Arithmetic processor accomodating different finite field size

The present invention relates to an arithmetic processor for performing cryptographic operations.

BACKGROUND OF THE INVENTION

Elliptic Curve (EC) cryptography over a finite field requires the arithmetic operations of addition, multiplication, squaring and inversion. Additionally, subtraction operations are also required if the field is not of characteristic two. Modular arithmetic operations are also required, for example in computing signatures, however these operations arc required less frequently than the finite field operations. EC cryptography as an example, requires the full complement of modular and finite field operations, addition, subtraction, multiplication and inversion.

Field sizes for cryptography tend to be relatively large, requiring fast, dedicated processors to perform the arithmetic operations in an acceptable time. Thus there have been numerous implementations of either fast modular arithmetic processors or dedicated processors for performing arithmetic operations in F_2η. The use of special purpose or dedicated processors is well known in the art. These processors are generally termed coprocessors and are normally utilized in a host computing system, whereby instructions and control is provided to the compressor from a main processor.

Traditionally RSA was the encryption system of choice; however with the advent of superior and more secure EC cryptography the need for processors that perform modular exponentiation exclusively is becoming less imperative. However, while users arc in transition from RSA cryptography to EC cryptography there is a need for an arithmetic processor that supports both these operations, with little or no penalty in performance and cost.

STATE OF THE ART

Such processors are known in the state of the art. Arithmetic processors are known from the <patcit id="pcit0001" dnum="EP0267836A"><text>EP-A 0 267 836</text></patcit> and the lectures of <nplcit id="ncit0001" npl-type="b"><text>De Win E et al: "A fast software implementation for arithmetic operations in GF (2)" Lecture Notes in Computer Science, Springer Verlag, New York, NY, US, 1996, pages 65-76, XP002981362 ISSN: 0302-9743</text></nplcit>. These known solutions do only describe processors involving redesigning the circuitry to accommodate a different field size.

Based on this, the application is concerned with the technical problem to provide an arithmetic processor for performing cryptographic operations being improved in terms of accommodating different field sizes.

This problem is solved by the teaching of claim 1.

One aspect of the present invention is an arithmetic processor as defined in independent claim 1. Further embodiments of the invention are specified in the appended dependent claims.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the invention will now be described by way of example only with reference to the accompanying drawings in which: <ul id="ul0001" list-style="none" compact="compact"><li><figref idref="f0001">Figure 1</figref> is a block diagram of an arithmetic processor architecture for performing finite field arithmetic and integer arithmetic;</li><li><figref idref="f0002">Figure 2</figref> is a block schematic diagram of the arithmetic logic unit ALU shown in <figref idref="f0001">figure 1</figref>;</li><li><figref idref="f0003">Figure 3</figref> is a block diagrams of an alternative embodiment of an arithmetic processor architecture for performing finite field arithmetic and integer arithmetic;</li><li><figref idref="f0004">Figure 4</figref> is a block schematic diagram of the ALU shown in <figref idref="f0003">figure 3</figref>;</li><li><figref idref="f0005">Figures 5(a)</figref>, <figref idref="f0006">(b) and (c)</figref> are block diagrams of an embodiment of a bit-slice of the ALU shown in <figref idref="f0002">figure 2</figref>;</li><li><figref idref="f0007">Figure 6</figref> is a circuit diagram of a finite-field multiplier of the bit-slice shown in <figref idref="f0005 f0006">figure 5</figref>;</li><li><figref idref="f0008">Figure 7</figref> is a block diagram of an arithmetic inverter;</li><li><figref idref="f0009">Figure 8</figref> is a circuit diagram of a combined finite-field/integer multiplier.</li><li><figref idref="f0010">Figure 9</figref> is a block schematic diagram showing an embodiment of a multi-bit ALU of <figref idref="f0001">figure 1</figref>; and</li><li><figref idref="f0011">Figure 10</figref> is a circuit diagram of the multi-bit finite-field multiplier of <figref idref="f0010">figure 9</figref>.</li></ul>

