Publication number | US20050246406 A9 |

Publication type | Application |

Application number | US 10/140,885 |

Publication date | Nov 3, 2005 |

Filing date | May 9, 2002 |

Priority date | Mar 22, 2002 |

Also published as | CN1729444A, DE60307089D1, EP1504338A1, EP1504338B1, US7167885, US20030182339, WO2003096182A1 |

Publication number | 10140885, 140885, US 2005/0246406 A9, US 2005/246406 A9, US 20050246406 A9, US 20050246406A9, US 2005246406 A9, US 2005246406A9, US-A9-20050246406, US-A9-2005246406, US2005/0246406A9, US2005/246406A9, US20050246406 A9, US20050246406A9, US2005246406 A9, US2005246406A9 |

Inventors | Erik Hojsted |

Original Assignee | Erik Hojsted |

Export Citation | BiBTeX, EndNote, RefMan |

Patent Citations (8), Referenced by (1), Classifications (7), Legal Events (4) | |

External Links: USPTO, USPTO Assignment, Espacenet | |

US 20050246406 A9

Abstract

An emod operation is a computational substitute for a traditional modulus operation, one that is computationally less expensive but also less precise. Where a modulus operation may be defined for some base number n, the emod operation determines a modulus of an operand using a “phantom modulus,” one that is an integer multiple of n. The phantom modulus is chosen to make emod calculations computationally inexpensive when compared to a modulus operation. Thus, the emod operation is particularly useful for multiplications or exponential operations using very large operands. Upon conclusion of interstitial processing associated with the multiplications or exponential operations, a single, traditional modulus operation may be used to obtain a final result.

Claims(5)

upon conclusion of each of a plurality of interstitial iterations, determining a modulus of a result thereof using a phantom modulus that is a multiple of n, the phantom modulus approximating 2^{k }for some arbitrary k, and

upon conclusion of a final iteration, determining a modulus of a result thereof using a true modulus n.

iteratively, for each of several words of B:

shifting a value c by a word length,

adding to the shifted value c, a value A·B[i], where B[i] is the ith word of B, performing an emod operation on a result of the addition, using a phantom modulus mn that is a multiple of n, and

following a last iteration, performing a modulus operation on the result of the addition from the last iteration.

iteratively, for each of several words of B:

generating a quotient q from a higher order portion of an earlier valuation of c,

generating a remainder r from a lower order portion of the value c,

revaluing c to be A·B[i]+r+d·q, where d is a precision value of a phantom modulus, where B[i] is the ith word of B,

following a last iteration, performing a modulus operation on the result of the revaluing from the last iteration.

iteratively, for each of several words of B:

generating a quotient q[i] from a higher order portion of an earlier valuation of c, where i represents the current iteration,

generating a remainder r[i] from a lower order portion of the value c,

shifting left a quotient q[i−1] by a word width

revaluing c to be A·B[i]+r[i]+d·q[i−1], where d is a precision value of a phantom modulus, where B[i] is the ith word of B and the revaluing uses the shifted quotient,

following a last iteration, performing a modulus operation on the result of the revaluing from the last iteration.

iteratively, for each bit position i of B:

performing a c=(c·c) emod n operation, and

if the ith bit position of B is a 1, performing a c=(c·A) emod n operation; and

following a last iteration, performing A mod N operation of c obtained from the last operation,

wherein the emod n operation calculates a modulus result from a phantom modulus mn, where mn is an integer multiple of n.

