A squaring circuit for a binary number x of n bits x0 to xn-1, includes a table of the squares of numbers p constituted by bits x1 to xn-2. An adder for adding numbers of 2n-3 bits receives at a first input a number constituted by bit xn-1, positioned on the left of the square p2 provided by the table. A first switching element receives the number p and provides same to the n-2 low weight lines of a second input of the adder if bit x0 is equal to 1. A second switching element receives number p and provides same to the n-2 high weight lines of the second input if bit xn-1 is equal to 1. An AND gate is connected to the remaining line of the second input and receives the bits x0 and xn-1. The square x2 of x is constituted by the adder output, to which a bit 0 and the bit x0 are positioned on the right.
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19. A squaring circuit comprising:
a circuit responsive to a second binary number p, wherein each number p includes bits of weight 1 to n-2 of a binary number x to be squared, wherein binary number x has n bits, each bit having a weight assigned thereto, wherein the weights increase from 0 to n-1, the circuit for generating the square p2 of the second binary number and including a table stored in a memory having at least one bit of the p2 ; an adder, coupled to the circuit, receiving at a first multi-bit input thereof, a bit of weight n-1 of the first number x and bits of the square p2 of the second binary number; a first switching element, coupled between the memory and the adder, receiving the second number p and providing the second number p to n-2 lines of low weight of a second input of the adder if a lowest weight bit of the first number x is equal to 1, wherein 1 is a predetermined logic level; a second switching element, coupled between the memory and the adder, receiving the second number p and providing p to n-2 high weight lines of the second input of the adder ira bit of weight n-1 of the first number x is equal to 1; and a third switching element, coupled between the memory and the adder, providing 1 to a remaining line of the second input of the adder if bits of weight 0 and n-1 of the first number x are both equal to 1; wherein the square x2 of the first number x is provided by an output of the adder, and to the output of the adder are added at predetermined positions at bit 0 and lowest weight bit of the first number x.
1. A squaring circuit for raising to the second power a first binary number x of n bits, each bit having a weight assigned thereto, wherein the weights increase from 0 to n-1, the circuit comprising:
a circuit responsive to a second binary number p including bits of weight 1 to n-2 of the first number x, for generating the square p2 of the second binary number, the circuit arrangement includes a table stored in a memory having at least one bit of the p2, an adder, coupled to the circuit, receiving at a first multi-bit input thereof, a number including bits of the square p2 of the second binary number and a bit of weight n-1 of the first number x; a first switching element, coupled between the memory and the adder, receiving the second number p and providing the second number p to n-2 lines of low weight of a second input of the adder of the bit of weight 0 of the first number x is equal to b 1; a second switching element, coupled between the memory and the adder, receiving the second number p and providing p to n-2 high weight lines of the second input of the adder if the bit of weight n-1 of the first number x is equal to 1; and a third switching element, coupled between the memory and the adder, providing a 1 to a remaining line of the second input of the adder if bits of weight 0 and n-1 of the first number x are both equal to 1, wherein 1 represents a first predetermined logic level; wherein the square x2 of the first number x is provided by an output of the adder, and to the output of the adder are added at predetermined positions a bit 0 and the bit of weight 0 of the fast number x.
10. A squaring circuit for raising to the second power a first binary number x of n bits, each bit having a weight assigned thereto, wherein the weights increase from 0 to n-1, the circuit comprising:
means responsive to a second binary number p including bits of weight 1 to n-2 of the first number x, for generating the square p2 of the second binary number, the means for generating including means for storing at least one bit of the p2, means for adding numbers, coupled to the means for generating, receiving at a first multi-bit input thereof, a number constituted by the bit of weight n-1 of the first number x and bits of the square p2 ; a first means for switching, coupled between the means for storing and the means for adding, receiving the second number p and providing the second number p to n-2 lines of low weight of a second input of the adder if a lowest weight bit of the first number x is equal to 1; a second means for switching, coupled between the means for storing and the means for adding, receiving the second number p and providing p to n-2 high weight lines of the second input of the adder if a bit of weight n-1 of the first number x is equal to 1; and a third means for switching, coupled between the means for storing and the means for adding, providing a 1 to a remaining line of the second input of the adder if bits of weight 0 and n-1 of the first number x are both equal to 1, wherein 1 represents a first predetermined logic level; wherein the square x2 of the first number x is provided by an output of the means for adding, and to the output of the means for adding are added at predetermined positions thereof a bit 0 and lowest weight bit of the first number x.