DETAILED DESCRIPTION OF A PREFERRED EMBODIMENT

Referring to <figref idref="f0001">figure 1</figref>, an embodiment of an arithmetic processor is shown generally by numeral 1. As will be appreciated it may be used alongside a general purpose processor in an integrated computing system, where data is exchanged between the computing system and the arithmetic processor. The arithmetic processor includes a group of general purpose registers 2, termed a register file (which may be used as intermediate storage for EC point additions, point doublings, etc.), which communicate with an arithmetic-logic unit (ALU) 4, via data input buses 6. The ALU 4 includes shared finite field and integer arithmetic circuitry. A data output bus or result bus 14 is provided from the ALU 4 to the register file 2 for writing results of computations performed in the ALU 4 to the register file 2.

Computational operations of the ALU 4 is controlled via micro-programmed instructions residing in a controller 8 section of the arithmetic processor 1. A mode selection control 10 is provided to select between either finite field computations or modular integer computations. A field size control 12 is also provided for initializing the ALU 4 to accommodate different operand vector sizes. Thus the controller 8 performs the following tasks amongst others: provides the appropriate arithmetic mode and operation to the ALU 4; coordinates data access between the register file 2 and the ALU 4; and provides to the ALU 4 the appropriate field size to be used.

The general-purpose registers are chosen to have a width large enough to handle at least the largest foreseeable F₂<i>ⁿ</i> EC cryptosystem. The registers may be combined to support larger lengths required for integer modular arithmetic. For example if a single register in the register file 2 is 512 bits wide, then four registers may be used to provide storage for a single 2048-bit RSA quantity. The GP registers are loaded with a block, e.g. 2048-bit, computation may be performed in blocks and then reassembled to obtain the full width result. Typically the arithmetic processor 1 is utilized in an existing host computer system and the controller 8 receives control signals from the host system and communicates data to the host data bus via a suitable host bus interface. Details of such an interface are well known in to those skilled in the art and will not be discussed further.

Turning now to <figref idref="f0002">figure 2</figref>, the ALU 4 includes several special purpose registers 16, combinatorial logic and arithmetic circuitry contained in a plurality of sub-ALU's 18, operate on one or more bits input data buses 28 to each of the sub ALU's from the special purpose registers; output data buses 30 to the special purpose registers 16 from the sub ALU's 18 and its own controller 20. The controller 20 performs the following tasks amongst others: sequences the ALU 4 through steps in a computational operation; monitors control bits from the special purpose registers 16; and implements a counter in its own control registers 22 for determining the size of a field being used, a feature which allows the processor 1 to be used for different field sizes without having to redesign the processor hardware. In order to provide these functions, the control bits 26 of the special purpose registers 16 are provided as control bit inputs 24 to the controller 20. The special purpose registers 16 are all individually addressable. The controller 20 also controls data input via the input buses 6 from and to the register file to the sub ALU's 18 or the special purpose registers 16. These sub-ALU's may operate on single bits at a time. Each of these components will be described in more detail below.

Referring to <figref idref="f0003">Figure 3</figref>, an alternative embodiment of an arithmetic processor is shown generally by numeral 1'. In this embodiment a separate finite field unit 34 and integer modular arithmetic unit 36 is provided. This processor also includes a register file 2', data input buses 6', data output buses 14', and a controller 8', however, separate controls 13a and 13 b are provided from the controller 8' to respective ALU's 34 and 36 respectively.

Referring to <figref idref="f0004">figure 4</figref>, the ALU's 34 and 36 of <figref idref="f0003">figure 3</figref> are shown in greater detail. Each of the ALU's 34 and 36 include their own respective special-purpose registers 16'a and 16'b and controller 20'a and 20'b. Each of the ALU's 34 and 36 contain their own sub ALU's 18'a and 18'b respectively. Thus it may be seen that in this embodiment special purpose registers 16'a and 16'b and arithmetic and control circuitry is not shared. One or more of the sub ALU's 18'a perform in concert the functions of Shift left/right, XOR-shift and one or more of the sub ALU's 18'b perform in concert the function of integer add and integer subtract, with the option of using carry save techniques, or carry propagation.

