MICROPROCESSOR EQUIPPED WITH AN ARITHMETIC AND LOGIC UNIT AND WITH A HARDWARE SECURITY MODULE

Abstract

This microprocessor is configured to compute a code C1, used to detect an execution fault, using a relationship Ci=P o Fα(Di), where: Fα(Di)=E0 o . . . o Eq o . . . o ENbE−1(Di),Eq(x)=Tαm,q o . . . o Tαj,q o . . . o Tα1,q o Tα0,q(X), andTαj,q is a conditional transposition, configured by a secret parameter αj,q, that permutes two blocks of bits B2j+1,q and B2j,q of the variable x only when the parameter aj,q is equal to a first value, the blocks B2j+1,q and B2j,q of all of the transpositions Tαj,q of the stage Eq being different from one another and not overlapping and the blocks B2j+1,q and B2j,q are placed within one and the same block of greater size permuted by a transposition of the higher stage Eq+1.

Description

The invention relates to a microprocessor equipped with an arithmetic and logic unit and with a hardware security module.

Numerous attacks are possible in order to obtain information about a binary code or to cause unexpected operation of the binary code. For example, attacks known under the name “fault injection” or “fault attack” may be implemented. These attacks involve disrupting the operation of the microprocessor or the memory containing the binary code, using various physical means such as modifying supply voltages, modifying the clock signal, exposing the microprocessor to electromagnetic waves, inter alia.

Using such attacks, an attacker is able to alter the integrity of machine instructions or data in order for example to recover a secret key of a cryptographic system, bypass security mechanisms such as verification of a PIN code during authentication, or simply prevent the execution of a function essential to the security of a critical system.

These attacks may notably cause three types of fault, called execution faults, when the binary code is executed:

1)altering the instructions of the machine code that is executed,

2) altering the data stored in the main memory or in registers of the microprocessor, and

3) altering the control flow of the machine code.

he control flow corresponds to the execution path that is followed when the machine code is executed. The control flow is conventionally depicted in the form of a graph, known under the name “control flow graph”.

To detect such execution faults, there has already been the proposal to associate an error correction code with each data item processed by the microprocessor. Next, the error correction code associated with the result of the instruction that processes these data is computed from the error correction codes of the processed data. In this way, if a fault occurs when this instruction is executed, the result obtained does not correspond to the computed error correction code. This allows this fault to be detected. Such a solution is for example disclosed in application FR3071082. The algorithm for constructing the error correction code associated with a data item is known. It is therefore possible for an attacker to inject faults in order to modify the error correction code computed for the result so that it corresponds to the faulted result. In this case, the execution fault is not detected.

To overcome the above disadvantage, it has been proposed that the error correction code be replaced by an integrity code. This integrity code is constructed from the data item and, in addition, using a secret key known only to the microprocessor. It is thus difficult for an attacker to modify an integrity code so that it corresponds to a faulted result, because he does not know the secret key. However, it should always be possible to construct the integrity code for the result using the integrity codes associated with the processed data and without using the result of the instruction executed by the arithmetic and logic unit. For example, such a solution is described in the following article: L. De Meyer, V. Arribas, S. Nikova, V. Nikov and V. Rijmen: “M&M: Masks and Macs against physical attacks”, IACR Transactions on Cryptographic Hardware and Embedded Systems, pages 25-50, 2019. This article is subsequently denoted by the term “DEMEYER2019”. The method described in this article for computing the integrity code for the result from the integrity codes of the processed data is complex. The reason is that it does this by using multiplications in a Galois field. Hardware circuits that rapidly compute multiplications in a Galois field are complex and slow. The method of DEMEYER2019 is therefore difficult to implement in a microprocessor, including in the case of Boolean operations.

Prior art is also known from EP3457620A1 and from the following article: Savry Olivier et al.: “Confidaent: Control Flow protection with Instruction and Data Authenticated Encryption”, 2020 23 RD Euromicro Conference On Digital System Design, 26/08/2020, pages 246-253.

The objective is to propose a microprocessor that has the same level of security for at least one executed arithmetic and logic operation as that described in the article of DEMEYER2019 but that is easier to produce. [oolo] The invention therefore relates to such a microprocessor.

The invention will be better understood on reading the description that follows, which is given solely by way of non-limiting example, with reference to the drawings, in which:

FIG. 1 is a schematic illustration of the architecture of an electronic apparatus capable of executing a binary code;

FIG. 2 is a schematic illustration of the structure of a register of the apparatus of FIG. 1,

FIG. 3 is a schematic illustration of a function F_α executed by the apparatus of FIG. 1;

FIG. 4 is a flowchart of a method for executing the binary code by means of the apparatus of FIG. 1,

FIG. 5 is a schematic illustration of a computation circuit for computing a code C_res−t in the case of an arithmetic shift operation for shifting one bit to the left,

FIGS. 6 and 7 are more detailed schematic illustrations of two components of the circuit of FIG. 5,

FIGS. 8 to 10 are schematic illustrations of various operating states of the circuit of FIG. 5,

FIG. 11 is a schematic illustration of a computation circuit for computing a code C_res−t in the case of an arithmetic shift operation for shifting 2^r bits to the left,

FIG. 12 is a schematic illustration of a computation circuit for computing a code C_res−t in the case of an arithmetic shift operation for shifting α_NbE−12^NbE−1+ . . . +α_q2^q+ . . . +α₀ bits to the left,

FIG. 13 is a schematic illustration of a computation circuit for computing a code C_res−t in the case of an arithmetic rotation operation for rotating one bit to the left,

FIG. 14 is a schematic illustration of a computation circuit for computing a code C_res−t in the case of an arithmetic addition operation, and

FIGS. 15 and 16 are more detailed schematic illustrations of two components of the circuit of FIG. 14.

SECTION I: CONVENTIONS, NOTATIONS AND DEFINITIONS

In the figures, the same references have been used to designate elements that are the same. In the rest of this description, features and functions that are well known to those skilled in the art will not be described in detail.

In this description, the following definitions have been adopted.

A “program” designates a set of one or more predetermined functions that it is desired to have executed by a microprocessor.

A “source code” is a representation of the program in a computer language, not being able to be executed directly by a microprocessor and being intended to be converted, by a compiler, into a machine code able to be executed directly by the microprocessor.

A program or a code is said to be “able to be executed directly” or “directly executable” when it is able to be executed by a microprocessor without this microprocessor needing to compile it beforehand by way of a compiler or to interpret it by way of an interpreter.

An “instruction” denotes a machine instruction able to be executed by a microprocessor. Such an instruction consists:

of an opcode, or operation code, that codes the nature of the operation to be executed, and

of one or more operands defining the value(s) of the parameters of this operation.

The registers in which the data item or data to be processed by an instruction are stored are typically identified by one or more operands of the instruction. Likewise, the register R_res−p in which the result D_res−p of the execution of an instruction needs to be stored can also be identified by an operand of this instruction.

“Logic instruction” is used to denote an instruction from the set of instructions of the microprocessor 2 that, when executed by the arithmetic and logic unit, stores the result of a Boolean operation in a register R_res−p of the microprocessor. The opcode of the logic instruction identifies the Boolean operation to be executed by the arithmetic and logic unit in order to modify or combine the data item or data D₁ to D_n.

The “&” symbol is used below to generically denote a Boolean operation. Thus, the notation D₁&D₂& . . . &D_n generically denotes a Boolean operation executed by the microprocessor 2 between the data D₁ to D_n. When n =1, the Boolean operation is the complement operation also known by the name “NOT”. When n is greater than or equal to two, the Boolean operation is chosen from the group made up of the following Boolean operations and their composition:

the “OR” logic operation,

the “EXCLUSIVE-OR” logic operation,

the “AND” logic operation.

The following notations are used to denote Boolean operations:

the “OR” logic operation is denoted by the symbol “ + ”,

the “EXCLUSIVE-OR” logic operation is denoted by the symbol “XOR”,

the “AND” logic operation is denoted by the symbol “ . ”,

the “NOT” Boolean operation is denoted by the symbol “ ′ ” placed after the variable for which the complement is computed.

“Arithmetic instruction” is used to denote an instruction from the set of instructions of the microprocessor 2 that, when executed by the unit 10, stores the result of an arithmetic operation in a register R_res−p of the microprocessor. An arithmetic operation is different from a Boolean operation. An arithmetic operation typically belongs to the group made up of bit shift operations, bit rotation operations, addition operations, multiplication operations and division operations.

“Arithmetic and logic instruction” denotes both a logic instruction and an arithmetic instruction. Unless indicated otherwise, the term “instruction” denotes an arithmetic and logic instruction below.

A “machine code” is a set of machine instructions. It typically is a file containing a sequence of bits with the value “0” or “1”, these bits coding the instructions to be executed by the microprocessor. The machine code is able to be executed directly by the microprocessor, that is to say without the need for a preliminary compilation or interpretation.

A “binary code” is a file containing a sequence of bits bearing the value “0” or “1”. These bits code data and instructions to be executed by the microprocessor. The binary code thus comprises at least one machine code and also, in general, digital data processed by this machine code.

The expression “execution of a function” is understood to designate execution of the instructions making up this function.

A block of bits of a data item or of a variable is a group of consecutive bits of this data item or of this variable. The size of a block of bits is equal to the number of bits contained in this block.

Section II: Architecture of the Apparatus

FIG. 1 shows an electronic apparatus 1 comprising a microprocessor 2, a main memory 4 and a mass storage medium 6. For example, the apparatus 1 is a computer, a smartphone, an electronic tablet, a chip card or the like.

The microprocessor 2 here comprises:

an arithmetic and logic unit 10;

a set 12 of registers;

a control module 14;

a data input/output interface 16,

an instruction loader 18 having a program counter 26,

a queue 22 of instructions to be executed, and

a hardware security module 28.

The memory 4 is configured so as to store instructions of a binary code 30 of a program to be executed by the microprocessor 2. The memory 4 is a random access memory. The memory 4 is typically a volatile memory. The memory 4 may be a memory external to the microprocessor 2, as shown in FIG. 1. In this case, the memory 4 is formed on a substrate that is mechanically separate from the substrate on which the various elements of the microprocessor 2, such as the unit 10, are formed.

