Abstract: A method of completing a two's complement operation includes receiving a plurality of byte values and splitting the plurality of byte values into a first portion and a second portion. Further, the method includes inputting the first portion to a first segment of a first four-to-two compressor, performing a first four-to-two compression operation on the first portion to generate a first set of results having a first row and a second row that is offset one bit from the first row, and carrying in a first value of one to complete a first two's complement operation. The method also includes inputting the second portion to a second segment of a second four-to-two compressor and adding two values of one immediately to the right of the second portion in order to carry in a second value of one to the second portion to complete a second two's complement operation.

Abstract: A sampling circuit includes a latch circuit which latches the digital signal S1 at a constant period, an addition register which adds the sampled data for the same input code, a divider which divides the added value by a predetermined divisor, a digital memory which stores the divided value and outputs it at an arbitrary timing for a predetermined reading out number, an operator which operates the output data from the digital memory in accordance with a previously set algorithm, a judgment circuit which judges the operation result with a predetermined judgment criterion, and a control logic part which controls such that the addition and outputting processing by the addition register and the division and outputting processing by the divider are carried out concurrently with the sampling processing by the latch circuit. This sampling circuit in an AD converter or a DA converter can reduce inspection cost.

Abstract: Techniques are described to determine N mod M, where N is a number having a width of n-bits, and M is a number having a width of m-bits. The techniques, generally, involve determining N?=Nrt2f mod M+NL and, subsequently, determining N? mod M.

Abstract: A first arithmetic operator (11) includes a first modular arithmetic coding encoder (11b) for encoding a numeric data transmitted by a command from a central controller (31) into a modular arithmetic code, a first arithmetic operation processor (11a) using the numeric data as modular arithmetic coded as an input operand, for executing an arithmetic operation based on a command from the central controller (13), to provide an output in the form of a modular arithmetic code, and a first modular arithmetic code decoder (11c) for determining presence or absence of a bit error in the numeric data output from the first arithmetic operation processor, correcting the bit error, if detected any, to output a decoded numeric data.

Abstract: A results reporting system can generate test results at one or more laboratory sites. Each of the generated test results can be in a conventional unit of measurement. The results reporting system can then convert each of the testing results to a normalized decimal equivalent unit system of measurement. The normalized decimal equivalent unit of measurement can be a unit of measurement that is normalized to an associated reference range of the test result. Once converted to normalized decimal equivalent unit, the test results can be stored in a mass storage device for later reporting. A triggering event (e.g., a request from a reviewing entity) can then trigger the reporting system to generate a results report using the stored test results. The results report can convey the test results to a reviewing entity using the normalized decimal equivalent unit of measurement in graphs, tables or other methods of conveying data.

Abstract: A computer executable method of performing a modal interval operation, and system for performing same is provided. The method includes providing representations of first and second modal interval operands. Each modal interval operand of the operands is delimited by first and second marks of a digital scale, each mark of the marks comprises a bit-pattern. Each bit-pattern of the bit-patterns of the marks of each of the modal interval operands are examined, and conditions of a set of status flags corresponding to each bit-pattern of the bit-patterns of the marks are set. A bit-mask is computed wherein the mask is based upon the set condition of the status flag sets and a presence/absence of an exceptional arithmetic condition, and a presence/absence of an indefinite operand are each represented by a bit of said bits of said bit mask.

Abstract: A method and a device having interleaving capabilities, the device comprises a first interleaver; the first interleaver comprises a first register, a second register, a first adder and a second adder; wherein the first register is coupled to the first adder and to the second adder; wherein the second register is coupled to the second adder; wherein the first adder is adapted to add a current first register value to a first coefficient to provide a next first register value that is stored at the first register; wherein the second adder is adapted to add a current first register value to a second coefficient, to a third coefficient and to a current second register value to provide an interleaved output value.

Abstract: The hash functions with elliptic polynomial hopping are based upon an elliptic polynomial discrete logarithm problem. Security using hash functions is dependent upon the implementation of a computationally hard problem, and the elliptic polynomial discrete logarithm problem provides enough relative difficulty in computation to ensure that the produced hash functions, as applied to message bit strings, are optimally secure. The hash functions are produced as functions of both the elliptic polynomial as well as the twist of the elliptic polynomial, particularly using a method of polynomial hopping.