Referring back to <figref idref="f0002">figure 2</figref>, the sub ALU's 18 perform the following logical functions on operands provided from the special purpose registers 16: XOR; Shift left/right, XOR-shift, integer add and integer subtract. These functions may be contained in one sub ALU 18 or across multiple sub ALUs. By providing multiple sub ALU's 18 the processor is capable of performing multiple operations, (e.g. for finite field inversion), simultaneously.

Turning now to <figref idref="f0005 f0006">figure 5</figref>, a bit-slice 41 of the ALU 4 shown in <figref idref="f0002">figure 2</figref> is shown in greater detail. In the following discussion, we shall refer to the interconnection of cells of respective special-purpose registers in conjunction with its associated logic circuitry as a bit-slice 41. The logic circuitry contained in a bit slice is generally represented schematically by one of the sub ALU's 18 as shown in <figref idref="f0002">figure 2</figref>. It is then intended that the configuration of a bit slice may be repeated N times for an N-bit register. Furthermore, for clarity, we define N to be the number of cells in a register, and we refer to individual cells in a register as, for example, A_i where 0 ≤ i ≤ N-1 and wherein A_N-1 is the left most cell of the special-purpose register. The contents of a register will be referred to by lower case letters, for example, a bit vector A of length n will have bits numbered from a₀...a_n-1 with a₀ being the LSB. It may also be noted that although the special-purpose registers have been given specific names, these registers may take on different functions depending on the arithmetic operation being performed as will be described below.

In <figref idref="f0005 f0006">figure 5</figref>, the special-purpose registers 16 include: a pair of operand registers <b>A</b> 42 and <b>B</b> 44, to hold, for example, the multiplicand and multiplier, respectively, in a multiplication operation; an accumulator register <b>C</b> 46; a modulus register <b>N</b> 48; and a carry extension register <b>C^ext</b> 50 (used in integer arithmetic). The registers each have N cells for holding the respective binary digits of bit vectors loaded therein. It is preferable that these registers are shift registers. A sub ALU 18 shown in <figref idref="f0002">figure 2</figref> may be implemented by the circuitry of block 52 in <figref idref="f0005 f0006">figure 5</figref>, and in a manner to be described below.

Multiplication

Operation of the ALU 4 may be best understood by reference to a specific arithmetic operation such as finite field multiplication. Consider the product C of two elements <b>a</b> and <b>b</b>, where <b>a</b> and <b>b</b> are bit vectors and wherein <b>b</b> will be of the form <b>b</b>=(b₀, ... b_n-1) in polynomial basis representation and <b>a</b> will be of the form <b>a</b>=(a₀,...a_n-1) in polynomial basis representation. A modulus bit vector <b>m</b> has the form <b>m</b>=(m₀,...m_n). As will be noted the modulus register has one bit more than the number of bits required to represent the modulus. Alternatively, since the most significant bit m_n is one, this bit might be implied and m represented by (m₀,...m_n-1). In <b>F</b>₂ⁿ, the multiplication may be implemented as a series of steps, which is more clearly set out by the following pseudo-code: <img file="EP1293891B2_D0001.tif" />

In performing the multiplication, partial products of the multiplicand and each of the bits of b_i of the multiplier, proceeding from the most significant bit (MSB) to the least significant bit (LSB), are formed. The partial products are reduced by the modulus if the MSB of the previous partial product is set.