By way of illustration, the binary code 30 notably comprises a machine code 32 of a secure function. Each secure function corresponds to a set of several lines of code, for example several hundred or thousand lines of code, stored at successive addresses in the memory 4. Each line of code corresponds here to a machine word. A line of code is thus loaded into a register of the microprocessor 2 in a single read operation. Likewise, a line of code is written to the memory 4 by the microprocessor 2 in a single write operation. Each line of code codes either a single instruction or a single data item.

By way of illustration, the microprocessor 2 is a reduced instruction set computer more commonly known by the acronym RISC.

The loader 18 loads the next instruction to be executed by the unit 10 into the queue 22 from the memory 4. More precisely, the loader 18 loads the instruction to which the program counter 26 points. To this end, the queue 22 comprises a succession of multiple registers.

The unit 10 is notably configured to execute one after another the instructions loaded into the queue 22. The instructions loaded into the queue 22 are generally consistently executed in the order in which these instructions were stored in this queue 22. The unit 10 is also capable of storing the result of these executed instructions in one or more of the registers of the set 12.

In this description, “execution by the microprocessor 2” and “execution by the unit 10” will be used synonymously.

The module 14 is configured to move data between the set 12 of registers and the interface 16. The interface 16 is notably able to acquire data and instructions, for example from the memory 4 and/or the medium 6 that are external to the microprocessor 2. To speed up transfers of data and instructions between the microprocessor 2 and the memory 4 here, the interface 16 comprises one or more cache memories. To simplify FIG. 1, only one cache memory 27 is shown. This cache memory 27 is used to temporarily store the data processed by the microprocessor 2 on the same chip as the unit 10.

The module 28 is capable of automatically executing the various operations described in detail in the sections that follow in order to make the execution of the arithmetic and logic instructions by the unit 10 secure. The module 28 operates independently and without using the unit 10. It is thus capable of processing the lines of code before and/or after they are processed by the unit 10. To this end, it notably comprises a secure nonvolatile memory 29 and various hardware computation circuits. This memory 29 can only be accessed via the module 28. In this embodiment, the module 28 is configured to execute operations such as the following operations:

verifying an integrity code,

constructing an integrity code from a data item,

constructing the integrity code for a result from the integrity codes of the processed data.

Each computation circuit constructs an integrity code C_res−t for a result D_res−p from the integrity codes C₁ to C_n of the data D₁ to D_n processed by the unit 10 and 30 without directly using the result D_res−p. Here, there is a computation circuit 110 for the logic instructions. There is also a computation circuit for each arithmetic instruction whose execution needs to be made secure. Here, the module 28 comprises five computation circuits 120, 130, 140, 149 and 160, each associated with a respective arithmetic instruction. These computation circuits are described in more detail in section IV that follows.

The memory 29 is used to store the secret information required for the operation of the module 28. Here, it therefore notably comprises a pre-stored secret key α.

In this example of embodiment, the set 12 comprises general registers that can be used to store any type of data. The size of each of these registers is sufficient to store a data item or a result and the integrity code associated therewith.

A data interchange bus 24 that connects the various components of the microprocessor 2 to one another is shown in FIG. 1 in order to indicate that the various components of the microprocessor are able to interchange data with one another.

The medium 6 is typically a non-volatile memory. It is for example an EEPROM or flash memory. Here, it contains a backup copy 40 of the binary code 30. It is typically this copy 40 that is automatically copied to the memory 4 to restore the code 30, for example after a power failure or the like or just before the execution of the code 30 starts.

Section III—Making Arithmetic and Logic Instructions Secure

By injecting faults while the unit 10 is operating, it is possible to disrupt its operation so that the result of the execution of the arithmetic and logic instruction does not correspond to that expected. The unit 10 is then said to have been caused to malfunction. This section describes a solution for detecting such a malfunction of the unit 10.

The registers R₁ to R_n denote the registers of the set 12 comprising the data D₁ to D_n, respectively, to be processed when the instruction is executed. The register R_res−p denotes the register of the set 12 in which the result, of the execution of the arithmetic and logic instruction, is stored.

The size, in terms of the number of bits, of each data item D₁, D₂ and D_res−p is equal to 2^d, where d is a whole number typically greater than four or five.

The structures of the registers R₁, R₂ and R_res−p are identical and shown in the specific case of the register R_i in FIG. 2. The register R_i comprises:

a bit range containing the data item D_i,

a range containing an integrity code C_i allowing the integrity and the authenticity of the data item D_i to be checked.

The code C_i is generated by the module 28 using a pre-programmed relationship defined generically by the following relationship: C_i=Q_α(D_i), where:

the subscript i identifies a register among the registers R₁, R₂ and R_res−p, and

the function Q_α is a function pre-programmed in the module 28 and configured by the secret key α.

The function Q_α is defined by the following relationship: Q_a(D_i)=P o F_α(D_i), where the symbol “o” denotes the function-composition operation. The function P is a predetermined function. In the first embodiments described below, the function P is the identity function. Thus, in these first embodiments, the function Q_α is equal to the function F_α. Examples where the function P is different from the identity function are given in the section dealing with variants.

The function F_α is a homomorphism of a set A equipped with the “&” Boolean operation towards a set B equipped with the same “&” Boolean operation such that F_α(D₁&D₂)=F_α(D₁) & F_α(D₂), and such is the case for all “&” Boolean operations. Here, the sets A and B are each the set of numbers that can be coded over 2^d bits, that is to say the set of possible data D₁ and D₂. Thus, using the notations introduced earlier, the function F_α is such that for any & Boolean operation, the circuit 110 simply computes the integrity code C_res−t associated with the result D_res−p of the Boolean operation D₁ & D₂ using the following relationship C_res−t=C₁ & C₂. When the Boolean operation executed is the complement operation of the data item D₁, the circuit 110 computes the code C_res−t associated with the result D_res−p using the following relationship C_res−t=C₁′, where the symbol “ ′ ” denotes the complement operation that returns a “1” when D₁=0 and that returns “0” when D₁=1.

FIG. 3 shows the function F_α used in this embodiment. The function F_α is defined by the following relationship: F_α(D_i)=E₀ o . . . o E_q o . . . o _ENbE−1(D_i), where:

each function E_q is a stage of transpositions that can be executed in parallel,

NbE is the number of stages of transpositions, and

the subscript q is an order number between zero and NbE−1.

The number NbE is greater than one and less than or equal to d. Preferably, the number NbE is equal to d. In FIG. 3, d=5 and NbE=5.

Each stage E_q of transpositions is defined by the following relationship: E_q(x)=T_αm,q o . . . o T_αj,q o . . . o T_α1,q o T_αo,q(X), where:

x is a variable whose size, in terms of the number of bits, is equal to the size of the data item D_i,

T_αj,q is a conditional transposition, configured by the parameter α_j,q, that permutes two blocks of bits B_2j+1,q and B_2j,q of the variable x when the parameter α_j,q is equal to “1” and that does not permute these two blocks of bits when the parameter α_j,q is equal to “0”,

“m+1” is the total number of transpositions T_αj,q of the stage E_q,

“j” is an order number identifying the transposition T_αj,q among the other transpositions of the stage E_q. The subscript “j” therefore also identifies the position of the blocks B_2j+1,q and B_2j,q in the variable x. In this application, the blocks are classified in ascending order of their subscript, which depends on the value of the subscript j.

Each transposition T_αj,q is distinguished from all of the other transpositions of the function F_α by the fact that it is the only one that permutes the two blocks B_2j+1,q and B_2j,q when the parameter α_j,q is equal to “1”. Moreover, the blocks B_2j+1,q and B_2j,q of all of the transpositions T_αj,q of the same stage E_q are different from one another and do not overlap. Thus, all of the transpositions T_αj,q of the stage E_q can be executed in parallel. Here, the stages E_q are executed one after the other in descending order of the subscripts q.

Moreover, this function F_α has the following characteristics:

the blocks B_2j+1,q and B_2j,q permuted by the transpositions T_αj,q are adjacent blocks,

the size of the blocks B_2i+1,q and B_2j,q is equal to 2^q,

the number m of transpositions T_αj,q per stage E_q is equal to 2^d−q−1.

The notations B_wj+1,q and B_2j,q indicate that these are the (2j+1)-th and 2j-th blocks of 2^q bits, respectively, of the variable x. In FIG. 3, the block B_o,q is the least significant block of bits, the block B_1,q the next block and so on. Each horizontal brace in FIG. 3 encompasses the two blocks and B_2j,q and B_2j,q of a transposition T_αj,q for which the parameter is α_j,q.

In the case of FIG. 3, for all of the stages E_q for which q is less than NbE−1 and for all of the transpositions T_αj,q of this stage, the blocks B_2j+1,q and B_2j,q are both placed within one and the same block B_l,q+1 of the higher stage E_q+1, where the subscript “l” denotes the block of the higher stage that contains the blocks B_2i+1,q and B_2j,q. For example, as can be seen in FIG. 3, the blocks B_1,q and B_o,q, of the stage E_q are consistently placed within the block B_0,q+1 of the higher stage E_q+1. Moreover, here, each block of a higher stage E_q+1 contains at most two blocks of the lower stage E_q.

The operation of the microprocessor 2 in order to make the execution of arithmetic and logic instructions secure will now be described in more detail with reference to FIG. 4.

The method begins by providing, in a step 86, the binary code 30. During this step, in this example, the binary code 30 is loaded into the memory 4 from the medium 6. Next, execution of the binary code 30 by the microprocessor 2 begins.

In a step 88, each time a data item D_i is stored in the cache memory 27, the module 28 computes the code C_i using the relationship C_i=F_α(D_i). Next, the data item D_i and the code C_i associated therewith are both stored in the memory 27.

Each time an instruction to load a data item into one of the registers R_i is executed by the unit 10, in a step 90, the data item D, and the code C_i are written to this register R_i.

Prior to the execution of an arithmetic and logic instruction between two data items D₁ and D₂, step 90 is executed once for the data item D₁ and once for the data item D₂.

Next, each time an arithmetic and logic instruction is about to be executed by the unit 10, just before it is executed, in a step 94, the module 28 checks whether there is an error in the data item D, contained in the register R_i identified by an operand of the instruction to be executed.

During this step, for each register R_i in question, the module 28 checks, using the code C_i contained in the register R_i, whether or not the data item D_i currently stored in this register has an error. For example, this involves the module 28 computing a code C_i* using the relationship C_i*=F_α(D_i) and without using the code C_i stored in the register R. If the code C,* computed in this way is identical to the code C_i stored in the register R_i, then the integrity and authenticity of the data item D_i are confirmed. In that case, the module 28 detects no error and proceeds to a step 96. Otherwise, the module 28 proceeds to a step 102.