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1. Field of the Invention
The present invention relates to a squaring circuit for raising to the second power a binary number of n bits and more particularly to such a circuit including a table of squared numbers, which is stored in a nonvolatile memory element (i.e. ROM).
2. Discussion of the Related Art
Conventionally, a circuit for raising to the second power a binary number X is achieved by multiplying X by itself with a binary multiplying circuit. A binary circuit for multiplying 8-bit numbers requires approximately 1,500 transistors in conventional technologies and architectures. Such a conventional circuit is called a multiplier and a common drawback includes the large surface area used.
An alternative for achieving a squaring circuit of an 8-bit binary number X is to store in a ROM memory element all of the squares of n-hi t numbers and to select one of these squares by addressing one of the squares. This addressing occurs by applying number X on the address lines of the ROM. If number X is composed of n bits, 2n numbers of 2n bits will have to be stored in the ROM (the square of an n-bit number is a number including, at the most, 2n bits). Hence, in this example, a ROM of 2n ×2n bits is required. A ROM including all the squares of 8-bit numbers includes approximately 1,900 transistors. Since these transistors are very orderly arranged, the ROM can be easily devised so as to occupy, for n<10, an equal or even smaller silicon surface than an equivalent conventional multiplier circuit (using the same technology).
Accordingly, an object of the present invention is to provide a squaring circuit including a ROM having a size at least four times smaller than the size of a ROM used in a conventional equivalent squaring circuit.
This object and other advantages and features of the invention is achieved with a circuit for providing the square of a binary number X having n bits with weights increasing from 0 to n-1, the circuit includes a table of the squares of binary numbers p. Each number p is constituted only by the bits of weight 1 to n-2 of number X. The circuit further includes: an adder, for adding numbers of 2n-3 bits, receiving at a first input a number constituted by the bit of weight n-1 of number X, positioned on the left of the square p2 provided by the table; a first switching element receiving the number p and providing same to the n-2 lines of low weight of a second input of the adder if the low weight bit of number X is equal to 1; a second switching element receiving the number p and providing same to the n-2 lines of high weight of the second input of the adder if the bit of weight n-1 of number X is 1; a third switching element providing a 1 to the remaining line of the second input of the adder if the bits of weight 0 and n-1 of number X are both 1. The square X2 of number X is constituted by the adder output, to which output are positioned on the right side a bit 0 and the lowest weight bit of number X.
According to an embodiment of the invention, the table is a memory element storing only the 2(n-2)-2 high weight bits of number p2. Number p2 is constituted by the memory output to which are positioned on the right side a bit 0 and the bit of weight 1 of number X.
According to this embodiment of the invention, the first switching dement includes n-2 AND gates, the first inputs of which receive the lowest weight bit of number X and the second inputs of which respectively receive one of the bits of number p.
According to this embodiment of the invention, the second switching element includes n-2 AND gates, the first inputs of which receive the bit of weight n-1 of number X and the second inputs of which respectively receive one of the bits of number p.
According to this embodiment of the invention, the third switching element is comprised of an AND gate, receiving at the inputs the bits of weight 0 and n-1 of number X.
The foregoing and other objects, features, aspects and advantages of the invention will become apparent from a reading of the following detailed description of the present invention when taken in conjunction with the accompanying drawings.
FIG. 1 symbolically and graphically illustrates a calculation method used by a squaring circuit according to the present invention; and
FIG. 2 illustrates in block diagram, schematic form, an embodiment of a squaring circuit according to the present invention.