Abstract: The hash functions using elliptic curve cryptography are hash functions that are produced using both an elliptic curve and a twist of the elliptic curve. Hash points are assigned values that either correspond to points on the elliptic curve or to points on the twist, depending upon whether the scalar value of the corresponding message block produces a quadratic residue or a quadratic non-residue when substituted as the x-value into the elliptic curve equation. The corresponding hash point x-coordinates are concatenated to form the hash bit string. The hash points may be doubled, and the hash functions may be applied to multimedia data by applying a media compression method to the message data before computing the hash points.

Abstract: A calculating unit for reducing an input number with respect to a modulus, wherein the input number has input number portions of different significances, wherein the input number portions represent the input number with respect to a division number, wherein the modulus has modulus portions of different significances, and wherein the modulus portions represent the modulus with respect to the division number, includes a unit for estimating a result of an integer division of the input number by the modulus using a stored most significant portion of the number, a stored most significant portion of the modulus and the number, and for storing the estimated result in a memory of the calculating unit, and a unit for calculating a reduction result based on a subtraction of a product of the modulus and a value derived from the estimated result from the number.

Abstract: Methods and a system are disclosed for one or more appliances including a controller for managing power consumption within a household. The controller is configured to receive and process a signal indicative of one or more energy parameters of an associated energy utility, including at least a peak demand period or an off-peak demand period. A generated serial number is obtained from an original serial number of the appliance or controller, which is configured for a signal to communicate to the appliance within a population and command the appliance to operate in an energy savings mode and a normal mode at various time periods. The generated serial number (GSN) is used to segregate a total population into segments to provide granularity in assigning DR activations and deactivations based upon the GSN.

Abstract: The message authentication code with elliptic polynomial hopping provides methods for the generation of message authentication codes (MACs) utilizing elliptic curves, which are based on the elliptic curve discrete logarithm problem. The elliptic curve discrete logarithm problem is well known to be a computationally “difficult” or “hard” problem, thus providing enhanced security for the MACs. Different elliptic polynomials are used for different blocks of the same plaintext, each elliptic polynomial for each message block being selected at random using an initial secret key and a random number generator.

Abstract: The cryptographic hash functions using of elliptic polynomial polynomials are based on the elliptic polynomial discrete logarithm problem, which is well known as a computationally hard problem. The hash functions are based on the elliptic polynomial equation in their generation, where different elliptic polynomials are used for different blocks of the same plain text. Particularly, the hash functions use an elliptic polynomial with more than one independent x-coordinate. More specifically, a set of elliptic polynomial points are used that satisfy an elliptic polynomial equation with more than one independent x-coordinate which is defined over a finite field F.

Abstract: A filter circuit includes: an adder/subtractor that performs at least addition; and a shifter that performs multiplication/division by a power of two through a shift operation. The adder/subtractors and the shifter are configured to obtain a first calculation result representing a pixel value of a target pixel included in image data multiplied by a first filter coefficient. At least the adder/subtractors and the shifter is configured to obtain a second calculation result representing pixel values of a plurality of peripheral pixels adjacent to the target pixel, with each of the pixel values being multiplied by a second filter coefficient. The adder/subtractor is configured obtain a third calculation result by adding the first and second calculation results. The shifter configured to divide the third calculation result by a power of two which is equivalent to a sum of the first and second filter coefficients, so as to output the division result.

Abstract: A processor or other device, such as a programmable and/or massively parallel processor or other device, includes processing elements designed to perform arithmetic operations (possibly but not necessarily including, for example, one or more of addition, multiplication, subtraction, and division) on numerical values of low precision but high dynamic range (“LPHDR arithmetic”). Such a processor or other device may, for example, be implemented on a single chip. Whether or not implemented on a single chip, the number of LPHDR arithmetic elements in the processor or other device in certain embodiments of the present invention significantly exceeds (e.g., by at least 20 more than three times) the number of arithmetic elements, if any, in the processor or other device which are designed to perform high dynamic range arithmetic of traditional precision (such as 32 bit or 64 bit floating point arithmetic).

Abstract: The elliptic-polynomial based Message Authentication Code (MAC) provides MAC generation methods based on the elliptic polynomial discrete logarithm problem. It is well known that an elliptic polynomial discrete logarithm problem is a computationally “difficult” or “hard” problem. The methods use both an elliptic polynomial and its twist, even if the polynomial and its twist are not isomorphic. Since both the polynomial and its twist are used, multiple x- and y-coordinates can be used to embed bit strings into a point that satisfies the elliptic polynomial, and the embedding process is non-iterative, so that the time required to embed the bit string is independent of the bit string content.