In step 102, the module 28 triggers signalling of an execution fault.

If the module 28 detects no error, in step 96, the microprocessor 2 decodes the arithmetic and logic instruction and then the unit 10 executes it and stores its result D_res−p in the register R_res−p.

When the executed instruction is an arithmetic and logic instruction whose execution is secure, in parallel with step 96 or after the execution of step 96, in a step 98, the module 28 computes the code C_res−t by using only the codes C_i associated 10 with the data D_i processed by the unit 10 in step 96. Thus, when it is the data D₁ and D₂ that are processed, the code C_res−t is computed by combining the codes C₁ and C₂ stored in the registers R₁ and R₂, respectively, prior to execution of the logic instruction.

More precisely, when the executed instruction is a logic instruction, the circuit 110 computes the code C_res−t using the following relationship: C_res−t=C₁ & C₂, where the “&” symbol denotes the Boolean operation executed by the unit 10 in step 96.

When the executed instruction is an arithmetic instruction for which the module 28 comprises a specific computation circuit for computing the code C_res−t, then this specific circuit is selected and the code C_res−t is computed by this specific circuit. Examples of such specific computation circuits are described in detail in the section that follows.

Next, in a step 100, the module 28 checks whether the computed code C_res−t corresponds to a code C_res−p computed from the result D_res−p stored in the register R_res−p. In the case of the circuits 110, 120, 130, 149 and 160, the code C_res−p is computed by implementing the relationship C_res−p=F_α(D_res−p). When the code C_res−t is computed by the circuit 140, the code C_res−p is equal to the result D_res−p.

Next, the module 28 compares the computed codes C_res−p ^{and C}_res−t. If these codes are different, the module 28 triggers the execution of step 102. Otherwise, this means that the code C_res−t corresponds to the code C_res−p and therefore that there was no fault during the execution of the instruction by the unit 10. In this last case, no signalling of an execution fault is triggered and the method continues with the execution of the next instruction in the queue 22.

The execution of steps 98 and 100 allows a malfunction in the unit 10 to be detected, because the computed codes C_res−p and C_res−t are identical only if the unit 10 has executed the arithmetic and logic instruction correctly. In the case of a logic instruction, this can be explained simply by the following relationship: C_res−p=F_α(D_res−p)=F_α(D₁&D₂)=F_α(D₁) & F_α(D₂)=C₁ & C₂=C_res−t. In the case of arithmetic instructions, this can be explained by the structure and the operations performed by the implemented computation circuit.

If the instruction executed in step 96 is the complement operation for the data item D₁, in step 98, the code C_res−t is computed using the following relationship: C_res−t=C₁′. The remainder of the method is then identical to what was described earlier. In the case of the complement operation, the codes C_res−p and C_res−t are identical only if the unit 10 has operated correctly. This can be demonstrated using the following relationship: C_res−p=F_α(D_res−p)=F_α(D_1′)=F_α(D₁)′=C₁′=C_res−t.

In response to an execution fault being signalled, in a step 104, the microprocessor 2 implements one or more countermeasures. A wide range of countermeasures are possible. The countermeasures implemented may have very different degrees of severity. For example, the countermeasures that are implemented may range from simply displaying or simply storing an error message without interrupting the normal execution of the machine code 32 as far as definitively taking the microprocessor 2 out of service. The microprocessor 2 is considered to be out of service when it is definitively put into a state in which it is incapable of executing any machine code. Between these extreme degrees of severity, there are many other possible countermeasures, such as:

using a human-machine interface to indicate detection of the faults,

immediately interrupting the execution of the machine code 32 and/or reinitializing it, and

deleting the machine code 32 from the memory 4 and/or deleting the backup copy 40 and/or deleting the secret data.

Section IV—Examples of Computation Circuits for Computing the Code C_res−t:

The function F_α described earlier allows the bit locality to be preserved. This denotes the property according to which the bits of a data item D_i that are placed within the block B_2j+1,q or B_2j,q still remain within this block. In other words, the transpositions T_αj,q−1 to T_αj,0 that apply to the bits of this block B_2j+1,q or B_2j,q cannot permute a bit of this block with a bit placed outside this block. On the other hand, the bits of the block B_4+1,q or B_2bq can be moved within this block by applying these transpositions T_αj,q−1 to T_αj,0. This stems from the fact that, for all of the stages E_q for which q is less than NbE−2 and for all of the transpositions T_αj,q of this stage, the blocks B_2j+1,q and B_2j,q are both placed within one and the same block B_l,q+1 of the higher stage E_q+1 . It is this particular property of the function F_α that allows simple and fast computation circuits for computing the code C_res−t to be produced, for each arithmetic instruction. This is illustrated below in the particular case of bit shift instructions, a rotation instruction and an addition instruction. However, on the basis of these examples, a person skilled in the art is capable of producing other computation circuits for computing the code C_res−t for other arithmetic instructions.

FIG. 5 shows the computation circuit 120 for computing the code C_res−t when the arithmetic instruction executed by the unit 10 is a logic shift instruction for shifting the bits of the data item D₁ 1 bit to the left. The code C_res−t is computed only from the code C₁. To simplify FIG. 5, the circuit 120 is illustrated in the particular case where the size of the codes C_i and C_res−t C is equal to 8 bits. In this case, the number NbE of stages is equal to three. However, the description provided for this particular size can easily be generalized for all possible sizes.

In FIG. 5, the eight consecutive bits of the code C₁ are denoted by the symbols a₀ to a₇, where a₀ is the least significant bit and a₇ is the most significant bit.

The bits of the code C_res−t are denoted by the symbols a′₀ to a′₇, these bits a′₀ to a′₇ being classified in the same order as the bits of the code C₁.

For each stage E_q of the function F_α, the circuit 120 comprises a corresponding stage ED_q. Each stage ED_q comprises a component CD_αj,q for each transposition T_αj,q of the stage E_q. More precisely, each component CD_αj,q is associated with a respective corresponding transposition T_αj,q. The components CD_αj,q are classified in ascending order of their subscript, which varies depending on the subscript j.

For q less than NbE−1, the components CD_αj,q are all structurally identical to one another. One of these components CD_αj,q is shown in more detail in FIG. 6. It has three outputs 122 to 124 and four inputs 126 to 129.

The output 122 delivers the result a.k′+c.k, where a, k and c are the values received at the inputs 126, 128 and 129, respectively.

The output 123 delivers the result b.k+c.k′, where b is the value received at the input 127.

The output 124 delivers the result a.k+b.k′.

The inputs 126 and 127 of each component CD_αj,0 of the stage ED₀ are connected to the 2j-th and (2j+1)-th bits, respectively, of the code C₁. For q greater than zero and q less than NbE−1, the inputs 126 and 127 of each component CD_αj,q are connected, respectively, to the outputs 124 of the components CD_α2j,q−1 and CD_{α(2j+1),q−1}, respectively, of the stage ED_q−1.

The input 128 of each component CD_αj,q receives the parameter α_j,q of the transposition T_αj,q with which it is associated.

For q greater than zero, the outputs 122 and 123 of each component CD_αj,q are connected to the input 129 of the components CD_{α(2j+1),q−1} and CD_α2j,q−1 , respectively. For q=0, the outputs 122 and 123 of each component CD_αj,0 deliver the (2j+1)-th and 2j-th bits, respectively, of the code C_res−t.

The stage ED_NbE−1 comprises a single component CD_α0,NbE−1. The component CD_α0,NbE−1 is shown in more detail in FIG. 7. It has two outputs 132 and 133 and three inputs 136 to 138. The outputs 132 and 133 deliver the results “a” and “0”, respectively, when the input 138 is equal to “0”, where “a” is the value received at the input 136 and “0” is the value zero. The outputs 132 and 133 deliver the results “0” and “b”, respectively, when the input 138 is equal to “1”, where “b” is the value received at the input 137.

The inputs 136 and 137 of the component CD_α0,NbEA−1 are connected to the outputs 124 of the components CD_α0,NbE−2 and CD_α1,NbE−2 respectively. The input 138 receives the parameter α_0,NbE−1. The outputs 132 and 133 are connected to the inputs 129 of the components CD_α1,NbE−2 and CD_α0,NbE−2, respectively.

When the instruction executed in step 96 is a shift instruction for shifting one bit to the left, the circuit 120 is selected in order to perform step 98. To this end, the bits of the code C₁ are delivered to the inputs 126 and 127 of the components CD_αj,0. In response, the components CD_αj,0 deliver the results present at their output 124 to the inputs 126 and 127 of the higher stage ED₁. This process is repeated stage by stage until the component CD_α0,NbE−1 is reached. The component CD_α0,NbE−1 then delivers the results a.k′ and b.k that are present at its outputs 132 and 133, respectively, to the inputs 129 of the components CD_{60 j,NbE−2}. In response, the component CD_{60 j,NbE−2} delivers the results a.k′+c.k and b.k+c.k′ to the inputs 129 of the components CD_αj,q of the lower stage E_q. The process is then repeated stage by stage until the components CD_αj,0 of the stage ED₀ are reached. The components CD_αj,0 then deliver the various bits of the computed code C_res−t at their outputs 122 and 123. Next, the method continues with step 100.

The operation of the circuit 120 is illustrated in FIGS. 8 to 10 in a simplified case where the size of the codes C₁ and C_res−t is equal to four bits. The number NbE of stages E_q is therefore equal to two. Moreover, in this particular example, the parameters α_0,0, α_1,0 and α_0,1 are equal to 0, 1 and 1, respectively. The bits a₀, a₁, a₂ and a₃ of the data item D₁ are classified in ascending order starting from the least significant bit to the most significant bit. In these conditions, with the values given above for the parameters α_0,0, α_1,0 and α_0,1, the code C₁ is equal to a₀, a₁, a₃, a₂. The result D_res−p itself is equal to a₂, a₁, a₀, 0. In FIGS. 8 and 9, the “?” symbol indicates that this value is unknown at this stage of operation.

FIG. 8 shows that when the values α_0,0 and α_1,0 are equal to 0 and 1, respectively, the components CD_α0,0 and CD_α1,0 deliver the values a₃ and a₁, respectively, at their output 124. Next, in response (FIG. 9), the component CD_α0,1 delivers the values 0 and a₁ at its outputs 132 and 133, respectively. Finally (FIG. 10), the components CD_α0,0 and CD_α1,0 deliver the bits of the computed code C_res−t at their outputs 122 and 123. In this example, the computed code C_res−t is 0, a₀, a₂, a₁. It can easily be checked that this computed code C_res−t is equal to F_α(D_res−p) in the absence of an execution fault.

