Efficient long division via Montgomery multiply

We present a novel right-to-left long division algorithm based on the Montgomery modular multiply, consisting of separate highly efficient loops with simply carry structure for computing first the remainder (x mod q) and then the quotient floor(x/q). These loops are ideally suited for the case where x occupies many more machine words than the divide modulus q, and are strictly linear time in the "bitsize ratio" lg(x)/lg(q). For the paradigmatic performance test of multiword dividend and single 64-bit-word divisor, exploitation of the inherent data-parallelism of the algorithm effectively mitigates the long latency of hardware integer MUL operations, as a result of which we are able to achieve respective costs for remainder-only and full-DIV (remainder and quotient) of 6 and 12.5 cycles per dividend word on the Intel Core 2 implementation of the x86_64 architecture, in single-threaded execution mode. We further describe a simple "bit-doubling modular inversion" scheme, which allows the entire iterative computation of the mod-inverse required by the Montgomery multiply at arbitrarily large precision to be performed with cost less than that of a single Newtonian iteration performed at the full precision of the final result. We also show how the Montgomery-multiply-based powering can be efficiently used in Mersenne and Fermat-number trial factorization via direct computation of a modular inverse power of 2, without any need for explicit radix-mod scalings.Comment: 23 pages; 8 tables v2: Tweak formatting, pagecount -= 2. v3: Fix incorrect powers of R in formulae [7] and [11] v4: Add Eldridge & Walter ref. v5: Clarify relation between Algos A/A',D and Hensel-div; clarify true-quotient mechanics; Add Haswell timings, refs to Agner Fog timings pdf and GMP asm-timings ref-page. v6: Remove stray +bw in MULL line of Algo D listing; add note re byte-LUT for qinv_

A high-performance inner-product processor for real and complex numbers.

A novel, high-performance fixed-point inner-product processor based on a redundant binary number system is investigated in this dissertation. This scheme decreases the number of partial products to 50%, while achieving better speed and area performance, as well as providing pipeline extension opportunities. When modified Booth coding is used, partial products are reduced by almost 75%, thereby significantly reducing the multiplier addition depth. The design is applicable for digital signal and image processing applications that require real and/or complex numbers inner-product arithmetic, such as digital filters, correlation and convolution. This design is well suited for VLSI implementation and can also be embedded as an inner-product core inside a general purpose or DSP FPGA-based processor. Dynamic control of the computing structure permits different computations, such as a variety of inner-product real and complex number computations, parallel multiplication for real and complex numbers, and real and complex number division. The same structure can also be controlled to accept redundant binary number inputs for multiplication and inner-product computations. An improved 2's-complement to redundant binary converter is also presented

A Solder-Defined Computer Architecture for Backdoor and Malware Resistance

This research is about securing control of those devices we most depend on for integrity and confidentiality. An emerging concern is that complex integrated circuits may be subject to exploitable defects or backdoors, and measures for inspection and audit of these chips are neither supported nor scalable. One approach for providing a “supply chain firewall” may be to forgo such components, and instead to build central processing units (CPUs) and other complex logic from simple, generic parts. This work investigates the capability and speed ceiling when open-source hardware methodologies are fused with maker-scale assembly tools and visible-scale final inspection. The author has designed, and demonstrated in simulation, a 36-bit CPU and protected memory subsystem that use only synchronous static random access memory (SRAM) and trivial glue logic integrated circuits as components. The design presently lacks preemptive multitasking, ability to load firmware into the SRAMs used as logic elements, and input/output. Strategies are presented for adding these missing subsystems, again using only SRAM and trivial glue logic. A load-store architecture is employed with four clock cycles per instruction. Simulations indicate that a clock speed of at least 64 MHz is probable, corresponding to 16 million instructions per second (16 MIPS), despite the architecture containing no microprocessors, field programmable gate arrays, programmable logic devices, application specific integrated circuits, or other purchased complex logic. The lower speed, larger size, higher power consumption, and higher cost of an “SRAM minicomputer,” compared to traditional microcontrollers, may be offset by the fully open architecture—hardware and firmware—along with more rigorous user control, reliability, transparency, and auditability of the system. SRAM logic is also particularly well suited for building arithmetic logic units, and can implement complex operations such as population count, a hash function for associative arrays, or a pseudorandom number generator with good statistical properties in as few as eight clock cycles per 36-bit word processed. 36-bit unsigned multiplication can be implemented in software in 47 instructions or fewer (188 clock cycles). A general theory is developed for fast SRAM parallel multipliers should they be needed

Implementation of a MSP430-based digital thermometer using the slope ADC of the timer port module

This report describes the slope A/D measurement of a resistance and the ease with which it can be applied to MSP430 microcontrollers. It describes a digital thermometer design that uses the slope ADC capabilities of the Timer Port module on the MSP430x3xx microcontrollers. It is used more generally as a reference on how to connect resistive sensors and reference resistors to the Timer Port module. All MSP430x3xx devices include the Timer Port module. The module allows several resistive sensors and reference resistors to be connected in an application. Unused module pins can be used as independent outputs. Slope A/D conversion is an analog-to-digital conversion technique that can be implemented with a comparator rather than a standalone ADC module or device. The technique is based on the charging/discharging of a capacitor with a known value. The number of clock cycles necessary to discharge the capacitor is then counted. Longer discharge times indicate larger voltages. The voltage is derived from the discharge time using the standard equation for capacitor discharge. In addition to digitizing voltages, a variation of the technique can be used to measure resistance. This is valuable in measuring any component that can have varying resistance, such as potentiometers and various types of transducers. Unlike voltage measurement, where the key relationship is between voltage and time while the resistance is constant, the key relationship in resistance measurement is between resistance and time, while the initial voltage remains constant. The R-relationship is linear, which means the calculation is easier and less- costly to implement in a microcontroller than for the exponential V-t relationship. The thermometer has been simulated by using a variable resistance instead of a thyristor. In addition care has been taken to optimize the power consumption by forcing the microcontroller to several low-power modes during the operation. The combination of the Timer Port module, the 16-bit CPU, and the ultra low power design provide unmatched MIPS per watt performance. The set up can be extended to provide a low power thermostat

A CAD Tool for Synthesizing Optimized Variants of Altera\u27s Nios II Soft-Core Processor

Soft-core processors offer embedded system designers the benefits of customization, flexibility and reusability. Altera\u27s NIOS II soft-core processor is a popular, commercially available soft-core processor that can be implemented on a variety of Altera FPGAs. In this thesis, the Nios II soft-core processor from Altera Corporation was studied and a VHDL implementation, called UW_Nios II, was developed. UW_Nios II was developed to enable us to perform design space exploration (DSE) for the Nios II processor. It was evaluated and compared with Altera Nios II and shown to be competitive. SCBuild is an existing CAD tool that was developed to enable DSE of soft-core processors. We modified SCBuild to automatically explore the design space of the UW_Nios II using a genetic algorithm. This tool can accurately estimate the area and critical path delay of different variants of the UW_Nios II on a field programmable gate array. Through experiments conducted using SCBuild, it was shown that employing a genetic algorithm to explore the design space of parameterized Nios II core, with a large design space, helps designers find optimized variants of UW_Nios II