Multiplication may be implemented by sequentially using a 1 x N multiplier in which case the inner "for" loops of the preceding pseudocode is done in parallel. The modulus register M is loaded with the modulus bit vector <b>m</b> stripped of its most significant bit m_n such that each cell contains a respective one of the binary digits m_i. In the implementation shown, the bits m_i are arranged from left to right with the MSB of the vector being the leftmost bit, i.e. cell M_n-1 contains bit M_n-1. If N ≠ n still bit M_n-1 is stored in M_n-1, that is the data is left justified. The shift registers <b>A</b> and <b>B</b> are loaded with the finite field elements bit vectors <b>a</b> and <b>b</b> respectively so that each cell contains one of the binary digits <b>a_i</b> or <b>b_i</b>. The finite field elements a and b are stored left justified, in their respective registers so that the topmost bit of the multiplier register b is always available at the left boundary cell bit, i.e. (a_n-1, a_n-2,...a₀) and (b_n-1, b_n-2, ...b₀). If the length of the vectors a and b are less than the length of the registers; the remaining cells are padded with zeros. The above is generally performed by the controller 20 shown in <figref idref="f0002">figure 2</figref>. Other arrangements of sequential multiplication are possible (such as sequentially reducing the multiplicand), but such arrangements do not allow flexible field sizes along with fixed control bit locations. Bit ordering from LSB to MSB is also possible with corresponding changes in the multiplication algorithm.

A bit-slice 41 of the ALU 4 for implementing multiplication in a finite field is now described. The bit-slice 41 includes first and second controllable adders 54 and 56, respectively, each having an XOR function. The topmost cell B_N-1 of the register <b>B</b> provides an add control signal b_n-1 57 to the first adder 54. Inputs 58 and 60 to the first adder 54 are derived from a register cell A_i and accumulator cell C_i. An output 62 from the first adder 54 is connected to an input of the second adder 56 along with an input 64 from the modulus register cell M_i. The adder 54 performs the operation output 62 equal to input 60 + (input 58 AND control 57) is shown in greater detail in <figref idref="f0006">figure 5(b)</figref>.

The output from the second adder 56 is then connected the accumulator cell C_i. A second add control signal 66 is derived from the topmost cell C_N-1 of the accumulator <b>C</b> 46. It may be seen that this signal implements the modular reduction of the partial product in the accumulator <b>C</b> by the modulus vector <b>m</b>, when the topmost bit C_N-1 of <b>C</b> is set. The adder 56 performs the operation output = input 62 + (input 64 AND control 66) as shown in greater detail in <figref idref="f0006">figure 5(c)</figref>. The <b>B</b> register is a clocked shift register. A clock signal CLK1 68, which may be provided by the controller 20 causes the contents of this register to be shifted left for each partial product, calculated.

Referring to <figref idref="f0007">figure 6</figref>, a detailed circuit implementation of the bit-slice 41 of <figref idref="f0005 f0006">figure 5</figref> for finite field multiplication is indicated by numeral 70. Referring to bit-slice i, 70 of <figref idref="f0007">figure 6</figref>, (only three bit-slices are shown for the purpose of illustration in <figref idref="f0007">figure 6</figref>), the cell a_i is ANDed with the add control signal b_n-1 by an AND gate 72. The output 74 of the AND gate 72 is connected to an input of an XOR gate 76 along with an input 78 from adjacent cell C_i-1 of the accumulator C. Thus implementing the calculation of the term "c_j-1 + b_ia_i". The term "c_n-1m_j" is implemented by ANDing the signal c_n 80 with m_i 82 utilizing an AND gate 84. The output 86 of the AND gate 84 is connected to the input of an XOR gate 84, along with the output 88 of XOR gate 76. The output 90 of XOR gate 84 is connected to cell C_i 92. Thus implementing the expression "c_j = c_j-1 + b_i(a_i) + c_n-1(m_j)". With this general sequential multiplier, the product of two n-bit finite field elements will be produced in n clock cycles. It is preferable that a synchronous counter, that may be contained in the controller 20, provides control of the number of iterations. The preceding description applies to integer modular multiplication when adder 54 is a bit slice of an integer adder and adder 56 is a bit slice of an integer subtractor, as will be described later.

Addition

Although the circuitry has been described with reference to multiplication in a finite field F₂ⁿ, other computational operations may also be performed with ease. Finite field addition has an advantage over integer arithmetic in that no carries are produced. The computation of a finite field sum requires only that an XOR gate be introduced at each cell of the registers in question since addition of two elements <b>a</b> and <b>b</b> in a finite field is simply a XOR b. Thus, referring back to <figref idref="f0005 f0006">figure 5</figref>, an input 100 is provided to the first adder 54 from cell B_i, and the second adder 56 is used for reduction. The output from adder 54 is then written directly into cell C_i. After the operands have been moved into registers <b>a</b> and <b>b</b>, the addition can be performed in a single clock cycle. It is also possible for the operation to be performed in the ALU and the result written back into a general register in the register file. For integer addition adder 54 is a bit slice of an integer adder and the result must be checked for modular overflow. If this condition arises adder 56 which is a bit slice of an integer subtractor is used to reduce the result.