FIG. 11 shows the computation circuit 130 for computing the code C_res−t when the arithmetic instruction executed by the unit 10 is a logic shift of 2^r bits to the left, 30 where r is a whole number greater than one and less than NbE−1. In FIG. 11, the circuit 130 is shown in the particular case where NbE=d =4 and r=2.

The following notations are used below to describe the circuit 130. The symbols BC_y,r, BCI_x,r, BCR_x,r and BD_z,r denote the blocks of 2^r bits at the position y in the code C₁, at the position x in an intermediate code CI, at the position x in the code C_res−p and at the position z in the data item D₁, respectively. The subscripts y, x and z here are order numbers that begin at 0 and increase by 1 each time there is a move from one block of 2^r bits to the next block of 2^r bits, moving towards the most significant bits.

The circuit 130 comprises a higher permutator 131 and a scheduler 134. The permutator 131 allows the position of the blocks of 2^r bits within the code C_res−t to be computed from the code C₁. To that end, here, the permutator 131 comprises stages ED_r to ED_NbE−1 , which are identical to the stages ED_r to _EDNbE−1 of the circuit 120 except that instead of manipulating blocks of 1 bit, it is blocks of 2^r bits that are manipulated.

More precisely, the inputs 126 and 127 of each component CD_αj,q, for q greater than or equal to r, receives blocks of 2^r bits rather than of a single bit. Thus, the outputs 122 and 123 of the components CD_αj,r deliver an intermediate code CI, in which each block BCI_x,r of 2^r bits is at the desired location, that is to say at the location that it needs to occupy in the code C_res−t, to the scheduler 134.

Moreover, each block BCI_x,r is identical to a corresponding block BC_y,r of the code C₁. The reason is that the permutator 131 is only able to move the blocks BC_y,r with respect to one another in order to obtain the intermediate code CI. The permutator 131 does not permute the bits placed within a block BC_y,r. The position of the block BC_y,r which is identical to the block BCI_x,r placed at the position x in the code CI, is denoted y below.

The order of the 2^r bits within the block BCI_x,r is identical to the order of the 2^r bits within the corresponding block BC_y,r. The arrangement of the 2^r bits within the block BC_y,r results from application of some of the transpositions of the stages E_r−1 to E₀ to a corresponding block BD_z,r when the code C₁ is calculated. The position of the block BD_z,r from which the positions of the bits placed within the block BC_y,r are computed is denoted z below. The position z of the block BD_z,r is not necessarily the same as the position y of the block BC_y,r because the transpositions of the stages E_NbE−1 to E_r may have moved this block BD_z,r before applying the transpositions of the next stages thereto. The composition of the transpositions T_αj,q of the stages E_r−1 to E₀ applied, during construction of the code C₁, to the bits placed within the block BD_z,r in order to obtain the block BC_y,r is denoted F_αcy. The key αc_y is a subset of the key a that contains only the parameters of the transpositions of the function F_αcy. The key αc_y is called the “current key” below because it is the one that explains the present arrangement of the bits within the block BC_y,r. The key αc_y is dependent on the position y of the block BC_y,r. The parameter of the transposition T_αj,q that is contained in the current key αc_y is also denoted αc_j,q below.

In the code C_resp−p, the arrangement of the 2^r bits within the block BCR_x,r results from application, to the corresponding block BD_z,r, of some of the transpositions of the stages E_r−1 to E_o. The corresponding block BD_z,r is the same as the one that corresponds to the block BC_y,r. The composition of the transpositions T_αj,q of the stages E_r−1 to E₀ applied, during construction of the code C_resp−p, to the bits placed within the block BD_z,r in order to obtain the block BCR_x,r is denoted F_αsx. The key αs_x is called the “desired key” below because it is the one that determines the arrangement of the bits within the block BCR_x,r. The key as_x is dependent on the position x of the block BCR_x,r. The parameter of the transposition T_αj,q that is contained in the desired key as_y is also denoted as_j,q below.

The result of the explanations above is that the arrangement of the 2^r bits within the block BCI_x,r is not necessarily identical to the arrangement of the 2^r bits within the block BCR_x,r that occupies the same position x in the code C_res−t. The reason is that the current key ac_y is not necessarily identical to the desired key αs_x. The scheduler 134 rearranges the order of the bits within each block BCI_x,r to obtain the desired block BCR_x,r.

To explain this, let us suppose that the data item D₁ comprises four blocks, of 2^r bits each, denoted in the order BD_3,r, BD_2,r, BD_1,r and BD_0,r. It is also supposed in this example that NbE=d=4 and r=2 and that the parameters α_0,3, α_1,2, α_0,2 are equal to 1, 1 and 0, respectively. The computation of the code C₁ is broken down into first and second successive phases. During the first phase, it is the transpositions of the stages E_NbE−1 to E_r that are applied. Thus, during this first phase, only the whole blocks BD_z,r of the data item D₁ are permuted. In this example, at the end of this first phase, the order of the blocks BD_z,r is as follows: BD_0,r, BD_1,r, BD_3,r and BD_2,r.

Next, during the second phase, it is the transpositions of the stages E_r−1 to E₀ that are applied. This second phase permutes only the bits within each of the blocks BD_z,r. During this second phase, no applied transposition moves a bit placed within a block BD_z,r to another block. At the end of this second phase, the code C₁ is obtained. The four blocks, of 2^r bits each, of the code C₁ obtained are denoted in the order BC_3,r, BC_2,r, BC_1,r and BC_0,r.

During the second phase, the bits of the block BD_0,r are permuted by applying the transpositions of the stages E_r−1 to E₀, which are applied only to the 2^r most significant bits. This stems from the fact that at the end of the first phase, the block BD_0,r is at the location of the most significant bits, that is to say at the position y=3 here. The composition of the transpositions of the stages E_r−1 to E₀ that apply only to the 2^r most significant bits is denoted F_αc3. Similarly, the compositions of transposition of the stages E_r−1 to E₀ that permute only the bits placed within the blocks BD_1,r, BD_3,r and BD_2,r respectively are denoted F_αc2, F_αc1 and F_αc0. It is thus noted that the block BC_3,r of the code C₁ is the result of application of the function F_αc3 to the bits of the block BD_0,r. Similarly, the blocks BC_2,r, BC_1,r and BC_0,r of the code C₁ are the results of application of the functions F_αc2, F_αc1 and F_αc0 to the blocks BD_1,r, BD_3,r and BD_2,r, respectively.

Following the logic shift of 2^r bits to the left, the result D_res−p is equal to the concatenation, in order, of the blocks BD_2,r, BD_1,r, BD_0,r and of a block [0] of 2^r null bits.

Application of the function F_α to the result D_res−p in order to compute the code C_res−p is broken down, similarly, into a first phase then a second phase. At the end of the first phase, the blocks of the result D_res−p are classified in the following order: [0], BD_0,r, BD_2,r and BD_1,r. During the second phase, the functions F_αs3 to F_αs0 are applied to the blocks [0], BD_0,r, BD_2,r and BD_1,r, respectively. Thus, the blocks BCR_3,r, BCR_2,r, BCR_1,4, BCR_0,r of the code C_res−p are the results of application of the functions F_αs3 to F_αs0 to the blocks [0], BD_0,r, BD_2,r and BD_1,r, respectively.

The intermediate code CI delivered by the permutator 131 is the concatenation, in order, of the blocks [0], BC_0,r, BC_2,r and BC_1,r. The block BCI_2,r is the result of application of the function F_αc3 to the block BD_0,r. The desired block BCR_2,r, which occupies the same position in the code C_resp−p, is the result of application of the function F_αs2 to the block BD_0,r. Thus, in the case of the block BCI_2,r, the role of the scheduler 134 is to cancel application of the transpositions of the function F_ac3 and to apply the transpositions of the function F₂ instead in order to obtain the desired block BCR_2,r.

For this, for example, for each stage E_r−1 to E₀ of the function F_α, the scheduler 134 comprises a stage corresponding to EO_r−1 to EO₀. Each stage EO_q comprises 2^d−q−1 comparators CO_j,q. Each comparator CO_j,q is associated with the corresponding transposition T_αj,q that permutes the blocks BCI_2j+1,q and BCI_2j,q of the intermediate code CI when the value of its parameter is equal to one. The size of the blocks BCI_2j+1,q and BCI_2j,q is equal to 2^q, To simplify FIG. 11, only some of these comparators are shown. Each comparator CO_j,q receives the block BCI_j,q+1, the parameter ac_j,q of the current key and the parameter αs_j,q of the desired key. If these two parameters are equal, the output of the comparator CO_j,q delivers the blocks BCI_2j+1,q and BCI_2j,q to the comparators CO_2j+1,q−1 and CO_2j,q−1, respectively. In other words, the transpositions T_αcj,q and T_αsj,q are identical and no change of order of the bits is necessary. If the two parameters αc_j,q and αs_j,q are different, the comparator CO_j,q permutes the positions of the blocks BCI_2j+1,q and BCI_2j,q before delivering them to the comparators CO_2j+1,q−1 and CO_2j,q−1, respectively. This replaces the transposition T_αcj,q with the transposition T_αsj,q. Proceeding thus by moving from the stage EO_r−1 to the stage EO₀, the code C_rest−t is obtained at the outputs of the comparators CO_j,0 of the stage EO₀. In FIG. 11, the bits of the code C_res−t-t computed by the circuit 130 are denoted by the references a₇ to a₀.

The current αc and desired as keys are obtained as follows, for example. The key α is divided into 2^r blocks BK_z,r of 2^r bits each. Each block BK_z,r contains only the parameters α_j,r−1 to α_j,0 of the transpositions to be applied, when the code C₁ is calculated, to the bits placed within the respective block BD_z,r of the data item D₁. Within each block BK_z,r, the parameters α_j,r−1 to α_j,0 are classified in a predetermined order. For example, here, they are first of all classified in descending order of stages and the various parameters α_j,q of one and the same stage E_q are also classified in descending order of subscript j. The desired key as is equal to the key a in which the parameters are classified as described above. Next, the various blocks BK_z,r are permuted, for example, by a circuit identical to the permutator 131. The key containing the blocks BK_z,r permuted in this way is equal to the current key αc. The parameters αc_j,q and as_j,q are the parameters that occupy the same position in the current key ac and the desired key as, respectively.

