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1Assistant Professor at GMU since Fall 1998Kris GajOffice hours: Monday, after classTuesday, 1:00-4:00 PM Wednesday, 5:00-7:00 PMResearch and teaching interests:• cryptography• computer arithmetic• VLSI design and testingContact:Science & Technology II, room [email protected], (703) 993-1575Computer ArithmeticECE 699:001 Spring 2000Advanced Topics in Electrical and Computer Engineering: Computer ArithmeticSince Fall 2000ECE 645 Computer Arithmetic: Implementations in Hardware and SoftwareECE 645Part of:MS in EEComputer EngineeringMS in CpEDigital Systems DesignNetwork and System SecurityCertificate in VLSI Design/ManufacturingPhD in ITPhD in ECE2Digital Systems DesignECE 545 ECE 586ECE 548 ECE 584ECE 645 ECE 680 ECE 681 ECE 682Network and System SecurityECE 542CS 571ECE 646ECE 545ECE 746ECE 642ECE 742ECE 645INFS 762INFS 766 INFS 767ECE 548ECE 545ECE 645ECE 511ECE 611ECE 612ECE 641ECE 546ECE 640Computer Engineering I3ECE 586ECE 681ECE 682ECE 545 ECE 537ECE 644ECE 744ECE 542ECE 646ECE 746Computer Engineering IIECE 511algorithmicDesign levelregister-transfergatetransistorlayoutdevicesCoursesComputerArithmeticIntroductionto VHDLDigitalIntegratedCircuitsPhysicalVLSI DesignVLSI Test ConceptsVLSI Design AutomationECE545ECE645ECE586ECE 680ECE681ECE682ECE684MOS Device ElectronicsContains basic information about the courseTo be extended in the futureCourse web pageECE web page →→→→ Courses →→→→ Course web pages→→→→ ECE 645http://teal.gmu.edu/courses/ECE645/index.htm4Computer ArithmeticLecture ProjectProject 115 %Project 230 %Homework20 %Midterm exam 1 (in class)20 %Midterm exam 2 (take-home)15 %Advanced digital circuit design course covering• addition and subtraction• multiplication• division and modular reduction• exponentiationEfficientIntegersunsigned and signedReal numbers• fixed point• single and double precisionfloating pointElementsof the Galoisfield GF(2n)• polynomial baseLecture topics (1)1. Applications of computer arithmetic algorithms2. Number representation• Unsigned Numbers• Signed NumbersINTRODUCTION51. Basic addition, subtraction, and counting2. Carry-lookahead adders3. Adders based on Parallel Prefix NetworksADDITION AND SUBTRACTIONMULTIOPERAND ADDITION1. Carry-save adders2. Wallece and Dadda Trees3. Adding multiple signed numbersMULTIPLICATION1. Basic hardware multipliers2. High-radix multipliers3. Tree multipliers4. Array multipliers5. Multiplication of signed numbers and squaringIn hardwareIn software6. Survey of software multiplication algorithms6DIVISIONIn hardwareIn software1. Basic hardware dividers2. High-radix dividers3. Array dividers4. Survey of algorithms for divisionmodular reduction, and modular exponentiationFLOATING POINT ARITHMETIC1. Floating-point number representations2. Floating-point operationsGALOIS FIELD ARITHMETIC1. Representations of elements of the Galois Field2. Galois Field operations• University of California, Santa Barbara, Behrooz Parhami, ECE252B: Computer Arithmetic. • University of Massachusetts, Amherst, Israel Koren,ECE666: Digital Computer Arithmetic • Lehigh University, Michael Schulte, ECE496: High-Speed Computer Arithmetic. • Worcester Polytechnic Institute, Berk Sunar, EE-579 V Computer Arithmetic Circuits.• Stanford University, Michael Flynn, EE486: Advanced Computer Arithmetic. • University of California, Davies, Vojin Oklobdzija, ECE278: Computer Arithmetic for Digital Implementation.Similar courses at other universities7New in this course• hardware vs. software algorithms• real life project based on VHDL or Verilog HDL• operations in the Galois Field (communications)Possible follow-up courseComputer ArithmeticECE 645 (Spring 2004)Advanced Computer ArithmeticSquare rootExponential and logarithm functionsTrigonometric functionsHyperbolic functionsFault tolerant arithmeticLow power arithmeticLiterature (1)Required textbooks:Behrooz Parhami, Computer Arithmetic: Algorithms and Hardware Design, Oxford University Press, 1999.Sundar Rajan, Essential VHDL: RTL Synthesis Done Right,S & G Publishing, 1998.Isreal Koren, Computer Arithmetic Algorithms, 2nd Edition,A. K. Peters, Natick, MA, 2002.Recommended textbooks:8Literature (2)Supplementary books:1. Milos D. Ercegovac and Tomas Lang, Digital Arithmetic, Morgan Kaufmann Publishers, 2004.2. E. E. Swartzlander, Jr., Computer Arithmetic, vols. I and II, IEEE Computer Society Press, 1990. 3. Amos Omondi, Computer Arithmetic Systems: Algorithms, Architecture and Implementation, Prentice Hall, 1994. 4. Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone, Handbook of Applied Cryptology, Chapter 14, Efficient Implementation, Zipped pdf file, 630k, CRC Press, Inc.,Boca Raton, 1998. 5. Christof Paar, Efficient VLSI Architectures for Bit Parallel Computation in Galois Fields, VDI Verlag, 1994. 6. Milos Ercegovac and Tomas Lang, Division and Square Root, Kluwer Academic Publishers, 1994.Literature (3)Proceedings of conferencesARITH - International Symposium on Computer ArithmeticASIL - Asilomar Conference on Signals, Systems, and Computers ICCD - International Conference on Computer DesignCHES - Workshop on Cryptographic Hardware and Embedded SystemsJournals and periodicalsIEEE Transactions on Computers,in particular special issues on computer arithmetic: 8/70, 6/73, 7/77, 4/83, 8/90, 8/92, 8/94. IEEE Transactions on Circuits and Systems IEEE Transactions on Very Large Scale Integration IEE Proceedings: Computer and Digital Techniques Journal of VLSI Signal Processing Homework• reading assignments (main textbook + articles)• analysis of hardware and software algorithmsand implementations• design of small hardware units using VHDLOptional assignmentsPossibility of tradingsoftware vs. hardwaretheory vs. practiceanalysis vs. design9Midterm examsExam 1 - 2 hrs 30 minutes, in classmultiple choice + short problemsExam 2 – 48 hrs, take-homeanalysis and design of arithmetic unitsPractice exams on the webExam 1 - March 22 or March 29Exam 2 - May 1 & 2Tentative days of exams:Project (1)Project I (15% of grade)Design and comparative analysis of fast adders(several hundred bits long)Due date for written reportMonday, March 15Optimization criteria:• minimum latency• maximum throughput• minimum area• minimum product latency · area• maximum ratio throughput/area• scalabilitySimilar for all students Done individuallyProject II (30% of grade)Fast • multiplication• squaring• division• modular


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