Squaring

Squaring a number can be performed in the same time as multiplication of two different numbers. Squaring in a polynomial basis can be performed in a single clock cycle, if the specific irreducible along with the squaring expansion is explicitly hardwired. As an alternative squaring may be performed with multiplication of identical inputs.

Inversion

Inversion of finite field elements in F₂<i>ⁿ</i> may be performed using the extended Euclidean algorithm and utilizing four of the special purpose registers with additional control logic. This will be completed in 2n cycles if the shifting is made concurrently to the adds (which is easily implemented by hard wiring the outputs of the add to the next register cell).

The registers used in the inversion are A, B, M and C. For convenience these registers are schematically shown in <figref idref="f0008">figure 7</figref> wherein they are assigned the following labels: M:UL; C:LL; A:UR; and B:LR. Once again the operation may be described with reference to a bit-slice 110.

The operands in an inversion are generally: an element to invert g; an irreducible polynomial <b>f</b> or modulus <b>m</b> (described later); a bit vector '0' and a bit vector 'l.' The UL register 116 is loaded with <b>f</b> or <b>m</b>. The LL register 118 is loaded with g, the UR register 112 is loaded with '0' and the LR register 114 is loaded with '1'. For the UR and LR registers 112 and 114, respectively, cells UR_i and LR_i are XORed together by XOR gate 120 to produce an output 122. A control signal 124 determines whether one of three possible inputs is written in cell UR_i and UL_i. The inputs are either a left or right shift from adjacent cells or the output 122. The control signal B determined by the state table to be described below. For the UL or LL registers 116 and 118, respectively, cells UL_i and LL_i are XORed together by XOR gate 126 to produce an output 128. A control signal 130 determines whether one of two possible inputs is written into cell UL_i and LL_i. The inputs are either a left shift from the adjacent cell (i - 1) or the output 128. Once again the control signal 130 is determined by the state table to be described below.

If we assume the control variables to be k_u - the length of the UL register and k_l - the length of the LL register. Then Δ = k_u - k₁. The values k₁ and k_u are implemented preferably with synchronous countdown counters, and Δ is implemented preferably with a synchronous up/down counter. Counter registers k_u, k_l and Δ are also provided. The UL and LL registers are left shift registers while the UR and LR registers are both left and right shift registers.

Furthermore, for the count registers, Δ is loaded with 0, <i>k</i>_u is initialized to <i>n</i>. A control bit latch provides a toggle function wherein a '1' designates an up count and a '0' designates a down count. The U/D control is initially set to '1.' Then a sequencer contained in the controller for performing the inversion in the ALU has the following outputs: <dl id="dl0001" compact="compact"><dt>deckl</dt><dd>Decrement <i>k_l kl</i></dd><dt>decku</dt><dd>Decrement <i>k</i>_u<i>ku</i></dd><dt>decDelta</dt><dd>Decrement Δ</dd><dt>incDelta</dt><dd>Increment Δ</dd><dt>toggle</dt><dd>Toggle UP/DOWN</dd><dt>lsUL</dt><dd>left-shift Upper Left register</dd><dt>lsLL</dt><dd>left-shift Lower Left register</dd><dt>lsUR</dt><dd>left-shift Upper Right register</dd><dt>lsLR</dt><dd>left-shift Lower Right register</dd><dt>rsUR</dt><dd>right-shift Upper Right register</dd><dt>rsLR</dt><dd>right-shift Lower Right register</dd><dt>outLR</dt><dd>Output Lower Right register</dd><dt>outUR</dt><dd>Output Upper Right register</dd><dt>dadd-lsLL</dt><dd>Down XOR and left-shift Lower Left register</dd><dt>uadd-lsUL</dt><dd>Up XOR and left-shift Upper Left register</dd></dl>