FIG. 12 shows the computation circuit 140 for computing the code C_res−t from the code C₁ when the arithmetic instruction executed by the unit 10 is a shift instruction for shifting the data item D₁ a_NbE−12^NbE−1+ . . . +a_q2^q+ . . . +a₀2⁰ bits to the left. The coefficients a_NbE−1 to a₀ are coefficients whose Boolean values are determined and fixed in advance. The circuit 140 computes a code C_res−t that is equal to the result D_res−p if no execution fault occurs. Thus, in this embodiment, in step 100, the code C_res−p is equal to the result D_res−p.

For each stage E_q of the function F_α, the circuit 140 comprises a stage ET_q. Each stage ET_q has four inputs 142 to 145 and three outputs 146 to 148. The input 142 receives a code CP_q to be permuted. Each code CP_q comprises 2^d−q blocks BCP_j,q, of 2^q bits each. These blocks BCP_j,q do not overlap and are immediately consecutive. The subscript j is equal to the order number of the block BCP_j,q counting from the block BCP_0,q that contains the least significant bits.

Each input 142 is connected to the output 146 of the previous stage ET_q+1, except the input 142 of the stage ET_NbE−1 , which receives the code C₁ at its input 142.

The input 143 receives the parameters α_j,q of the stage E_q of the function F_α. Here, to this end, the input 143 of the stage ET_q is connected to the output 147 of the previous stage ET_q+1.

The input 144 receives a permutation key K_ETq that contains only the parameters α_j,q−1 to α_j,0 that are required for implementing the transpositions of the stages E_q−1 to E₀ of the function F_α. This key K_ETq is divided into 2^d−q blocks BK_j,q of 2^q bits each. Each block BK_j,q contains only the parameters α_j,q−1 to α_j,0 of the transpositions to be applied to the bits placed within the block B_j,q of the data item D₁. Within each block BK_j,q, the parameters α_j,q−1 to α_j,0 are classified in a predetermined order. For example, here, they are first of all classified in descending order of stages and the various parameters of one and the same stage E_q−1 are also classified in descending order of subscript j. For example, let us suppose that q=3, the transpositions to be applied to the block B_2,3 are, successively, the transpositions T_α2,2, T_α5,1, T_α4,1, T_α11,0, T_α10,0, T_α9,0 and T_α8,0. This follows from the organization of the transpositions T_αj,q in the function F_α described with reference to FIG. 3. From then on, the block BK_2,3 comprises the parameters α_2,2, α_5,1, α_4,1, α_11,0, α_10,0, α_9,0 and α_8,0 and an additional 0 to reach the size of 2³ bits.

The input 145 receives the coefficient a_q of the shift to be applied to the data item D₁.

The stage ET_q comprises two permutators 150 and 151, two shift registers 154 and 155 and two multiplexers 158 to 159.

The permutator 150 executes the transpositions of the stage E_q on the data received at its input 142. In other words, the permutator 150 executes the following function: T_αm,qo . . . oT_αj,qo . . . oT_α0,q(CP_q), where m is equal to 2^d−q−1.

To do this, the permutator 151 is connected to the input 142 in order to receive the code CP_q to be permuted and to the input 143 in order to receive the parameters α_m,q to α_0,q.

The code permuted by the permutator 150 is transmitted directly to a first input of the multiplexer 158 and, in parallel, to an input of the register 154.

The register 154 performs a logic shift of 2^q bits to the left on the bits received at its input in order to obtain a permuted and shifted code, which is delivered to a second input of the multiplexer 158.

The multiplexer 158 connects the first input directly to the output 146 if the coefficient α_q received at the input 145 is equal to zero. If the coefficient a_q received is equal to one, the multiplexer 158 connects its second input directly to the output 146.

The permutator 151 and the register 155 are identical to the permutator 150 and the register 154, respectively. They therefore perform the same operations as the permutator 150 and the register 154, respectively, but applied to the key K_ETq, that is to say to the key received at the input 144.

The multiplexer 159 selects the permuted key delivered by the permutator 151 if the coefficient α_q is equal to zero. Otherwise, it selects the permuted and shifted key delivered by the register 155. Moreover, using the key selected on the basis of the coefficient a_q, the multiplexer 159 delivers to the output 147 the parameters α_j,q−1 required for configuring the transpositions of the stage ET_q−. It also delivers the key K_ETq−1 to the output 148.

The operation of the circuit 140 is as follows: the permutator 150 of the stage ET_q cancels the effect of the transpositions T_j,q of the stage E_q that is applied to the data item D₁ when the code C₁ is calculated. It should be remembered here that T_j,q o T_j,q is the identity function. Thus, in the permuted code delivered by the permutator 150, the blocks BCP_j,q occupy the same position as the one that they had in the data item D₁. However, within each block BCP_j,q, the position of the bits corresponds to the result of application of the transpositions of the stages E_j,q−1 to E_j,0 to the bits placed within the block B_j,q of the data item D₁. Thus, the order of the bits within the blocks BCP_j,q is not the same as the order of the bits within the block B_j,q of the data item D₁. However, this is not important for the application of a logic shift of 2^q bits to the left, because such a shift shifts only blocks of 2^q bits. Thus, applying the shift of 2^q bits in the stage ET_q allows the code C_res−t to be computed without this requiring:

the data item D₁ to be found from the code C₁, then

the various logic shifts of 2^q bits to be applied to this data item D₁ that has been found.

In this embodiment, the parameters α_j,q−1 to α_j,0 required for configuring all of the transpositions T_j,q−1 to T_j,0 to be applied to the bits placed within a block BCP_j,q received at the input 142 are placed within the block BK_j,q of the same size and occupying the same position j in the key K_ETq received at the input 144. At the output 148, this relationship between the position of the blocks BCP_j,q−1 and the position of the blocks BK_j,q−1 is preserved. In other words, in the key delivered at the output 148, the block BK_j,q−1 contains all of the parameters α_j,q−2 to α_j,0 required for configuring the transpositions to be applied to the bits placed within the block BCP_j,q−1 delivered at the output 146.

To preserve this relationship, the permutator 151 and the register 155 apply the same transpositions and the same shifts, respectively, as those applied to the code CP_q, but this time to the key K_ETq received. Next, the multiplexer 159 extracts from the selected key the parameters α_j,q−1 to be delivered to the output 147. The multiplexer 159 also extracts the parameters α_j,q−2 to α_j,0 and generates the key K_ETq−1 that is delivered to the output 148. Here, the multiplexer 159 identifies the parameters α_j,q to be extracted on the basis of their position in the selected key.

The output 146 of the stage ET₀ delivers the code C_res−t to be compared with the code C_res−p in order to check, in step 100, that the execution of the shift instruction has taken place without a fault.

FIG. 13 shows a computation circuit 149 for computing the code C_res−t when the arithmetic instruction executed by the unit 10 is a rotation instruction for rotating the bits of the data item D₁ one bit to the left. This rotation instruction is identical to the shift instruction for shifting one bit to the left, except that the most significant bit of the data item D₁ is reinjected on the right and therefore becomes the least significant bit after this rotation has been executed.

For each stage E_q of the function F_α, the circuit 149 comprises a corresponding stage ER_q. Each stage ER_q comprises a component CC_αj,q for each transposition T_αj,q of the stage E_q. More precisely, each component CC_αj,q is associated with a respective corresponding transposition T_αj,q.

For q less than NbE−2, each component CC_αj,q is identical to the component CD_αj,q of the circuit 120. The component CC_α0,NbE−1 is shown in FIG. 13 in the particular case where NbE is equal to three. In FIG. 13, it is therefore denoted by the reference CR_α0,2. The component CC_α0,NbE−1 has two inputs 152 and 153 and two outputs 156 and 157. The inputs 152 and 153 are connected to the outputs 124 of the components CC_α0,NbE−2 and CC_α1,NbE−2 respectively. The outputs 156 and 157 are connected to the inputs 129 of the components CC_α1,NbE−2 and CC_α0,NbE−2, respectively.

The output 156 consistently delivers the same value as that received at the input 152. The output 157 consistently delivers the same value as that received at the input 153. In other words, the component CC_α0,NbE−1 consistently reverses the position of the bits received at its inputs. This allows the most significant bit to be reinjected at the location intended to receive the least significant bit.

The operation of the circuit 149 is derived from the explanations given for the circuit 120.

FIG. 14 shows the computation circuit 160 for computing the code C_res−t when the arithmetic instruction executed by the unit 10 is an addition instruction for adding the data D₁ and D₂. This time, the code C_res−t is computed from the codes C₁ and C₂. The circuit 160 computes the code C_res−t, which, in the absence of an execution fault, is equal to F_α(D_res−p).

FIG. 14 shows the circuit 160 in the particular case where the size of the codes C₁ and C₂ and of the code C_res−t is equal to 8 bits. In this figure, the 8 bits of the codes C₁ and C₂ are denoted, in the order towards the most significant bit, α₀ to α₇ and b₀ to b₇, respectively. The 8 bits of the computed code C_res−t are denoted, in the same order, s₀ to s₇.

The circuit 160 comprises a stage 162 of adders and a carry look-ahead unit 164.

The stage 162 comprises an adder AD_p for each pair of bits a_p, b_p to be added. The subscript p denotes the position of the bits in the codes C₁, C₂ and C_res−t. These adders AD_p are structurally identical to one another and differ from one another only in the bits a_p and b_y that they add.

The adder AD_p is shown in more detail in FIG. 15. The adder AD_p has three inputs 170 to 172 and three outputs 174 to 176. The inputs 170 and 171 receive the bits a_p and b_p, respectively.

The input 172 receives a carry c_p to be used in the addition of the bits a_p and b_p.

The output 174 delivers the result a_p.b_p. The output 175 delivers the result a_p+b_p. The output 176 delivers the bit s_p. The bit s_p is computed by the adder AD_p using the following relationship: s_p=a_pXOR b_pXOR c_p.

The function of the unit 164 is to rapidly propagate the various carries c_p to be used by the adders AD_p. To that end, here, the unit 164 computes the carries c_p from the information delivered at the outputs 174 and 175 of each adder AD_p.

In this embodiment, the unit 164 comprises a stage EA_q for each stage E_q of the function F_α. Each stage EA_q comprises 2^d−q−1 components CA_αj,q, where the subscript j is the order number of the component CA_αj,q in the stage EA_q. Each component CA_αj,q is associated with a respective transposition T_αj,q of the function F_α.