Abstract: A cell for an arithmetic logic unit includes a first input; a second input; a carry-in input; a first control input and a second control input; and a circuit connected to the first input, the second input, the carry-in input, the first control input, and the second control input. The circuit has a first output and a second output, the second output having a first value as a function of the first input and the second input when the first control input and the second control input are supplied values equal to a value at the carry-in input, and having a second value as a function of the first input and second input when the values at the first control input and the second control input are independent of the value at the carry-in input.

Abstract: A system includes a serial connection mode for obtaining a first approximation to a zero error result by means of a negative rough precision for manufacturing a plurality of first semi-finished products, and a measurement apparatus for measuring a precision value of each first semi-finished product, and a Full-9 Principle for sifting the first semi-finished products. A parallel connection mode is used for obtaining a second approximation to the zero error result by means of a positive rough precision by division to manufacture a plurality of second semi-finished products, and the measurement apparatus is used to measure a precision value of each second semi-finished product, and an error sift formula is utilized to sift the second semi-finished products.

Abstract: A computing method and circuit for computing a modular operation with at least one operand having a binary representation. Iteratively for each bit of this operand, doubling the value of an intermediate result stored in a first memory element by shifting the bits of the intermediate result towards the most significant bit and, while the most significant bit of the intermediate result is one, updating this intermediate result by subtracting a modulus stored in a second memory element.

Abstract: A method and apparatus perform many different types of algorithms that utilizes a calculation unit capable of utilizing the same multipliers for different algorithms. The calculation unit preferably includes a processor that has a plural number of arithmetic logic unit circuits that are configured to process data in parallel to provide processed data outputs and an adder tree configured to add the processed data outputs from the arithmetic logic circuits. A shift register that has more parallel data outputs then the processor's inputs is controlled to selectively output data from the parallel outputs to the data inputs of the processor. A communication device preferably includes the calculation unit to facilitate processing of wireless communication signals.

Abstract: A functional unit is added to a graphics processor to provide direct support for double-precision arithmetic, in addition to the single-precision functional units used for rendering. The double-precision functional unit can execute a number of different operations, including fused multiply-add, on double-precision inputs using data paths and/or logic circuits that are at least double-precision width. The double-precision and single-precision functional units can be controlled by a shared instruction issue circuit, and the number of copies of the double-precision functional unit included in a core can be less than the number of copies of the single-precision functional units, thereby reducing the effect of adding support for double-precision on chip area.

Abstract: An arithmetic logic and shifting device is disclosed and includes an arithmetic logic unit that has a first input to receive a first operand from a first register port, a second input to receive a second operand from a second register port, and an output to selectively provide a memory address to a memory unit in a first mode of operation and to selectively provide an arithmetic output in a second mode of operation. Further, the arithmetic logic and shifting device includes a programmable shifter device that has a first input to receive data from the memory unit, a second input to receive the arithmetic output, a third input to receive an operation code of a computer execution instruction, and a shifted output to provide shifted data.

Abstract: A multiple-word multiplication-accumulation circuit suitable for use with a single-port memory. The circuit is composed of a multiplication-accumulation (MAC) operator and surrounding registers. The MAC operator has multiplicand and multiplier input ports with different bit widths to calculate a sum of products of multiple-word data read out of a memory. The registers serve as buffer storage of multiple-word data to be supplied to individual input ports of the MAC operator. The amount of data supplied to the MAC operator in each clock cycle is adjusted such that total amount of data consumed and produced by the MAC operator in one clock cycle will be equal to or smaller than the maximum amount of data that the memory can transfer in one clock cycle. This feature enables the use of a bandwidth-limited single-port memory, without causing adverse effect on the efficiency of MAC operator usage.

Abstract: The present invention provides extended precision in SIMD arithmetic operations in a processor having a register file and an accumulator. A first set of data elements and a second set of data elements are loaded into first and second vector registers, respectively. Each data element comprises N bits. Next, an arithmetic instruction is fetched from memory. The arithmetic instruction is decoded. Then, the first vector register and the second vector register are read from the register file. The present invention executes the arithmetic instruction on corresponding data elements in the first and second vector registers. The resulting element of the execution is then written into the accumulator. Then, the resulting element is transformed into an N-bit width element and written into a third register for further operation or storage in memory. The transformation of the resulting element can include, for example, rounding, clamping, and/or shifting the element.