A state table outlining the action of the inverter follows, wherein M_u and C_l are the upper bit of registers UL and LL respectively and wherein M_u and C_l determine the current state. When an action is performed on the registers and counters which places the inverter in a new state. The process is repeated until either <i>k_u</i> or <i>k_l</i> are zero and one of the right register RL or RU will contain g^-1, the other will contain the modulus itself, which may be restored to register <i>m</i> for use in multiplication or inversion operations to follow. <tables id="tabl0001" num="0001"><table frame="all"><tgroup cols="7"><colspec colnum="1" colname="col1" colwidth="11mm" /><colspec colnum="2" colname="col2" colwidth="9mm" /><colspec colnum="3" colname="col3" colwidth="8mm" /><colspec colnum="4" colname="col4" colwidth="8mm" /><colspec colnum="5" colname="col5" colwidth="10mm" /><colspec colnum="6" colname="col6" colwidth="9mm" /><colspec colnum="7" colname="col7" colwidth="57mm" /><thead><row><entry valign="top">U/ D</entry><entry valign="top"><i>k_u</i></entry><entry valign="top"><i>k_l</i></entry><entry valign="top">Δ</entry><entry valign="top">M_u</entry><entry valign="top">C_l</entry><entry valign="top">Action</entry></row></thead><tbody><row><entry>X</entry><entry>0</entry><entry>X</entry><entry>X</entry><entry>X</entry><entry>X</entry><entry>OutLR</entry></row><row><entry>X</entry><entry>X</entry><entry>0</entry><entry>X</entry><entry>X</entry><entry>X</entry><entry>OutUR</entry></row><row><entry>1</entry><entry><o ostyle="single">0</o></entry><entry><o ostyle="single">0</o></entry><entry>0</entry><entry><o ostyle="single">0</o></entry><entry>1</entry><entry>Deck_u, dec Δ, lsUL, lsUR, toggle</entry></row><row><entry>1</entry><entry>0</entry><entry><o ostyle="single">0</o></entry><entry><o ostyle="single">0</o></entry><entry>0</entry><entry>1</entry><entry>Deck_u, dec Δ, lsUL, rsLR</entry></row><row><entry>0</entry><entry><o ostyle="single">0</o></entry><entry><o ostyle="single">0</o></entry><entry>X</entry><entry>0</entry><entry>1</entry><entry>Deck_u, dec Δ, lsUL, lsUR</entry></row><row><entry>0</entry><entry><o ostyle="single">0</o></entry><entry><o ostyle="single">0</o></entry><entry>0</entry><entry>1</entry><entry>0</entry><entry>Deck_l, inc Δ, lsLL, lsLR toggle</entry></row><row><entry>0</entry><entry><o ostyle="single">0</o></entry><entry><o ostyle="single">0</o></entry><entry><o ostyle="single">0</o></entry><entry>1</entry><entry>0</entry><entry>Deck_l, inc Δ, lsLL, rsUR</entry></row><row><entry>1</entry><entry><o ostyle="single">0</o></entry><entry><o ostyle="single">0</o></entry><entry>X</entry><entry>1</entry><entry>0</entry><entry>Deck_l, inc Δ, lsLL, lsLR</entry></row><row><entry>0</entry><entry><o ostyle="single">0</o></entry><entry><o ostyle="single">0</o></entry><entry>0</entry><entry>1</entry><entry>1</entry><entry>Deck_l, inc Δ, Dadd-lsLL, lsLR, toggle</entry></row><row><entry>0</entry><entry><o ostyle="single">0</o></entry><entry><o ostyle="single">0</o></entry><entry><o ostyle="single">0</o></entry><entry>1</entry><entry>1</entry><entry>Deck_l, inc Δ, Dadd-lsLL, rsUR</entry></row><row><entry>1</entry><entry><o ostyle="single">0</o></entry><entry><o ostyle="single">0</o></entry><entry>0</entry><entry>1</entry><entry>1</entry><entry>Deck_u, dec Δ, Uadd-lsUL, lsUR; toggle</entry></row><row><entry>1</entry><entry><o ostyle="single">0</o></entry><entry><o ostyle="single">0</o></entry><entry>0</entry><entry>1</entry><entry>1</entry><entry>Deck_u, dec Δ, Uadd-lsUL, rsLR</entry></row></tbody></tgroup></table></tables>

