statically or dynamically. Another embodiment relating to a computer program product comprises computer executable instructions corresponding to each of the means of at least one of the systems and/or products set forth.
[0189]
[0190] A computer readable medium, e.g., computer readable medium 1000, may comprise an increment table, and/or a decoding table, and/or an encoding table.
[0191]
[0192] 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.
[0193] 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.
In the claims references in parentheses refer to reference signs in drawings of embodiments or to formulas of embodiments, thus increasing the intelligibility of the claim. These references are not exhaustive and shall not be construed as limiting the claim.