Abstract: A programmable element for data processing comprises a crosspoint switch (318), a mathematical operation module (320), and a plurality of data hold modules (604,606). Each of the data hold modules (604,606) receives data from the crosspoint switch (318) and communicates the data to an input of the mathematical operation module (320) such that data arrives at the inputs of the mathematical operation module (320) substantially simultaneously. A first data hold module (604) communicates a first data valid signal to a second data hold module (606) upon receipt of first valid data, and the second data hold module communicates a second data valid signal to the first data hold module upon receipt of second valid data.

Abstract: The present disclosure describes implementations for processing instructions and data across multiple Arithmetic Logic Units (ALUs). In one implementation, a graphics processing apparatus comprises a plurality of ALUs configured to process independent instructions in parallel. Pre-processing logic is configured to receive instructions and associated data to be directed to one of the plurality of ALUs for processing from a register file, the pre-processing logic being configured to selectively format received instructions for delivery to a plurality of the ALUs. In addition, post-processing logic is configured to receive data output from the plurality of the ALUs and deliver the received data to the register file for write-back, the post-processing logic being configured to selectively format data output from a plurality of the ALUs for delivery to the register file as though the data had been output by a single ALU.

Abstract: A general-purpose register file including a plurality of general-purpose registers stores parallel arithmetic data. A plurality of pattern registers store a plurality of items of pattern data indicating the rearrangement of data in bytes, in half words, in words, or in a combination of these units. A data select circuit selects one of the items of pattern data stored in the plurality of pattern registers according to specifying data included in an instruction. A rearranging circuit rearranges parallel arithmetic data according to the item of pattern data selected by the data select circuit.

Abstract: A radix-2k Montgomery multiplier including an input coefficient generation unit to receive a multiplier, a multiplicand, a modulus, a sum and a previous sum, to generate and to output a partial product and a multiple modulus by using at least one of the multiplier, the multiplicand, the modulus and the sum, and to divide and to output the received previous sum into units of k bits, an accumulator circuit to receive the partial product, the multiple modulus and k bits of the previous sum from the input coefficient generation unit, and to generate and to output a carry and a sum by summing the partial product, the multiple modulus and the previous sum, and a carry propagation adder (CPA) circuit to generate and to output an ultimate sum by using the carry and the sum.

Abstract: An interface to a dynamically configurable arithmetic unit can include data alignment modules, where each data alignment module receives input variables being associated with one or more arithmetic expressions. The interface can include multiplexers coupled to the data alignment modules, wherein a data alignment module has outputs coupled to a first multiplexer. The first multiplexer can have a selection line and an output coupled to an input port of the dynamically configurable arithmetic unit. The interface can include a second multiplexer having input instructions and the selection line, where each instruction is associated with one of the arithmetic expressions and has an operation to be performed by the dynamically configurable arithmetic unit. The second multiplexer is configurable to provide selected ones of the input instructions to the dynamically configurable arithmetic unit through an output of the second multiplexer responsive to the selection line.

Abstract: A method for instruction processing may include adding a first operand from a first register, a second operand from a second register and a carry input bit to generate a sum and a carry out bit, loading the sum into a third register and loading the carry out bit into a most significant bit position of the third register to generate a third operand, performing a single bit shift on the third operand via a shifter unit to produce a shifted operand and loading the shifted operand into the fourth register, loading a least significant bit from the sum into the most significant bit position of the fourth register to generate a fourth operand, generating a greatest common divisor (GCD) of the first and second operands via the fourth operand and generating a public key based on, at least in part, the GCD. Many alternatives, variations and modifications are possible.

Abstract: Modulus scaling applied a reduction techniques decreases time to perform modular arithmetic operations by avoiding shifting and multiplication operations. Modulus scaling may be applied to both integer and binary fields and the scaling multiplier factor is chosen based on a selected reduction technique for the modular arithmetic operation.

Abstract: A method of operating an arithmetic logic unit (ALU) by inverting a result of an operation to be executed during a current cycle in response to control signals from instruction decode logic which indicate that a later operation will require a complement of the result, wherein the result is inverted during the current cycle. The later operation may be a subtraction operation that immediately follows the first operation. The later instruction is decoded prior to the current cycle to control the inversion in the ALU. The ALU includes an adder, a rotator, and a data manipulation unit which invert the result during the current cycle in response to an invert control signal. The second operation subtracts the result during a subsequent cycle in which a carry control signal to the adder is enabled, and the rotator and the data manipulation unit are disabled. The ALU may be used in an execution unit of a microprocessor, such as a fixed-point unit.