Here, the components CA_αj,q are all structurally identical to one another and are distinguished only by their connection to the other components of the circuit 160.

The component CA_αj,q is shown in more detail in FIG. 16. The component CA_αj,q has six inputs 180 to 185 and four outputs 188 to 191. The output 188 delivers the result (G_r+P_r.G_I).k+(G_I+P_I.G_r).k′, where G_r, P_r, G_I, P_I and k are the values received at the inputs 180, 181, 182, 183 and 185, respectively.

The output 189 delivers the result P_I.P_r. The output 190 delivers the result c_i.k′+(G_I+P_I.c_i).k, where c_i is the value received at the input 184. The output 191 delivers the result c_i.k+(G_r+P_r.C₁).k′.

The input 185 receives the parameter α_j,q of the transposition T_αj,q associated with this component CA_αj,q.

The inputs 180 and 181 of each component CA_αj,0 of the stage EA₀ are connected to the outputs 174 and 175, respectively, of the adder AD_2j. The inputs 182 and 183 of each component CA_αj,0 of the stage EA₀ are connected to the outputs 174 and 175, respectively, of the adder AD_2j+1.

The outputs 190 and 191 of each component CA_αj,0 of the stage EA₀ are connected to the inputs 172 of the adders AD_2j and AD_2j+1 , respectively.

For q greater than zero:

the inputs 180 and 181 of each component CA_αj,q of the stage EA_q are connected to the outputs 188 and 189, respectively, of the component CA_α2j,q−1 of the lower stage EA_q−1,

the inputs 182 and 183 of each component CA_αj,q are connected to the outputs 188 and 189 of the component CA_α(2j+1)_,chi of the lower stage EA_q−1,

the outputs 190 and 191 of each component CA_αj,q are connected to the inputs 184 of the components CA_α2j,q−1 and CA_{α(2j+1),q−1}, respectively, of the stage ET_q−1

The outputs 188 and 189 of the component CA_α0,NbE−1 are not used. The input 184 of the component CA_α0,NbE−1 allows the carry computed by another circuit, for example identical to the circuit 160, to be received. This allows multiple circuits 160 to be linked to one another, so as to perform additions on data of greater size.

The unit 164 functions as a conventional carry look-ahead computation unit. Here, however, this conventional unit is modified to take account of the transpositions T_αj,q and therefore the parameters α_j,q that are used for computing the code C_res−t. In summary, the components CA_αj,q propagate the carry to the right when the parameter α_j,q is equal to zero and to the left when the parameter α_j,q is equal to one.

Section IV—Variants

Variants of the Function Q_α:

In the relationship Q_a(D_i)=P o F_α(D_i), the function P is not necessarily the identity function. For example, the function P is a compression function that constructs, from each of the bits of the result F_α(D_i), a code C₁ whose size, in terms of the number of bits, is less than 2^d. The reason is that when the function P is the identity function, the size of the code C_i is equal to the size of the data item D_i, that is to say equal to 2^d. Now, in some contexts, it is desirable to reduce the size of the code C. For example, this is desirable in order to reduce the space that it can take up in the cache memory 27. By way of illustration, to this end, the function P is the 30 function that performs the following operations:

1) the function P divides the result F_α(D_i) into two blocks P₀ and p₁ of bits of the same size, then,

2) the function P performs an “EXCLUSIVE-OR” between the blocks P₀ and p₁. In this case, the size of the code C_i is halved and is equal to 2^d−1.

Many other compression functions P are possible. The function P can also be different from the identity function and from a compression function. For example, the function P is an encryption or other function.

When the function P is different from the identity function, each of the computation circuits described here is broken down into a first and a second subcircuit. The first subcircuit is identical to one of the computation circuits described earlier. This first subcircuit therefore delivers a code C_res−int, which, in the absence of an execution fault, is consistently equal to the result F_α(D_res−p). The second subcircuit applies the predetermined function P to the code C_res−int in order to obtain the code C_res−t.

The various variants are described below in the particular case where the function P is equal to the identity function. However, these variants also apply to the case where the function P is different from the identity function.

As a variant, the transposition T_αj,q permutes the blocks B_2j+1,q and B_2j,q when the parameter α_j,q =0 and does not permute them when the parameter α_j,q=1.

The function F_α has been described in the particular case where the stages of transpositions first transpose the blocks of greater size and end by transposing the blocks of smaller size. However, as a variant, the stages E_q of transpositions can be executed and classified in reverse order. In this case, the transpositions of smaller size are applied first, ending by applying the transposition T_α0,NbE−1 of greater size. The order in which the various stages E_q are classified does not modify the bit locality property described earlier. Thus, even when the order of the stages E_q is reversed, it is possible to construct fast and simple computation circuits for computing the code C_res−t for arithmetic instructions.

As a variant, one or more stages of the function F_α are omitted.

Some of the transpositions T_αj,q can be omitted. In this case, at least one of the stages comprises fewer than 2^d−q−1 transpositions T_αj,q.

Variants of the Computation Circuits for Computing the Code C_res−t:

The teaching provided here in the case of a few arithmetic instructions, such as shifts, rotations and additions, can be applied to other arithmetic instructions. In particular, it is possible to take the teaching provided in these particular cases as a basis for developing computation circuits for computing a code C_res−t for other arithmetic instructions. For example, the circuit 120 can be modified to compute the code C_res−t corresponding to a logic shift instruction for shifting 1 bit to the right. In practice, it is sufficient, for this purpose, to retain the same circuit as the circuit 120, but to send the complement of the parameter α_j,q, that is to say the parameter α_j,q′, to the input 128 or 138 of each of the components CD_αj,q.

Equally, it is possible to construct a computation circuit for computing the code C_res−t for an arithmetic shift instruction for shifting one bit to the left or to the right. An arithmetic shift is distinguished from the logic shifts described earlier by the fact that the most significant bit remains unchanged and is therefore not shifted, unlike the other bits.

In the circuit 140, each 2^q-bit shift register can be replaced by a register that performs a rotation of 2^q bits. The circuit thus obtained computes the result C_res−t, which corresponds to a rotation instruction for rotating the bits of the data item D₁ a_NbE−12^NbE−1+ . . . +a_q2^q+ . . . +a₀ bits to the left. By replacing these registers with registers that perform a shift to the right or a rotation to the right, the circuit obtained computes the code C_res−t corresponding to a shift or rotation instruction for shifting or rotating a_NbE−12^NbE−1+ . . . +a_q2^q+ . . . +a₀ bits to the right.

It is also possible to link multiple computation circuits in order to compute the code C_res−t for an operation that corresponds to the composition of multiple suboperations for each of which there is already a computation circuit for computing the code C_res−t. For example, if the executed operation is a logic shift instruction for shifting two bits to the left, the code C_res−t is computed by applying the circuit 120 twice. To that end, the code C₁ is first injected at the inputs of the circuit 120 and a first intermediate code CI_cres−t is obtained. Next, the code CI_cres−t is injected at the inputs of this same circuit 120 in order to obtain the desired code C_res−t.

Similarly, the circuits 120 and 130 can be linked in order to compute the code C_res−t for a logic shift to the left for any number of bits.

If the coefficients a_q are consistently constant, as a variant, the inputs 145 of the circuit 140 are omitted. In this case, the coefficients α_q are wired inside each stage ET_q. For example, the multiplexer 158 is omitted. If the value of the coefficient a_q is consistently equal to one, then the output of the register 154 is directly connected to the output 146. The multiplexer 159 is also simplified, since it consistently selects the output of the register 155. Conversely, if the coefficient a_q is consistently equal to zero, then the output of the permutator 150 is directly connected to the output 146 and the register 154 is omitted. Equally, the register 155 is omitted.

As a variant, the component CA_α0,NbE−1 of the circuit 160 is devoid of the outputs 188 and 189, which are not used.

As a variant, the output 175 of the component AD_p delivers the result α_p XOR b_y rather than the result α_p+b_p.

In a simplified embodiment, only the execution of some arithmetic or logic instructions is secure. For example, only the execution of one of the following instructions is made secure by implementing the method described here: the bit shift instruction, the bit rotation instruction and the bit addition instruction. The execution of the other instructions, such as the logic instructions, is thus not secure. In this latter case, this means that, for these other instructions, no code _res−t is calculated and step 100 is omitted.

Other Variants:

The module 28 is not necessarily a hardware module of a single block. As a variant, it is made up of multiple hardware submodules that each perform one of the specific functions of the module 28. These hardware submodules are thus preferably embedded as close as possible to the data that they process. For example, in this case, the hardware submodule that computes the code C_i associated with each data item D₁ is embedded in the cache memory 27. From then on, the code C_i associated with each data item D_i stored in the cache memory 27 is computed locally in this cache memory.

As a variant, each instruction of the machine code is also associated with an integrity code F_α(I_i) computed from the value of the loaded instruction I_i. This code F_α(I_i) is verified just before the unit 10 executes the instruction I_i. This allows the signalling of an execution fault to be triggered if the instruction I_i is modified in the queue 22.

It is possible to associate the code C_i with the data item D_i in various ways. For example, instead of storing the code C_i in the same register R_i as the one that contains the data item D₁, the code C_i is stored in a register RC_i associated with the register R_i rather than in the register R_i.

The secret key a can be modified, for example, at regular intervals.

Other embodiments of step 100 are possible. For example, the module 28 computes, as in the case of the circuit 140, a code C_res−t that is equal to the result D_res−p in the absence of an execution fault. In this case, the code C_res−t is computed using the following relationship: C_res−t=F_a^-1(C I_res−t), where:

the function F_α⁻¹ is the inverse of the function F_α, and

CI_res−t is the code computed, for example, by the circuits 120, 130, 149 or 160.

The various computation circuits for computing the code C_res−t that are described here can be implemented independently of one another.

Section V—Advantages of the Described Embodiments

Computing the code C; using a secret key α makes the method for executing the machine code more robust in the face of attempted attacks. The reason is that the attacker then has greater difficulty in falsifying the code C_res−t so that it corresponds to an expected code when an execution fault has been deliberately introduced. Thus, the methods described earlier have the same advantages in terms of robustness as the one described in the article by DEMEYER2019. Moreover, the use of a function F_α that has the locality property described earlier makes it possible to obtain computation circuits for computing the code C_res−t that are simpler and faster than those required for implementing the method of the article DEMEYER2019.