Description

- [0001]The present invention relates to modulus calculations. In particular, it relates to modulus calculations that may be performed with high degrees of efficiency.
- [0002]A modulus calculation (colloquially, a “mod” calculation) determines the remainder of a division operation. Thus, the expression A mod N determines a result that is the remainder obtained by dividing the number A by N. Example: 17 divided by 3 is 5 with a remainder of 2. “17 mod 3” yields a result having value 2.
- [0003]Mod calculations are performed in many computing applications including key negotiation conducted between two parties before engaging in encrypted communication. In the key negotiation context, evaluation of equations having the form (A
^{B}) mod n is performed at two terminals. Often, the A and B values may be quite large—from 1024 to 2048 bits long. Of course, when two operands having length l are multiplied, the result may have a length of up to 2l. With such large operands, it is impractical to build result registers in a processor that have the full width of the multiplication result. Instead, as multiplication results are generated, they typically are truncated by applying the mod calculation to each product. Because the mod calculation divides each product by a modulus having its own length (say, j), the result always has a length less than j. - [0004]Assuming operands of length l and an equal number of 0s and 1s therein, evaluation of A
^{B }mod n may require l multiplications and l mod operations. This involves considerable computational expense. The expense associated with such computations becomes particularly severe in high-load environments such as computer servers where it can be expected that several thousand key negotiation requests (maybe more) would be received per hour. - [0005]Accordingly, there is a need in the art for a fast, computationally inexpensive technique for resolving mod operations with large operands.
- [0006]
FIG. 1 illustrates a method according to an embodiment of the present invention. - [0007]
FIG. 2 illustrates a method according to an embodiment of the present invention. - [0008]
FIG. 3 illustrates another method according to an embodiment of the present invention. - [0009]
FIG. 4 is a block diagram illustrating an interstitial product generator according to an embodiment of the present invention. - [0010]
FIG. 5 is a block diagram illustrating an IPG according to an alternate embodiment of the present invention. - [0011]
FIG. 6 illustrates a multiplier according to an embodiment of the present invention. - [0012]
FIG. 7 illustrates a multiplier circuit according to an embodiment of the present invention. - [0013]
FIG. 8 illustrates a multiplier circuit according to another embodiment of the present invention. - [0014]
FIG. 9 illustrates a multiplier circuit according to an embodiment of the present invention. - [0015]
FIG. 10 illustrates a multiplier circuit according to another embodiment of the present invention. - [0016]
FIG. 11 illustrates another method according to an embodiment of the present invention. - [0017]Embodiments of the present invention introduce an “emod” operation for use in mod calculations. The emod is a computational substitute for a traditional mod operation, one that is computationally less expensive but also less precise. The emod operation may be used in connection with interstitial multiplications that may be generated during evaluation of an A
^{B }mod n calculation or an (A·B) mod n calculation. At the end, when a final product is available, a traditional mod operation may be performed to obtain a final result. In this way, the embodiments of the present invention avoid the computational expense of perhaps thousands of mod operations that might otherwise be performed at interstitial stages of operation. - [0018]Although computers perform arithmetic operations having binary values (base 2), the advantages of the emod operation might best be understood with an example using traditional decimal numbers (base 10). To evaluate the operation 23754 mod 3331, it would be conventional to divide 23754 by the modulus 3331 to obtain the remainder 437. However, such division is computationally expensive. It would be far easier to use some multiple of the modulus that is closer to 10
^{k }for some arbitrary number k. Using such a multiplier (say 9993 instead of 3331), one may employ a series of subtractions instead of a division operation. After twice subtracting 9993 from 23754, one is left with a residual of 3768 (which includes the correct remainder 437 plus 3331). This residual is sufficient for use with the interstitial products obtained at intermediate stage of computation. When a final result is obtained and a true mod operation is employed, the correct remainder will be isolated from any multiples of the modulus that may have been carried over from the intermediate stages. Use of this “phantom modulus,” however, improves processing speed. - [0019]The example illustrated above also works in a binary scheme. In the base 2 domain, for some modulus n, a multiple is chosen that closely approximates some 2
^{k }for some arbitrary k. Just as the decimal example above included a consecutive series of 9s in the most significant bit positions, in the binary example, the phantom modulus will include a consecutive series of 1s in the most significant bit positions. This property simplifies the subtraction that takes place when reducing the operand A by the phantom modulus. - [0020]Given a modulus n, a phantom modulus mn may be chosen such that m·n=2
^{k}−d, where d<n. Then the emod operation may be employed as a recursive subtraction in which mn is subtracted from the source operand until the residual is less than mn. These two parameters, d and m, control the emod operation. - [0021]Evaluation of (A·B) emod n
- [0022]According to an embodiment, evaluation of:

*c=*(*A·B*)*emod n*(1)

may be performed by parsing the operand B into multiple words of w bits each. Thus:$\begin{array}{cc}\begin{array}{c}B=B\left[M-1\right]B\left[M-2\right],\dots \text{\hspace{1em}},B\left[0\right]\\ =\sum _{i=0}^{M-1}{2}^{w\xb7i}B\left[i\right]\end{array}& \left(2\right)\end{array}$

Equation 1, then becomes:$\begin{array}{cc}\begin{array}{c}c=\left(A\xb7\sum _{i=0}^{M-1}{2}^{w\xb7i}B\left[i\right]\right)\mathrm{emod}\text{\hspace{1em}}n\\ =((\dots \text{\hspace{1em}}\left(A\xb7B\left[M-1\right]\xb7{2}^{w}+A\xb7B\left[M-2\right]\right)\xb7\\ {2}^{w}+\dots \text{\hspace{1em}})\xb7{2}^{w}+A\xb7B\left[0\right])\mathrm{emod}\text{\hspace{1em}}n\end{array}& \left(3\right)\end{array}$