Abstract: Time to perform scalar point multiplication used for ECC is reduced by minimizing the number of shifting operations. These operations are minimized by applying modulus scaling by performing selective comparisons of points at intermediate computations based on primality of the order of an ECC group.

Abstract: The computation time to perform scalar point multiplication in an Elliptic Curve Group is reduced by modifying the Barrett Reduction technique. Computations are performed using an N-bit scaled modulus based a modulus m having k-bits to provide a scaled result, with N being greater than k. The N-bit scaled result is reduced to a k-bit result using a pre-computed N-bit scaled reduction parameter in an optimal manner avoiding shifting/aligning operations for any arbitrary values of k, N.

Abstract: A binary arithmetic decoding device may include a first and second lookup table each receive signal output from a first register. A third lookup table receives signal output from the first lookup table and a fourth lookup table receives signal output from the second lookup table. A first multiplexer receives signal output from the third and fourth lookup tables. A second multiplexer receives signal output from the first and second lookup tables and from the second lookup table where the first and second multiplexers are controlled by a same first signal. The proposed decoder may further include a second register, a first adder, a third adder, and a first comparison module coupled in series and output of the first comparison module is the first signal.

Abstract: A scalar multiplication can be performed on an elliptic curve cryptosystem at a high speed. P is set as an initial value of Q[0], and 2×P is set as an initial value of Q[1]. An elliptic curve doubling ECDBL of Q[d[i]] is performed, and an arithmetic result is stored in Q[2]. An elliptic curve addition ECADD of Q[0] and Q[1] is performed, and an arithmetic result is stored in Q[1]. Q[2?d[i]] is stored in Q[0]. Q[1+d[i]] is stored in Q[1]. The elliptic curve addition ECADD and the elliptic curve doubling ECDBL are concurrently performed in the respective processors.

Abstract: A circuit element includes a plurality of computation blocks connected at least partially in series for processing multi-bit numbers. Each of the computation blocks includes a plurality of transistors having characteristic threshold voltages. The circuit element is configured so that the transistors will each operate at a voltage below its threshold voltage. The circuit element includes a plurality of circuit sub-elements each having an output. The circuit sub-element outputs are connected together.

Abstract: Provided are an arithmetic method and device of a reconfigurable processor. The arithmetic device includes: an Arithmetic Logic Unit (ALU) for performing an addition and subtraction operation and a logic operation of a binary signal; a multiplier for performing a multiplication operation of the binary signal; a shifter for changing an arrangement of the binary signal; a first operand selector and a second operand selector each for selecting one of values output from the ALU, the multiplier, and the shifter; and an adder for adding the values selected by the first operand selector and the second operand selector.

Abstract: Disclosed herein is a floating-point number arithmetic circuit for efficiently supplying data to be performed arithmetic operation. The floating-point number arithmetic circuit includes an floating-point number arithmetic unit for performing a predetermined floating-point number arithmetic operation on a floating-point number of a predetermined precision, and a converting circuit for converting data into the floating-point number of predetermined precision and supplying the floating-point number of the predetermined precision to at least either one of input terminals of the floating-point number arithmetic unit.

Abstract: A logic circuit computes various modal interval (MI) arithmetic values using a plurality of arithmetic function units (AFUs), each dedicated to compute a specific MI arithmetic operation. The AFUs receive first and second MI operand values each encoded in first and second operand signals. Each AFU provides a MI result value encoded in a result signal to a multiplexer. The multiplexer receives a selector signal specifying the MI arithmetic operation desired, and provides to a result register, an output signal encoding the MI result value specified by the selector signal. The result register stores the MI result value.

Abstract: A content delivery system, enabling a ciphertext to be reduced in size when using the ElGamal cipher, includes a content delivery device performing elliptic curve encryption on a content key, generating an encrypted content key that includes an x coordinate of an elliptic curve point obtained by the elliptic curve encryption, and outputting the encrypted content key. Further, the content delivery system includes content reception device receiving the encrypted content key, calculating a y coordinate of the elliptic curve point using the x coordinate included in the encrypted content key, and performing elliptic curve decryption using the elliptic curve point and other information included in the encrypted content key, to generate a decrypted content key.