The circuits 120, 130, 140, 149 and 160 each allow simple and fast computation of the code C_res−t corresponding to a specific arithmetic instruction.

The circuit 110 also allows simple and fast computation of the code C_rest−t for all Boolean operations.

Claims

1. Microprocessor equipped with an arithmetic and logic unit and with a hardware security module, wherein: a) the arithmetic and logic unit is capable of executing an arithmetic instruction, comprising an opcode and one or more operands, that, when executed by the arithmetic and logic unit of the microprocessor, causes a mathematical operation D1*D2* . . . *Dn to be performed and the result of this operation to be stored in a register Rres−p where:the subscript n is equal to the number of data items D1 processed by the arithmetic instruction, the subscript n being greater than or equal to one,D1 to Dn are data items that are stored in registers R1 to Rn, respectively, of the microprocessor, the size, in terms of the number of bits, of each of these data items Di being equal to 2d, where d is an integer greater than two,the registers R1 to Rn are the registers denoted by the operands of the arithmetic instruction,the symbol “*” is the arithmetic operation denoted by the opcode of the arithmetic instruction,b) the microprocessor is configured to perform the following operations:1) for each data item Di, computation of a code Ci using a relationship Ci=Qa(Di) and association of the computed code Ci with the data item Di, the function Qα being a pre-programmed function configured by a secret key a that is pre-stored in the microprocessor and known only to the microprocessor,2) each time an instruction for loading a data item Di into a register Ri of the microprocessor is executed by the microprocessor, the loaded data item Di is stored in the register Ri and the code Ci associated therewith is stored in the same register Ri or in a register associated with the register Ri, then3) execution of the arithmetic instruction and storage of the result Dres−p of this execution in the register Rres−p, and computation, by the security module, of a code Cres−t using the codes C1, C2, . . . , Cn and without using the result Dres-p, then4) checking that the computed code Cres−t corresponds to a code Cres−p obtained from the result Dres−p and triggering of the signalling of an execution fault if the code Cres−t does not correspond to the code Cres−p and, otherwise, suppressing this signalling, wherein the function Qα is defined by the following relationship: Qa(Di)=P o Fα(Di), where P is a predetermined function and Fα is a function defined by the following relationship: Fα(Di)=E0 o . . . o Eq o . . . o ENbE−1(Di), where each function Eq is a stage of transpositions and the index q is an order number between zero and NbE−1, where NbE is a whole number greater than one and less than or equal to d, each stage Eq of transpositions being defined by the following relationship: Eq(x)=Tαm,q o . . . o Tαj,q o . . . o Tα1,q o Tα,q(x), where:x is a variable whose size, in terms of the number of bits, is equal to the size of the data item Di,Tαj,q is a conditional transposition, configured by the parameter αj,q, that permutes two blocks of bits B2j+1,q and B2j,q of the variable x when the parameter αj,q is equal to a first value and that does not permute these two blocks of bits when the parameter αj,q is equal to a second value, the transposition Tαj,q being distinguished from all of the other transpositions of the function Fα by the fact that it is the only one that permutes the two blocks B2j+1,q and B2j,q when the parameter αj,q is equal to the first value, the blocks B2j+1,q and B2j,q of all of the transpositions Tαj,q of the stage Eq being different from one another and not overlapping in such a way that all of the transpositions Tαj,q of the stage Eq can be executed in parallel,“m+1” is the total number of transpositions Tαj,q of the stage Eq,“j” is an order number identifying the transposition Tαj,q among the other transpositions of the stage Eq,the symbol “o” denotes the function-composition operation,the concatenation of the bits of all of the parameters αj,q of all of the stages Eq is equal to the value of the secret key α, andfor all of the stages Eq for which q is less than NbE−1 and for all of the transpositions Tαj,q of this stage, the blocks B2j+1,q and B2j,q are placed within one and the same block of greater size permuted by a transposition of the higher stage Eq+1 when the parameter of this transposition of the higher stage Eq+1 is equal to the first value.

2. Microprocessor according to claim 1, wherein the number NbE is equal to d,the sizes of the blocks permuted by all of the transpositions Tαj,q of one and the same stage Eq are equal to 2q,the blocks B2j+1,q and B2j,q permuted by each transposition Tαj,q are adjacent.

3. Microprocessor according to claim 2, wherein the arithmetic instruction is a shift instruction for shifting the bits of the data item D1 one bit to the left or to the right and the code Cres−t is equal to Qa(Dres−p) in the absence of an execution fault, the security module comprising, to this end, a computation circuit for computing the code Crest−t from the code C1 and without using the result Dres−p, this computation circuit comprising: for each stage Eq of the function Fα, a stage EDq comprising components CDαj,q, each component CDαj,q being associated with a respective transposition Tαj,q of the stage Eq,the components CDαj,q of the stages ED0 to EDNbE−2 all being structurally identical to one another and each having a first, a second, a third and a fourth input and a first, a second and a third output: the first output delivering the result a.k′+c.k, where: “a”, “k” and “c” are the values received at the first, third and fourth inputs, respectively, of this component,the symbol “ + ” denotes the “OR” logic operation,the symbol “ . ” denotes the “AND” logic operation, andthe symbol “ ′ ” denotes the “NOT” logic operation,the second output delivering the result b.k+c.k′, where “b” is the value received at the second input of this component,the third output delivering the result a.k+b.k′,the component CDαj,NbE−1 of the stage EDNbE−1 having a first, a second and a third input and a first and a second output: the first and second outputs delivering the results “a” and “0”, respectively, when the third input is equal to “0”, where “a” is the value received at the first input and “0” is the value zero, andthe first and second outputs delivering the results “0” and “b”, respectively, when the third input is equal to “1”, where “b” is the value received at the second input,for q equal to zero, the first and second inputs of each component CDαj,0 of the stage ED0 being connected to the 2j-th bit and to the (2j+1)-th bit, respectively, of the code C1, and for q greater than zero, the first and second inputs of each component CDαj,q of the stage EDq being connected, respectively, to the third outputs of the 2j-th and (2j+1)-th components, respectively, of the stage EDq−1,the third input of each component CDαj,q is connected to the parameter αj,q when the instruction is a left shift and to the complement αj,q′ of the parameter αj,q when the instruction is a right shift,for q greater than zero, the first and second outputs of each component CDαj,q are connected to the fourth input of the (2j+1)-th and 2j-th components, respectively, of the stage Eq−1, and for q equal to zero, the first and second outputs of each component CDαj,0 deliver the (2j+1)-th and 2j-th bits, respectively, of the code Crest−t.

4. Microprocessor according to claim 2, wherein the arithmetic instruction is a shift instruction for shifting the bits of the data item D1 2r bits to the left or to the right, where r is a whole number greater than one and less than NbE, and the code Crest−1 is equal to Qa(Dres−p) in the absence of an execution fault, the security module comprising, to this end, a computation circuit for computing the code Crest−t from the code C1 and without using the result Dres−p, this computation circuit comprising a block permutator and a bit scheduler, the permutator comprises, for each stage Eq of transpositions included in the group of stages Er to ENbEt that permute blocks of size greater than or equal to 2r, a stage EDq comprising components CDαj,q, each component CDαj,q being associated with a respective transposition Tαj,q of the stage Eq,the components CDαj,q of the stages EDr to EDNbE−2 all being structurally identical to one another and each having a first, a second, a third and a fourth input and a first, a second and a third output: the first output delivering the result a.k′+c.k, where: “a”, “k” and “c” are the values received at the first, third and fourth inputs, respectively, of this component,the symbol “ + ” denotes the “OR” logic operation,the symbol “ . ” denotes the “AND” logic operation, andthe symbol “ ′ ” denotes the “NOT” logic operation,the second output delivering the result b.k+c.k′, where “b” is the value received at the second input of this component,the third output delivering the result a.k+b.k′,the component CDαj,NbE−1 of the stage EDNbE−1 having a first, a second and a third input and a first and a second output: the first and second outputs delivering the results “a” and “0”, respectively, when the third input is equal to “0”, where “a” is the value received at the first input and “0” is the value zero, andthe first and second outputs delivering the results “0” and “b”, respectively, when the third input is equal to “1”, where “b” is the value received at the second input,the first and second inputs of each component CDαj,r of the stage EDr being connected to the 2j-th block of 2r bits and to the (2j+1)-th block of 2r bits, respectively, of the code C1, and for q greater than r, the first and second inputs of each component CDαj,q of the stage EDq being connected, respectively, to the third outputs of the 2j-th and (2j+1)-th components, respectively, of the stage EDq−1,the third input of each component CDαj,q is connected to the parameter αj,q when the instruction is a left shift and to the complement αj,q′ of the parameter αj,q when the instruction is a right shift,for q greater than r, the first and second outputs of each component CDαj,q are connected to the fourth input of the (2j+1)-th and 2j-th components, respectively, of the stage Eq−1 , and for q equal to r, the first and second outputs of each component CDαj,r deliver the (2j+1)-th intermediate block BCI2j+1,r of 2r bits and the 2j-th intermediate block BCI2j,r of 2r bits, respectively, of an intermediate code CIrest−t, each block BCIx,r being identical to a respective block BCy,r of the code C1, the order of the bits within this block BCy,r having been obtained, during computation of the code C1, by applying a first set of transpositions of the stages Er−1 to E0 to a respective block BDz,r of 21 bits of the data D1, where the subscripts x, y and z are each an identifier of a position of a block of 21 bits in a data item or a code of 2d bits, the subscript x being equal to 2j+1 if the order number of the block of 2r bits is odd and equal to 2j otherwise, and the first set of transpositions being dependent on the position y,the scheduler capable of replacing each intermediate block BCIx,r of the intermediate code with a block BCRx,r of 2r bits that is equal to the result of the application of a second set of transpositions of the stages Er−1 to E0 to the block BDx,r of 2r bits of the data item D1, the second set of transpositions being dependent on the position x.

5. Microprocessor according to claim 4, wherein the scheduler is capable, for each block BCIx,r: of comparing each parameter of a current permutation key with the corresponding parameter of a desired permutation key, the current key containing the parameters of all of the transpositions of the first set of transpositions used, during computation of the code C1, to permute the bits placed within the block BDz,r, the desired key containing the parameters of all of the transpositions of the second set of transpositions used, during a computation of Fα(Dres−p), to permute the bits placed within the block BDx,r, and,each time the compared parameters are different, of executing the transposition configured by this parameter on the bits of the block BCIx,r and,each time the compared parameters are identical, of keeping the order of the bits of the block BCIx,r unchanged.