Integer arithmetic

The extreme similarity of polynomial and integer representations allows for the sharing of hardware in the ALU. For addition, the integer arithmetic is only complicated by the requirement for carries. The integer arithmetic operations of the ALU are best illustrated by way of example utilizing a multiplication operation.

Multiplication in <b>Z</b> is illustrated by way of reference to the following sequence of steps represented in pseudo-code, wherein as earlier, <b>a</b> and <b>b</b> are bit vectors to be multiplied and <b>c</b> is the product of <b>a</b> and <b>b</b>, and wherein c = (c₀, c₁, ...c_n-1). <img file="EP1293891B2_D0002.tif" />

Analogously, this may be used to invert integers modulo p if the XOR's are replaced with subtractors and the m register is loaded with the prime. As a refinement carry - save methods may be employed to delay carry propagation.

It may be observed that the bit-slices 70 for finite field multiplication illustrated in the embodiment of <figref idref="f0007">figure 6</figref>, may be modified to include multiplication for integer representations. It may also be noted that for integer multiplication, the registers are loaded with the bit vectors in reverse order from that of F₂m i.e. the leftmost cell of a register contains the LSB of the bit vector. In integer number multiplication, it is necessary to implement carries between successive partial products, furthermore as the partial products are not being reduced by a modulus the carries from the addition of successive partial products must be provided for. Thus the accumulator register <b>C</b> is extended and a new register <b>C</b>^ext 49 is provided as shown in <figref idref="f0005 f0006">figure 5</figref>. Before each partial product is formed, the lowest bit of the accumulator C (cell C_M) is shifted into the topmost bit of the extension register C^ext (cell C^{ext<sub2>1</sub2>}) and then both the accumulator C and C^ext are shifted toward the LSB by one bit. The final result is obtained in C and C^ext, wherein C^ext contains the low order bits of the product. This is represented by the operation C^ext ← C above.

Referring now to <figref idref="f0009">figure 8</figref>, a bit-slice 170 is shown, and which is similar to the bit-slice 70 of <figref idref="f0007">figure 6</figref>. Accordingly the reference numerals used in the description of <figref idref="f0007">figure 6</figref> will be used to identify like components with a prefix 100 added i.e. reference numeral 70 will become 170. The arrangement of <figref idref="f0009">figure 8</figref> differs from <figref idref="f0007">figure 6</figref> in two important ways; the modulus register <b>m</b> is used as a carry register, and a mode selection signal <b>Z F₂m</b> 171 is provided.

Now the terms c_j = c_j-1 + b_ia_i + c_n-im_j are implemented as before for the finite field multiplication with the product of the control signal b_m and the contents of register cell A_i, implemented by AND gate 172. The output 174 of the AND gate 172 is XORed with the contents of register cell c_j-1 by XOR gate 176 to produce an output term c_j-1 + b_i(a_i) indicated by numeral 158. This output signal is XORed using XOR gate 184 with the term 'c_n-1(m_j)' indicated by numeral 185, derived from the AND gate 160 to produce the term c_j. In addition, a carry term m_i is produced from the sum of the respective products 'b_i(a_i). c_j-1' 162 and 'c_j-1 + b_ia_i.m_j' 163 and written into cell m_i 182. The product terms 162 and 163 are implemented by AND gates 164 and 166 respectively. The sum of the terms 162 and 163 are implemented by OR gate 167.

The mode selection signal <b>Z</b> 171, is ORed with the carry input signal c_n 180 and is also ANDed 168 with clock signal 169. Thus by setting <b>Z</b> = 0, will implement finite field arithmetic and by setting <b>Z</b> = 1 will implement integer arithmetic.