The emod operation is distributive and may be replicated within the parenthetical. This property leads to the method illustrated inFIG. 1 . - [0023]
FIG. 1 illustrates a method**1000**according to an embodiment of the present invention. According to the method**1000**, a variable c may be initialized to be zero (box**1010**). Thereafter, the method**1000**iteratively may consider each word of the multiplicand B, starting with the word corresponding to the most significant bit position of B and working toward the word corresponding to the least significant bit position. During each iteration, the method**1000**may shift left the c value from a prior iteration by the length of a word (box**1020**). The value of A multiplied by the new B word (labeled, “B[i]”) may be added to the shifted value of c (box**1030**). Thereafter, the emod operation may be performed on the result obtained from box**1030**(box**1040**). The result of the emod may be used as the initial value c of a subsequent iteration. Following the last iteration, the c value obtained is the result of the calculation. - [0024]In one embodiment, the method of
FIG. 1 may be implemented in software according to the following pseudocode. - [0000]Where shiftleft(w,c) merely shifts left the c operand by w bits. This is equivalent to a multiplication by 2
^{w }in binary. - [0025]The emod operator operates based on a phantom modulus mn=m·n, yielding a precision factor d=2
^{k}−mn. If the operand c were split into two parts, a quotient q and a remainder r, so that:

*c=q·*2^{k}*+r*(4)

then the emod operation may be defined as:$\begin{array}{cc}\begin{array}{c}\left(c\text{\hspace{1em}}\mathrm{emod}\text{\hspace{1em}}n\right)=\left(\left(q\xb7{2}^{k}+r\right)\mathrm{emod}\text{\hspace{1em}}n\right)\\ =\left(q\xb7{2}^{k}+r-q\xb7\mathrm{mn}\right)\\ =\left(r+q\xb7d\right).\end{array}& \left(5\right)\end{array}$

This emod function may be integrated into the method ofFIG. 1 as shown in the embodiment ofFIG. 2 . - [0026]
FIG. 2 illustrates a method**1100**according to an embodiment of the present invention. According to the method**1100**, a dummy variable c may be initialized to be zero (box**1110**). Thereafter, the method iteratively may consider each word of the multiplicand B, starting with the word corresponding to the most significant bit position of B and working to the word corresponding to the least significant bit position. The method may calculate a quotient value q and a remainder value r from the value c obtained from a prior iteration (boxes**1120**,**1130**). The quotient q may be taken as a span of bits from c extending from the most significant bit position to the k^{th }bit position, shifted left by w bits. The remainder r may be taken as the remaining bits of c, extending from the k−1^{th }bit position to the 0^{th }bit position, shifted left by w bits. Thereafter, the c value may be evaluated as:

*c=A·B[i]+r+d·q*(6)

(box**1140**). The c value obtained at the last iteration may be taken as the result of the emod function. - [0027]In one embodiment, the method of
FIG. 2 may be implemented in software according to the following pseudocode. - [0028]This implementation requires that the product d·q is available immediately. In practice, since this product may take some time to generate, the method effectively becomes stalled until the product becomes available.
- [0029]In an alternate embodiment, the method may complete a current iteration without having the d·q product available. Instead, it may advance to the next iteration of i and integrate the d·q product from a previous iteration.
FIG. 3 illustrates this embodiment. - [0030]
FIG. 3 illustrates a method**1200**according to an embodiment of the present invention. According to the method**1200**, a variable c may be initialized to be zero (box**1210**). Thereafter, the method iteratively may consider each word of the multiplicand B, starting with the word corresponding to the most significant bit position of B and working to the word corresponding to the least significant bit position. At each iteration, the method**1200**may calculate a quotient value q[i] and a remainder value r[i] from the value c obtained from a prior iteration (boxes**1120**,**1130**). The quotient q[i] may be taken as a span of bits from c extending from the most significant bit position to the k^{th }bit position, shifted left by w bits. The remainder r[i] may be taken as the remaining bits of c, extending from the k−1^{th }bit position to the least significant bit position, shifted left by w bits. The quotient q[i−1], obtained from a prior iteration, also may be shifted left by w bits (box**1240**). Thereafter, the c value may be evaluated as:

*c=A·B[i]+r[i]+d q[i−*1], (7)

where the q[i−1] value is the shifted value obtained in box**1240**(box**1250**). - [0031]Following the final iteration, the quotient from the final iteration may be added to c (box
**1260**). The value obtained from this operation may be taken as the result from the emod operation. - [0032]In one embodiment, the method of
FIG. 3 may be implemented in software according to the following pseudocode. - [0033]In this embodiment, the d·q product from a prior iteration (relabeled as d·q1) is shifted left to account for positional differences between the two words.
- [0034]As noted above, the embodiment of
FIG. 3 need not be stalled while waiting for evaluation of a d·q operation. This embodiment may find application in a high-load application where avoidance of computational latencies may be at a premium. - [0035]Multiplication of Large Numbers with Small Numbers
- [0036]As described above, the multiplicand B may be parsed into a plurality of smaller words B[w], w=0 to M−1, and the words may be used as a basis on which to perform the multiplication with the multiplier A. A discussion of a circuit implementation for this embodiment follows.
- [0037]
FIG. 4 is a block diagram illustrating an interstitial product generator (“IPG”)**100**according to an embodiment of the present invention. The IPG**100**generates an interstitial product from a multiplicand A. It may include a multiplicand register (called, an “A register” herein)**110**, a pair of shifters**120**,**130**(labeled “shift 1” and “shift 2 respectively) and a “3A” register”**140**. The A and 3A registers are illustrated in phantom because they may (but need not) be placed within the IPG**100**itself; alternatively, they may be provided in some other circuit but their contents may be provided as an input to the IPG**100**. The IPG**100**also may include a pair of multiplexers (colloquially, “MUXes”)**150**,**160**and an inverter**170**. - [0038]The shifters
**120**,**130**each present values representing the value stored in the A register shifted by a predetermined number of bit positions. The first shifter**120**may present an A value shifted one bit position toward the most significant bit position. It is labeled “shift 1.” The second shifter**130**may present the A value having been shifted two bit positions toward the most significant bit position, labeled “shift 2.” In binary data systems, a single or double bit shift causes a two-fold or four-fold multiplication of a source data value respectively. - [0039]The shifters
**120**,**130**may be provided as any number of embodiments. Perhaps the simplest embodiment is to provide a shifter as a wired interconnection between the A register**110**and the MUX**150**. For example, each bit position i in the A register**110**may be connected to a position i+1 of the MUX**150**to constitute the “shift 1” shifter**120**. Similarly, each bit position i in the A register may be connected to a position i+2 of the MUX**150**to satisfy the “shift 2” shifter**130**. The least significant bit position of the shift 1 shifter may be grounded. The two least significant bit position of the shift 2 shift inputs to the MUX 150 also may be grounded. This architecture provides the desired shift functions with the least implementation cost in terms of area or control hierarchy. - [0040]Alternatively, the shifters
**120**,**130**could be provisioned as formal shift registers complete with storage cells (not shown) for storage of shifted values. Although there is no performance advantage for this alternative, it may find use in applications where such shift registers are employed for other purposes. - [0041]The 3A register
**140**, as its name implies, is a register to store a value representing three times the value in the A register. The values stored in this register may be obtain from a straightforward addition of the values from the A register**110**and the shift 1 register**120**or, alternatively, from a subtraction of the values in the shift 1 register**120**from the value in the shift 2 register**140**. Circuitry to implement these functions is straightforward and omitted from the illustration ofFIG. 1 to keep the illustration simple. In an embodiment, the 3A register**140**also may be provided in a location external to the IPG**100**; it is illustrated in phantom accordingly. - [0042]Outputs from the A register
**110**, the two shifters**120**and**130**and the 3A register**140**may be input to the first MUX**150**. An output from the first MUX**150**may be input to both the second MUX**160**and the inverter**170**. An output from the inverter**170**may be provided as a second input to the second MUX**160**. The inverter**170**may generate a two's complement inversion of the multibit output from the first MUX**150**. The second MUX**160**may have a third input coupled directly to a zero value “∅.” Alternatively, the zero value could be input to the first MUX**150**. Thus, given an input value A, the IPG**100**may generate any of the following outputs: A, {overscore (A)}, 2A, {overscore (2A)} 3A, {overscore (3A)}, 4A, 4{overscore (A)} and ∅. - [0043]The IPG
**100**may include a controller**180**that governs operation of the two MUXes**150**,**160**. As discussed below, given an input “segment,” the controller**180**may generate a control signal (labeled c_{i}) that causes the MUXes**150**,**160**to output a selected one of the possible outputs on each cycle of a driving clock (not shown). - [0044]
FIG. 5 is a block diagram illustrating an IPG**200**according to an alternate embodiment of the present invention. According to an embodiment, the IPG**200**may include a plurality of inverters**210**,**220**,**230**-**1**,**230**-**2**, a 3× multiplier**240**, a pair of shifters**250**-**1**,**250**-**2**and a multiplexer**260**. In this embodiment, the IPG**200**is illustrated as connected to an external multiplicand register rather than including the multiplicand register as part of it. The multiplicand may be input to the IPG**200**on a first terminal**270**thereof. One of the inverters**210**may be coupled to the first terminal**270**, to invert a multiplicand when presented. - [0045]The 3× multiplier, as its name implies, may generate a value that is three times a multiplicand when presented at the input terminal. A second inverter
**220**may be coupled to the 3× multiplier**240**to invert the output thereof. - [0046]The shifters
**250**-**1**,**250**-**2**provided shifted versions of the multiplicand as in the embodiment ofFIG. 2 . One of them (say, shifter**250**-**1**) shifts the input multiplicand by a single bit position; the other**250**-**2**shifts the multiplicand by two bit positions. Inverters**230**-**1**,**230**-**2**from the respective shifters**250**-**1**,**250**-**2**may generated inverted shifted values of the multiplicand. The shifters**250**-**1**,**250**-**2**may be provided according to any of the embodiments described above. - [0047]Outputs from the inverters
**210**,**220**,**230**-**1**,**230**-**1**, the 3× multiplier**240**and the shifters**250**-**1**,**250**-**2**may be input to the multiplexer**260**. The multiplexer**260**also may be controlled to output none of the inputs from the IPG**200**. In this condition, the multiplexer**260**causes the IPG**200**to generate a zero output therefrom. - [0048]According to an embodiment, when it is desired to perform a multiplication based on a long multiplicand A and a shorter multiplier B, the multiplicand A may be input to the IPG
**200**. Values of A, 3A, {overscore (A)} and {overscore (3A)} will be available to the multiplexer**260**after a short initialization period. Similarly, shifted values of A and {overscore (A)} also will be available to the multiplexer**260**. Once these values are available, they may be retrieved from the IPG and forwarded to a remainder of a multiplication circuit (not shown inFIG. 5 ) based on values of multiplier segments. - [0049]The IPG may include a controller
**290**responsive to these multiplier segments to cause the multiplexer**260**to retrieve one of the previously stored values from within the IPG. As is known, many multiplication circuits include controllers for other purposes. The controller**290**may be integrated into these known controllers or may be provided as a separate element as may be desired.FIG. 5 illustrates the controller**290**as being separate from the IPG**200**for convenience only. - [0050]In an embodiment, the multiplier B may be parsed into several four bit segments s
_{i}. Each segment s_{i }includes the bits B_{3i+2}-B_{3i−1 }from the multiplier B. From these segments, a control signal CTRL may be generated to determine which value from within the IPG should be output from the multiplexer. In an embodiment, the IPG may generate outputs according to the scheme shown in Table 1 below.TABLE 1 Input Pattern of Segment s _{i}CTRL Interstitial Product 0000 0 0 0001 1 A 0010 1 A 0011 2 A << 1 0100 2 A << 1 0101 3 3A 0110 3 3A 0111 4 A << 2 1000 −4 A >> 2 1001 −3 3A 1010 −3 3A 1011 −2 A << 1 1100 −2 A << 1 1101 −1 {overscore (A)} 1110 −1 {overscore (A)} 1111 0 0