Assuming that it is desired to provide the square of a number X including n bits,
X=2n-1 xn-1 + . . . +21 x1 +20 x0,
where x0 . . . xn-1 designate respectively the bits of weights 0 to n-1 of number X.
A circuit according to the invention provides the square of this number, as described below.
Number X can be written in the form:
X=2n-1 xn-1 +2p+20 x0, (1)
where p=2n-3 xn-2 + . . . +21 x2 +20 x1.
Thus, by raising equation (1) to the second power and by simplifying the resulting equation, taking into account that xi2 =xi since xi =1 or 0, one obtains: ##EQU1##
FIG. 1 symbolically and graphically illustrates equation (2). Various binary numbers are shown in the form of rectangles divided into cells corresponding to bits. Each cell of the rectangles is disposed in a column which cot-responds to the weight of the bit of the particular cell. In FIG. 1, columns 0 to 2n-1 are numbered from right to left, the bits with the lowest weight corresponding to the rightmost column. An ordinary binary number expressed in the form 2i z, where z is a binary value of nz bits, can be represented by a rectangle having nz cells, the cell having the lowest weight being in column i. the cells of the rectangle are occupied by the bits of value z, which is indicated by writing this value in the rectangle. The column(s) that are not occupied by a rectangle correspond to null bits.
Rectangle X represents equation (1). The bits x0 to xn-1 of number X appear in columns 0 to n-1. Number p is formed by bits x1 to xn-2 of number X and appears in the rectangle X within the appropriate columns.
The six following rectangles represent terms T1-T6 of the right side of equation (2); the last rectangle represents result X2.
Rectangle T1 represents the term
T1=2n+1 xn-1 p
of equation (2). This rectangle, as shown, occupies the n-2 columns n+1 to 2n-2. Value xn-1 p includes as many bits as number p since this value is obtained by a simple logic ANDing of each of the bits of number p and bit xn-1.
Rectangle T2 represents the term T2=4x0p=22 x0p
of equation (2). As shown, rectangle T2 occupies the n-2 columns 2 to n-1.
Rectangle T3 represents the term T3=2n x0 xn-1.
As shown, rectangle T3 occupies column n only (x0 xn-1 corresponds to one bit).
In viewing rectangles T1, T2 and T3, it can be seen that the sum T1+T2+T3 is in fact equal to a number formed by the juxtaposition of values xn-1 p, x0 xn-1 p and x0 p followed by two 0s. Hence, a binary adder is not required to sum terms T1-T3 since, physically, a corresponding juxtaposition of the columns carrying the above value bits simply forms the sum.
Rectangle T4 represents the term
T4=4p2 =22 p2
of equation (2). As shown, rectangle T4 occupies the 2(n-2) columns 2 to 2n-3.
Rectangle T5 represents the term T5=22 (n-1)xn-1.
As shown, rectangle T5 occupies column 2n-2. Rectangle T6 represents the term T6=x0 and, as shown, occupies column 0. The sum of terms T4, T5 and T6 is obtainable by a simple juxtaposition of values xn-1, p2, 0 and x0, which operation avoids the use of a binary adder.
In addition, FIG. 1 shows that for any value of number X, the bit of weight 1 of number X2 (represented by rectangle X2) is always null and that the bit of weight 0 is always equal to bit x0 of weight 0 of number X. Thus, it is not necessary to store the bits of weight 0 and 1 of number p2 because they are known to be respectively equal to the bit x1 of number p (which is the bit of weight 1 of number X) and to 0. Number p2 can thus be obtained, as shown in rectangle T4, by positioning the numbers 0 and x1 on the right of its 2(n-2)-2 bits of higher weight.
The sum of the six terms T1-T6 is reduced to a single sum of the number corresponding to the juxtaposition of rectangles T1-T3 and of the number corresponding to the juxtaposition of rectangles T4 and T5. A null bit and bit x0 (T6) are then positioned on the fight of this sum to provide number X2. The last two bits are known to exist in their respective positions for all values of X, as described above.