Thus the modifications necessary to convert the finite field multiplier given previously in <figref idref="f0007">figure 6</figref> into a combined finite field/integer multiplier are shown in <figref idref="f0009">Figure 8</figref>. Note that the output register <b>C</b> is extended to collect the low order bits of the multiplication. As computations in <b>Z</b> are performed without a modulus, The modulus register M is not used to reduce the partial products but as a holder of the carries. The control signal <b>Z/F₂^M</b> 171 enables the integer multiplication circuitry for the ALU.

A final carry propagation may be provided by a Manchester ripple chain, possibly extended by a carry-skip mechanism of one or two layers owing to the long register length. It is also possible to clock for n more cycles, allowing the carry save adders to completely merge the carries.

Two's complement subtraction can be implemented in the carry propagation adder provided that one input can be conditionally complemented at its input and that a 'hot' carry-in is made at the LSB of the adder.

When multiplying, the ripple-carry will be intolerable even if improved by the carry-skip, but this carry propagation can be almost entirely removed by using a carry-save adder, which provides a redundant representation of the partial product, which is only resolved after the multiplication is complete.

In a further embodiment the ALU 4 may be modified to provide a linear increase in computation speed as shown in <figref idref="f0010">figure 9</figref>. This is achieved by processing consecutive bits from the special-purpose registers 16' at once, and implementing additional circuitry indicated by the modified sub ALU's 190 to process the incremental additions as schematically illustrated in <figref idref="f0010">figure 9</figref>. Processing multiple bits then results in a linear increase in speed. For example, where a computation is performed sequentially two or more steps in the sequence may be performed simultaneously. In this case the controller 20' will process two or more control bits 194 from the special-purpose registers 16', and the inputs 192 to the controller are indicated in <figref idref="f0010">figure 9</figref> as multi-bit lines.

A circuit diagram of a two-bit at a time multiplier for finite fields is shown in <figref idref="f0011">Figure 10</figref>. In this implementation, the bit-slices 200 have twice the number of XOR gates 210, implementing two terms of the addition, the circuit takes two bits of multipliers and adds in two adjacent shifts of the multicand a_i and a_i-1, and reduces with two adjacent shifts of the modulus M_i and M_i-1. This has the effect of simultaneously producing two consecutive partial products with modulus reduction, thus halving the total computation time.

It should also be noted that the top-bits of the special-purpose registers are used as control bits for the controllers 20' or 20. This has the advantage that when the operands are loaded into the registers, they are aligned left; thus control is always obtained from a fixed bit location. However, other bits may be used as a control bits, e.g. the bottom bits; however, this may additionally increase the complexity of the hardware.

Again, multi-bit operation potentially providing improved linear increase in computation speed, since such options as Booth (or modified-Booth) recoding become possible.

It is assumed that the ALU will also be able to perform simple arithmetic operations on general registers. An alternative is to have all arithmetic performed on ALU internal registers, with the general-purpose registers able only to read and write these registers.

The functionality of the ALU will include integer addition, utilizing some carry propagation method, such as a ripple carry or the combination of carry skip addition and carry completion.

The ALU will also provide simple XOR functionality for use in finite field addition. Since the integer and finite field representations (bit orders) are reversed, it is beneficial to provide a bit reversal mechanism for use in field to integer and integer to field conversions. The tops of two shift registers are connected to provide for this facility in n clock cycles, where n is the length of the arithmetic operands.

The general architecture given here has the potential not only to share the register file between EC and modular exponential arithmetic, but also to share special purpose registers and even combinational logic, in addition to shared control registers.

It may be noted that in the embodiments described, reference is made to specific logic circuits, however equivalent circuits may be used, for example by using de Morgans Rule or if inverted logic is implemented then complementary circuits may be used. In addition, when referring to the orientation of the registers and bit vectors, i.e. left, right, top, bottom, other arrangements of these directions are also implied.

The terms and expressions which have been employed in the specification are used as terms of description and not of limitations, there is no intention in the use of such terms and expressions to exclude any equivalents of the features shown and described or portions thereof, but it is recognized that various modifications are possible within the scope of the invention as claimed.