6. Microprocessor according to claim 2, wherein the arithmetic instruction is a rotation instruction for rotating the bits of the data item D1 one bit to the left or to the right and the code Cres−t is equal to Qa(Dres−p) in the absence of an execution fault, the security module comprising, to this end, a computation circuit for computing the code Crest-t from the code C1 and without using the result Dres−p, this computation circuit comprising: for each stage Eq of the function Fα, a stage EDq comprising components CCαj,q, each component CCαj,q being associated with a respective transposition Tαj,q of the stage Eq,the components CCαj,q of the stages ED0 to EDNbE−2 all being structurally identical to one another and each having a first, a second, a third and a fourth input and a first, a second and a third output: the first output delivering the result a.k′+c.k, where: “a”, “k” and “c” are the values received at the first, third and fourth inputs, respectively, of this component,the symbol “ + ” denotes the “OR” logic operation,the symbol “ . ” denotes the “AND” logic operation, andthe symbol “ ′ ” denotes the “NOT” logic operation,the second output delivering the result b.k+c.k′, where “b” is the value received at the second input of this component,the third output delivering the result a.k+b.k′,the component CCαj,NbE−1 of the stage EDNbE−1 having a first and a second input and a first and a second output, the first and second outputs delivering the results “a” and “b”, respectively, where “a” and “b” are the values received at the first and second inputs, respectively, andfor q equal to zero, the first and second inputs of each component CCαj,0 of the stage ED0 being connected to the 2j-th bit and to the (2j+1)-th bit, respectively, of the code C1, and for q greater than zero, the first and second inputs of each component CCaj,q of the stage EDq being connected, respectively, to the third outputs of the 2j-th and (2j+1)-th components, respectively, of the stage EDq−1,the third input of each component CCαj,q is connected to the parameter αj,q when the instruction is a left rotation and to the complement αj,q′ of the parameter αj,q when the instruction is a right rotation,for q greater than zero, the first and second outputs of each component CCαj,q are connected to the fourth input of the (2j+1)-th and 2j-th components, respectively, of the stage Eq−1, and for q equal to zero, the first and second outputs of each component CCαj,0 deliver the (2j+1)-th and 2j-th bits, respectively, of the code Crest−t.

7. Microprocessor according to claim 2, wherein the arithmetic instruction is a shift instruction for shifting the bits of the data item D1 aNbE−12NbE−1+ . . . +aq2q+ . . . +a0 bits to the left or to the right, where the coefficients aq are predetermined coefficients, and the security module comprises a computation unit for computing the code Crest−t from the code C1 and without using the result Dres−p, this code Crest−t being identical to the result Dres−p in the absence of an execution fault during the execution of this shift instruction by the arithmetic and logic unit, this computation circuit comprising, for each stage Eq of the function Fα, a stage ETq, each stage ETq having: a first input for receiving a code to be permuted, this code to be permuted being formed by a juxtaposition of blocks BCPy,q of 2q bits each, the subscript y being the order number of the block of 2q bits in this juxtaposition, this first input receiving the code C1 when q=NbE−1,a second input for receiving the parameters αm,q to α0,q of the transpositions of the stage Eq, this second input receiving the parameters α0,NbE−1 when q=NbE−1,a third input for receiving a permutation key for the next stages, this permutation key being formed by a juxtaposition of blocks BKy,q of 2q bits each, each block BKy,q containing only the parameters of the transpositions of the next stages Echi to E0 that permute the bits placed within the block BCPy,q of the code received at the first input, this third input receiving the private key a of the parameter α0,NbE−1 when q=NbE−1,a first permutator capable of performing the permutation Tαm,q o . . . o Tαj,q o . . . o Tα1,q o Tαo,q(x) in order to obtain a permuted code, where the variable x is equal to the code to be permuted that is received at the first input and the permutation parameters αm,q to α0,q are those received at the second input,a first shift register capable of performing a shift of 2q bits for the permuted code in order to obtain a permuted and shifted code,a first multiplexer configured to select the permuted code if the coefficient aq is equal to “0” and to select the permuted and shifted code if the coefficient aq is equal to “1”, this first multiplexer having an output that delivers to the first input of the next stage ETq−1 the code selected from the permuted code and the permuted and shifted code,a second permutator capable of executing the permutation Tαm,q o . . . o Tαj,q o . . . o Tα2,q o Tα0,q(k) in order to obtain a permuted key, where the variable k is equal to the permutation key received at the third input,a second shift register capable of performing a shift of 2q bits for the permuted key in order to obtain a permuted and shifted key, the first and second shift registers shifting the bits to the left if the shift instruction is a shift instruction for shifting the bits to the left and, otherwise, shifting the bits to the right if the shift instruction is a shift instruction for shifting the bits to the right,a second multiplexer configured to select the permuted key if the coefficient aq is equal to “0” and to select the permuted and shifted key if the coefficient aq is equal to “1”, this second multiplexer having: a first output that delivers, to the second input of the next stage ETq−1, the parameters αm,q−1 to α0,q−1 extracted from predetermined locations of the key selected from among the permuted key and the permuted and shifted key, anda second output that delivers, to the third input of the next stage ETq−1, a permutation key for the next stages comprising the parameters αm,q−2 to α0,0 extracted from predetermined locations of the key selected from among the permuted key and the permuted and shifted key, the permutation key delivered at the second output being formed by a juxtaposition of blocks BKy,q−1 of 2q−1 bits each, each block BKy,q−1 containing the parameters of the transpositions of the next stages Eq−2 to E0 to be applied to the block BCPy,q−1 of the data item received at the first input of the next stage ETq−1, andduring operation 4), the microprocessor is configured to compare the code Cres−t with the result Dres−p and to trigger the signalling of a fault only if they are not identical.

8. Microprocessor according to claim 2, wherein the arithmetic instruction is an addition instruction for adding the data item D1 to a data item D2, and the security module comprises a computation circuit for computing the code Crest−t from the codes C1 and C2 and without using the result Dres−p, this computation circuit comprising: an adder stage comprising, for each pair of bits of the codes C1 and C2 to be added, an adder ADp, where the subscript p is the order number of the two bits of the codes C1 and C2 to be added by this adder, this order number varying from zero to 2d−1, each adder ADp having: a first, a second and a third input, the first and second inputs being intended to receive the bit of the code C1 and the bit of the code C2 to be added, respectively,a first output delivering the result a.b, where: “a” and “b” are the values received at the first and second inputs, respectively, andthe symbol “ . ” denotes the “AND” logic operation,a second output delivering the result a+b, where the symbol “+ ” denotes the “OR” logic operation,a third output delivering the result a XOR b XOR c, where “c” is the value received at the third input and XOR is the symbol of the “EXCLUSIVE-OR” logic operation, the concatenation of the bits delivered at the third outputs of the various adders ADp of the addition stage forming the code Cres−t,a carry look-ahead unit having an input at which the permutation key a is received, this carry look-ahead unit being capable of determining the values of the third inputs of the adders of the adder stage that need to have a carry delivered to them from the results delivered at the first and second outputs of each adder and from the parameters αj,q, of the key a received at its input.

9. Microprocessor according to claim 8, wherein: the carry look-ahead unit comprises, for each stage Eq of the function Fα, a stage EAq comprising components CAαj,q, each component CAαj,q being associated with a respective transposition Tαj,q of the stage Eq,the components CAαj,q of the stages EA0 to EANbE−2 all being structurally identical to one another and each having a first, a second, a third, a fourth, a fifth and a sixth input and a first, a second, a third and a fourth output: the first output delivering the result (Gr+Pr.GI).k+(GI+PI.Gr).k′, where: Gr, Pr, GI, PI and k are the values received at the first, second, third, fourth and sixth inputs, respectively, of this component,the symbol “ + ” denotes the “OR” logic operation,the symbol “ . ” denotes the “AND” logic operation, andthe symbol “ ′ ” denotes the “NOT” logic operation,the second output delivering the result PI. Pr,the third output delivering the result ci.k′+(GI+PI.ci).k, where ci is the value received at the fifth input,the fourth output delivering the result ci.k+(Gr+Pr. Ci_.k′,the component CAαj,NbE−1 of the stage EANbE−1 has a first, a second, a third, a fourth, a fifth and a sixth input and a first and a second output: the first output delivering the result ci.k′+(GI+PI.ci).k, where ci is the value received at the fifth input,the second output delivering the result ci.k+(Gr+Pr.ci).k′,the first and second inputs of each component CAαj,0 of the stage EA0 are connected to the first and second outputs, respectively, of the adder AD2j, where the adder AD2j is the adder ADp, the subscript p of which is equal to 2j,the third and fourth inputs of each component CAαj,0 of the stage EA0 are connected to the first and second outputs, respectively, of the adder AD2j+1, where the adder AD2j+1 is the adder ADp, the subscript p of which is equal to 2j+1,the third and fourth outputs of each component CAαj,0 of the stage EA0 are connected, respectively, to the third inputs of the adders AD2j and AD2j+1, respectively,for q greater than zero, the first and second inputs of each component CAαj,q of the stage EAq are connected to the first and second outputs, respectively, of the component CAa2j,q−1 of the stage EAq−1,for q greater than zero, the third and fourth inputs of each component CAαj,q of the stage EAq are connected to the first and second outputs, respectively, of the component CAαa(2j+1),q−1 of the stage EAq−1,for q greater than zero, the third and fourth outputs of each component CAαj,q of the stage EAq are connected, respectively, to the fifth inputs of the components CAα2j,q−1 and CAα(2j+1),q−1, respectively,the sixth input of each component CAαj,q is capable of receiving the parameter αj,q.

10. Microprocessor according to claim 1, wherein: the arithmetic and logic unit is capable of executing a logic instruction that, when executed, causes a Boolean operation D1&D2& . . . &Dn to be performed and the result of this Boolean operation to be stored in the register Rres−p, where the “&” symbol denotes the Boolean operation, andthe microprocessor is configured to perform operations 1) to 4) for this logic instruction and, during the execution of operation 3), to compute the code Cres−t using the following relationship: Cres−t=C1 & C2 & . . . &Cn.

11. Microprocessor according to claim 1, wherein the arithmetic operation is chosen from the group made up of a bit shift, a bit rotation and an addition.