Where {overscore (A)} is the two's complement of an input A. The control value CTRL may be related to the four bits input pattern by:

*CTRL=−*4*s*_{i3}+2*s*_{i2}*+s*_{i1}*+s*_{i0}(8)

where s_{ij }represents the j^{th }bit position of segment s_{i}. For the segment s_{0}, the zero^{th }bit position, which would be a fictional bit position “B_{−1},” may be set to 0 to render the response of the controller to segment s_{0 }consistent with Table 1. - [0051]It may be observed from
FIG. 6 that a multiplier B will not fill all segments completely unless the length of the multiplier is a multiple of 3. In an embodiment, when the length of the multiplier is not a multiple of 3, it may be sign extended in sufficient length to fill an otherwise unused portion of the last segment. This typically involves copying the sign bit, the most significant bit B_{MSB}, to fill fictional bit positions beyond the most significant bit. - [0052]The IPG embodiments described above may be applied to multiplier circuits of a variety of architectures. In each application, use of an IPG permits the multiplier circuit to achieve faster operation essentially by permitting additions to occur once every three bit positions rather than once per bit position as is conventional.
- [0053]
FIG. 7 is a diagram of an application of an IPG in a combinatorial multiplier**300**according to an embodiment of the present invention. The combinatorial multiplier**300**may include first and second registers**310**for storage of a multiplicand A and a multiplier B respectively. It may include an IPG**330**, a controller**340**, a plurality of interstitial product registers**350**.**1**-**350**.L, an adder tree**360**a product register**370**. - [0054]During operation, the IPG
**330**may be initialized to create the values A, {overscore (A)}, 3A and {overscore (3A)} and shifted values of A and {overscore (A)}. The controller**340**may parse the multiplier B into segments and, responsive to the bit pattern in each segment, cause the IPG**330**to load one of the values into a corresponding interstitial product (say,**350**.**1**). In an embodiment, the number of interstitial products**350**.**1**-**350**.L may be based upon the number of segments supported by the multiplier B. Thus, the number of interstitial products**350**.**1**-**350**.L may be tied to the length of the multiplier register**320**. - [0055]Once values are loaded in each of the interstitial product registers
**350**.**1**-**350**.L, the combinatorial multiplier**300**may cause a final product to be stored in the product register**370**by summing across all the interstitial product registers**350**.**1**-**350**.L. An adder tree**360**accepts the interstitial product values from each product registers and sums them in a manner that is cognizant of the respective bit offsets among the interstitial product registers. Adder trees are well known per se and may be modified for this purpose. - [0056]In an embodiment for a multiplicand A of length m and a multiplier B of length n, the product register
**370**may have a length n+m just as in the traditional combinatorial multiplier. Interstitial product registers**350**.**1**-**350**.L may have a length m+2 whereas in traditional combinatorial multipliers, they would have a length m. - [0057]Traditional combinatorial multipliers include one interstitial product register for each bit position of a multiplier B. In the foregoing embodiment, there need be only one interstitial product register (say,
**350**.**1**) for every three bit positions of the multiplier B. Thus, because the combinatorial multiplier**300**of the foregoing embodiments include approximately one-third the number of interstitial registers than conventional counterparts, the adder tree include one-third the number of adders to compute the final summation. Products are expected to be generated faster in the present embodiment because the final addition is carried out across fewer interstitial values and, therefore, is available more quickly. This embodiment, therefore, yields higher throughput with less logic. - [0058]
FIG. 8 is a block diagram of an application of an IPG integrated with a shift-add multiplier circuit**400**according to an embodiment of the present invention. The multiplier circuit**400**may include a pair of registers**410**,**420**for storage of the multiplicand A and the multiplier B. The shift-add multiplier circuit**400**also may include an IPG**430**, a controller, a carry save adder**450**and a product register**460**. - [0059]During operation, the multiplier circuit