Abstract: A cell for an arithmetic logic unit includes a first input; a second input; a carry-in input; a first control input and a second control input; and a circuit connected to the first input, the second input, the carry-in input, the first control input, and the second control input. The circuit has a first output and a second output, the second output having a first value as a function of the first input and the second input when the first control input and the second control input are supplied values equal to a value at the carry-in input, and having a second value as a function of the first input and second input when the values at the first control input and the second control input are independent of the value at the carry-in input.

Abstract: A division and square root arithmetic unit carries out a division operation of a higher radix and a square root extraction operation of a lower radix. A certain bit number (determined on the basis of a radix of an operation) of data selected from upper bits of the output of a carry save adder and the output of the adder are input to convert the data into twos complement representation data, and the twos complement representation data is shifted a certain bit number (determined on the basis of the radix of the operation) to use the shifted data for a partial remainder of the next digit. Hence, a large number of parts such as registers of a divisor and a partially extracted square root can be commonly used in a divider and a square root extractor to realize an effective and high performance arithmetic unit.

Abstract: A plurality of general-purpose registers each has a first bit width. A computing unit has a first and a second input end, at least the first input end having a second bit width wider than the first bit width, and performs an arithmetical operation on data supplied from the general-purpose registers to the first and second input ends. An overflow register having a bit width narrower than the first bit width holds data on figures overflowed as a result of calculation by the computing unit as overflow data and supplies the held overflow data as higher-order bits to at least one input end of the computing unit.

Abstract: A method for transmitting the value of a parameter in a compressed form, the method comprising the steps of: accepting successive numbers representing the value of a parameter; manipulating each number, the manipulation comprising placing the number in a form comprising a mantissa and an exponent, and defining a transmission mantissa to be transmitted; transmitting to a receiver, in turn, the transmission mantissas only of the successive numbers; and receiving the transmission mantissas of the successive numbers at the receiver, characterised by the steps of maintaining a record, at the receiver, of a receiver variable, the receiver variable initially corresponding to the exponent of an initial number; formulating at the receiver, for each received transmission mantissa, a reconstructed number comprising at least the transmission mantissa and an exponent corresponding to the receiver variable; and altering the receiver variable in a first manner if the transmission mantissa of the current number fulfils a fir

Abstract: An extension of the serial/parallel Montgomery modular multiplication method with simultaneous reduction as previously implemented by the applicants, adapted innovatively to perform both in the prime number and in the GF(2q) polynomial based number field, in such a way as to simplify the flow of operands, by performing a multiple anticipatory function to enhance the previous modular multiplication procedures.

Abstract: A digital signal processing circuit including: a multiplier circuit; a plurality of multiplexers coupled to the multiplier circuit and controlled by a first opcode; and an arithmetic logic unit coupled to plurality of multiplexers and controlled by a second opcode.

Abstract: A digital signal processing block having: 1) a first digital signal processing element including: a first multiplexer of a first plurality of multiplexers, the first multiplexer selecting between a first data input and a first zero constant input; and a first arithmetic unit coupled to the first plurality of multiplexers, the first arithmetic logic unit configured for addition; and 2) a second digital signal processing element including: a second multiplexer of a second plurality of multiplexers, the second multiplexer selecting between a second data input and a second zero constant input; and a second arithmetic unit coupled to the second plurality of multiplexers and to a third multiplexer of the first plurality of multiplexers, the second arithmetic unit configured for addition.

Abstract: A method and an apparatus for adjusting a display parameter are provided. The method includes the steps of: (a) calculating a scene change value between a current frame and a previous frame; (b) setting a first weight according to the scene change value; (c) calculating an original parameter of the current frame; (d) providing a display parameter of the previous frame; (e) calculating a display parameter of the current frame according to the first weight, the original parameter, and the display parameter of the previous frame.

Abstract: A processor or other device, such as a programmable and/or massively parallel processor or other device, includes processing elements designed to perform arithmetic operations (possibly but not necessarily including, for example, one or more of addition, multiplication, subtraction, and division) on numerical values of low precision but high dynamic range (“LPHDR arithmetic”). Such a processor or other device may, for example, be implemented on a single chip. Whether or not implemented on a single chip, the number of LPHDR arithmetic elements in the processor or other device in certain embodiments of the present invention significantly exceeds (e.g., by at least 20 more than three times) the number of arithmetic elements, if any, in the processor or other device which are designed to perform high dynamic range arithmetic of traditional precision (such as 32 bit or 64 bit floating point arithmetic).