This analysis teaches that instead of storing 2n squares of X, it is merely necessary to store 2n-2 squares of p. Thus, the amount of squares necessary to store is one quarter of that of the conventional binary multiplier. Moreover, only the 2(n-2)-2=2n-6 high weight bits of numbers p2, as opposed to the 2n bits of number X2, as is done with conventional binary multipliers, are stored. The total amount of memory bits is therefore 2n-2 (2n-6)=2n-1 (n-3), which provides a gain of surface area ("chip real estate") of 4n/(n-3) over the conventional binary multiplier. This gain approaches 4 as n approaches infinity. For 8-bit numbers X (a=8), this gain is 6.4.
According to the invention, in addition to the memory needed, additional circuits are required to carry out the digital logic. Such circuits include a (2n-3)-bit adder for summing values (T1+T2+T3) and (T4+T5) and logic gates for calculating the values xn-1 p, x0 p and x0 xn-1. Those skilled in the art will appreciate that these circuits occupy a small surface area in comparison to the surface spared by the use of a smaller memory.
FIG. 2 is a block diagram showing a preferred embodiment of the squaring circuit according to the present invention. Bits x0 to xn-1 of a number X, the square of which is to be calculated, are introduced into the circuit at lines x0 to xn-1. The n-2 lines x1 to xn-2, which carry the bits of number p, are connected to the address lines of ROM 20. ROM 20 outputs the 2n-6 high weight bits (p2)2 to (p2)2n-5 of number p2. The remaining low weight bit lines (p2)1 and (p2)0 are connected to line 0 and line x1, respectively.
The circuit includes an adder ADD for adding the 2n-3 bit numbers. Adder ADD comprises 2n-3 elementary adders ADD2 to ADD2n-2 which respectively provide the bits of weight 2 to 2n-2 of number X2 Each elementary adder includes two bit inputs a and b and each carries an output c for transmitting a carry bit to an elementary adder having an immediately higher weight. The carry output of adder ADD2n-2 provides the bit of weight 2n-1 of number X2. The carry outputs of the other adders are not shown.
Bits (p2)0 to (p2)2n-5 are provided to inputs b of adders ADD2 to ADD2n-3. Input b of adder ADD2n-2 is connected to line xn-1. The bits of weight 0 and 1 of number X2 are provided by line x0 and a line connected to 0, respectively.
Input a of adders ADD2 to ADD2m-2 are connected to the output of AND gates &2 to &2n-2, respectively. A first input of AND gates &2 to &n-1 is connected to line x0. Bits x1 to xn-2 of number p are provided to a second input of AND gates &2 to &n-1, respectively. AND gates &2 to &n-1 provide value x0 p to adders ADD2 to ADDn-1, respectively.
A first input of AND gates &n+1 to &2n-2 is connected to line xn-1, and bits x1 to xn-2 of number p are respectively provided to a second input of these AND gates AND gates &n+1 to &2n-2, respectively, provide value xn-1 p to adders ADDn+1 to ADD2n-z.
The two inputs of gate &n are connected to line x0 and line xn-1. AND gate &n provides the number x0 xn-1 to adder ADD1.
FIG. 2 shows that the arrangement of the AND gates and elementary adders is very orderly, permitting ease of placement of such elements on a small surface.
As will be apparent to those skilled in the art, various modifications can be made to the above disclosed preferred embodiment. In particular, AND gates &2 to &n-1 can be replaced with a multiplexer controlled by line x0 and receiving a null binary number as one input and the number p as another input. A similar multiplexer, controlled by line xn-1, can be substituted for AND gates &n+1 to &2n-2. The ROM can be replaced with any equivalent circuit, such as a programmable logic array (PLA), the output bits of which correspond to preprogrammed equations of the bits present at the input (on the address lines). Adder ADD2 can be replaced with an AND gate receiving bit x1 and the complementary of bit x0 since (X2)2=x1.x0 .
The foregoing description is provided by way of example only and in no way is intended on being limiting. The scope of the present invention is defined by the appended claims and equivalents thereto.
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