**400**may be initialized. In this embodiment, the product register**460**may be cleared to zero and the interstitial product generator**430**may be initialized with the value of the multiplicand A. Thereafter, during operation, the controller**440**may shift each segment out of the multiplier register**420**and, responsive to the new segment, may cause a selected value to be output from the IPG**430**to a first input of the carry save adder**450**. The most significant bits from the product register**460**may be shifted 3 places and input to a second input of the carry save adder. The carry save adder**450**may add the values presented on each of its two inputs together and write the value back to the product register**460**. This process may repeat in an incremental fashion for as many segments as are supported by the multiplier B. - [0060]As in the embodiment of
FIG. 4 , the embodiment ofFIG. 8 provides for improved performance over traditional shift-add multiplier circuits. The traditional circuits perform an addition for each bit position of a multiplier B. By contrast, the embodiment shown inFIG. 8 , provides an addition only once for every three bit positions of the multiplier B. Again, a fewer number of additions permit the shift-add multiplier ofFIG. 8 to generate multiplication products in a shorter amount of time than would be available from traditional circuits. - [0061]In another embodiment, a multiplier circuit may omit use of a multiplier register (such as the multiplier
**420**ofFIG. 8 ).FIG. 9 illustrates a multiplier**500**populated by a multiplicand register**510**, an IPG**520**, a controller**530**, a carry save adder**540**and a product register**550**. A multiplicand value (A) may be input to the IPG**520**as an initialization step. Additionally, a multiplier value (B) may be loaded into the least significant bit positions of the product register**550**. - [0062]On each clock cycle, the contents of the product register
**550**may be downshifted by three bit positions. When the least significant bits of the product register are shifted out of the product register, they may be input to the controller**530**as a new segment. In response to these three bits (and one bit from the shift of a prior clock cycle), the controller**530**may cause the IPG**520**to generate an output as shown above in Table 1. The IPG output may be provided to a first input of the carry save adder**540**. The downshifted value from the product register**550**may be provided to a second input of the carry save adder**540**. The carry save adder**540**may add the two input values and store them to the product register in the most significant bit positions. Again, this embodiment provides improved performance over other shift-add multipliers that operate a single bit shift at a time. As compared to the embodiment ofFIG. 4 , this embodiment also provides better register utilization because the register**420**(FIG. 4 ) may be omitted. - [0063]
FIG. 10 illustrates a multiplication circuit**600**according to another embodiment of the present invention. This embodiment may be used to evaluate equation 7 above. In this embodiment, the multiplier**600**may include a pair of IPGs**610**,**620**. A first IPG**610**may evaluate the A·B[i] term of equation of equation 7, the second IPG**620**may evaluate the d·q[i−1] term. The first IPG**610**may accept as inputs values of A and 3A from respective registers**630**,**640**. The first IPG**610**may be controlled by a controller**650**which in turn may be controlled by the respective word of B (input not shown). - [0064]The second IPG
**620**may accept values of d and 3d from respective sources**660**,**670**. The second IPG**620**may be controlled by a q value**680**. - [0065]The multiplication circuit
**600**may include a carry save adder**690**that accepts, as inputs, outputs from the two IPGs**610**,**620**and a result value from a product register**700**. An output of the carry save adder**690**may be input to the product register**700**. The product register**700**may be provided as a shift register to implement the shifts described in connection withFIG. 3 . A span of bits from the least significant bit positions may output to the carry save adder**690**as the r[i] value. - [0066]As is known, a carry save adder generates a result in the so-called “redundant form.” Carry save adders are faster than other types of adders because they generate results of addition operations without performing a traditional carry propagation (a time consuming operation). Instead, the addition results are stored using multiple bits per “bit position.” Multiple additions can be performed in redundant form. After a final addition, a single carry propagation may be performed to obtain a result in non-redundant form.
- [0067]According to an embodiment, a portion of the product register
**700**corresponding to the quotient q may be input to a second adder**710**such as a carry lookahead adder. The second adder**710**may generate a non-redundant result which may be fed back to the second IPG**610**as the q value. - [0068]The foregoing embodiments have been presented in connection with an evaluation of a (A·B) emod n operation. If done in connection with an (A·B) mod n operation, the methods illustrated in
FIGS. 1-3 may be followed by a single mod operation to complete the evaluation. These embodiments may be used advantageously, however, for the evaluation of other operations, such as an A^{B }emod n operation, in which case the mod n operation may be deferred. - [0069]Evaluation of A
^{B }emod n - [0070]Embodiments of the emod operation are described below in the context of an A
^{B }mod n calculation. Resolution of the calculation may proceed as a nested loop process, constituting an outer loop and an inner loop. - [0071]The outer loop may scan the bits of the exponent and to control the multiplications.
- [0072]Each pass of the outer loop may include squaring operation:

*c=*(*c·c*)*emod n,*(9)

and, depending on the current bit in the exponent, an additional multiplication:

*c=*(*c·a*)*emod n,*(10)

This embodiment is illustrated inFIG. 11 . - [0073]
FIG. 11 illustrates another method**1300**according to an embodiment of the present invention. According to the method**1200**, a dummy variable c may be initialized to be zero (box**1210**). Thereafter, the method iteratively considers each bit i of the exponent B. During each iteration, the method may evaluate equation 9 using an inner loop (box**1320**). Thereafter, the method**1300**may determine whether the i^{th }bit of the exponent B (B[i]) is 1. If so, the method**1300**may evaluate equation 10 using an inner loop (box**1330**). When the operation of box**1330**concludes or if the ith bit was not a 1, the operation may advance to a next successive iteration. - [0074]Upon conclusion of the last iteration, the method
**1300**may invoke a traditional mod operation upon the value c (box**1340**). This result obtains the result of A^{B }mod n. - [0075]The inner loop of the operation may be performed using any of the methods described hereinabove with respect to
FIGS. 1-3 . During processing of the inner loop, both cases may be implemented using the same hardware. There need be no special handling to accommodate the squaring case. Typically the operands c and mn are of the same size and a is somewhat smaller. - [0076]Determining d and m.
- [0077]To determine the parameters d and m, the number of significant bits in n, size(n), may be determined so that:

2^{size(n)−1}*≦n<*2^{size(n)}. (11)

Also, k may be defined as:

*k=size*(*n*)+*precision*(12)

for some desired precision value. Further, mn may be defined to be m·n, an integer multiple of n, so that:

*d=*2^{k}*−mn<n*(13)

The parameter mn, when viewed in binary format, has a number of leading 1s equaling or exceeding the precision value. - [0078]To find m and d from n and 2k, note that:
$\begin{array}{cc}\mathrm{mn}=m\xb7n=\mathrm{int}\text{\hspace{1em}}\left(\frac{{2}^{k}}{n}\right)\xb7n& \left(14\right)\end{array}$

where int(x) finds the largest integer of some value x. To find m, one may use the Newton method:

*u*_{i+1}=(*u*_{i}(2^{k+1}*−u*_{i}*·n*))/2^{k}. (15) - [0079]Equation 15 has a property that:

u_{i}→(2^{k}/n) as u →∞. (16)

As it is desirable to find only the integer part of (2^{k}/n), this iteration converges quite fast. To keep the number of bits in u_{i }to a minimum, it is possible also to take the integer part after each iteration and throw away the fractional part:

*u*_{i+1}=(*u*_{i}·(1^{k+1}*−*_{i}*·n*))/2^{k}(17)

This implementation at times can lead to a problem where the iteration stalls before reaching the correct number. Modifying the final iteration slightly however cures the problem.

*u*_{Final}=((*u*_{Final−1}+1)·(2^{k+1}*−u*_{Final−1}*·n*)−2^{k})*div*2^{k } - [0080]In another embodiment, the calculation of (2
^{k+1}−u_{i}·n) where (u_{i}·n)<2^{k+1 }can be done simply by setting the leading ones of the negative number −(u_{i}·n) to zero from position (k+1) and up. That is equivalent to keeping the bits from position k down to 0. - [0081]In one embodiment, the calculation of m and d may be implemented in software according to the following pseudocode.
- [0000]where the function selectbits(max downto min, c) returns the bits in c from and including positions max and down to min.
- [0082]The embodiments presented hereinabove provide a computational substitute for a mod operation, labeled “emod,” that incurs much less computational expense at the cost of lost precision. It is useful when performing a mod operation in connection with multiplications or exponential operations. By applying an emod operation to interstitial products, the length operands may be maintained to be within some predetermined length window. When a final product is obtained, a traditional mod operation may be applied to obtain a final result. This scheme obtains a final result with much less processing than would be possible using only the mod operation.
- [0083]Several embodiments of the present invention are specifically illustrated and described herein. However, it will be appreciated that modifications and variations of the present invention are covered by the above teachings and within the purview of the appended claims without departing from the spirit and intended scope of the invention.

Patent Citations

Cited Patent | Filing date | Publication date | Applicant | Title |
---|---|---|---|---|

US4208722 * | Jan 23, 1978 | Jun 17, 1980 | Data General Corporation | Floating point data processing system |

US4864529 * | Oct 9, 1986 | Sep 5, 1989 | North American Philips Corporation | Fast multiplier architecture |

US5008850 * | May 25, 1990 | Apr 16, 1991 | Sun Microsystems, Inc. | Circuitry for multiplying binary numbers |

US5373560 * | Aug 4, 1993 | Dec 13, 1994 | Schlafly; Roger | Partial modular reduction method |

US5402369 * | Jul 6, 1993 | Mar 28, 1995 | The 3Do Company | Method and apparatus for digital multiplication based on sums and differences of finite sets of powers of two |

US5644695 * | Jul 15, 1994 | Jul 1, 1997 | International Business Machines Corporation | Array combinatorial decoding with multiple error and erasure detection and location using cyclic equivalence testing |

US20020154768 * | Apr 12, 2002 | Oct 24, 2002 | Lenstra Arjen K. | Generating RSA moduli including a predetermined portion |

US20040010530 * | Jul 10, 2002 | Jan 15, 2004 | Freking William L. | Systolic high radix modular multiplier |

Referenced by

Citing Patent | Filing date | Publication date | Applicant | Title |
---|---|---|---|---|

US7966361 * | Jan 24, 2007 | Jun 21, 2011 | Nvidia Corporation | Single-cycle modulus operation |

Classifications

U.S. Classification | 708/491 |

International Classification | G06F7/72 |

Cooperative Classification | G06F7/722, G06F2207/728, G06F7/72 |

European Classification | G06F7/72, G06F7/72C |

Legal Events

Date | Code | Event | Description |
---|---|---|---|

May 9, 2002 | AS | Assignment | Owner name: INTEL CORPORATION, CALIFORNIA Free format text: ASSIGNMENT OF ASSIGNORS INTEREST;ASSIGNOR:HOJSTED, ERIK;REEL/FRAME:012881/0055 Effective date: 20020421 |

Aug 30, 2010 | REMI | Maintenance fee reminder mailed | |

Jan 23, 2011 | LAPS | Lapse for failure to pay maintenance fees | |

Mar 15, 2011 | FP | Expired due to failure to pay maintenance fee | Effective date: 20110123 |

Rotate