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Kris Gaj

Learn computer arithmetic algorithms, FPGA design, VHDL coding, and more. Covers topics like addition, multiplication, division, floating-point arithmetic, and Galois field arithmetic. Prerequisite knowledge required.

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Kris Gaj

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  1. Kris Gaj • Research and teaching interests: • cryptography • computer arithmetic • FPGA design and verification • Contact: • Engineering Bldg., room 3225 • kgaj@gmu.edu • (703) 993-1575 Office hours:Monday, 6:00-7:00 PM, Tuesday 7:30-8:30 PM, Thursday, 4:30-5:30 PM, and by appointment

  2. ECE 645 Part of: MS in CpE Digital Systems Design– pre-approved course Other concentration areas – elective course MS in EE Certificate in VLSI Design/Manufacturing PhD in ECE PhD in IT

  3. DIGITAL SYSTEMS DESIGN 1. ECE 545 Digital System Design with VHDL– K. Gaj, project, FPGA design with VHDL, Xilinx & Altera FPGAs 2. ECE 645 Computer Arithmetic– K. Gaj, project, FPGA design with VHDL, Xilinx & Altera FPGAs 3. ECE 586 Digital Integrated Circuits – D. Ioannou, homework/small projects 4. ECE 681 VLSI Design for ASICs– TK Ramesh, project/lab, front-end and back-end ASIC design with Synopsys tools 5. ECE 682 VLSI Test Concepts– T. Storey, homework

  4. Prerequisites ECE 545 Digital System Design with VHDL or Permission of the instructor, granted assuming that you know High level programming language (preferably C) RTL design with VHDL

  5. Prerequisite knowledge This class assumes proficiency with FPGA CAD tools from ECE 545 You are expected to be proficient with: Synthesizable VHDL coding Advanced VHDL testbenches, including file input/output FPGA synthesis and post-synthesis simulation FPGA implementation and timing simulation Reading and interpreting all synthesis and implementation reports

  6. Course web page ECE web page  Courses  Course web pages  ECE 645 http://ece.gmu.edu/coursewebpages/ECE/ECE645/S12/

  7. Computer Arithmetic Lecture Project Homework 10 % Midterm exam (in class) 15 % Final Exam (in class) 25 % Project 1 20 % Project 2 30 %

  8. Advanced digital circuit design course covering Efficient • addition and subtraction • multiplication • division and modular reduction • exponentiation • Elements • of the Galois • field GF(2n) • polynomial base Integers unsigned and signed Real numbers • fixed point • single and double precision • floating point

  9. Lecture topics INTRODUCTION • Applications of computer arithmetic algorithms.

  10. ADDITION AND SUBTRACTION • Basic addition, subtraction, and counting • Addition in Xilinx and AlteraFPGAs • 3. Carry-lookahead, carry-select, and hybrid adders • 4. Adders based on Parallel Prefix Networks • 5. Pipelined Adders

  11. MULTIOPERAND ADDITION • Sequential multi-operand adders • Carry Save Adders • Wallace and Dadda Trees

  12. NUMBER REPRESENTATIONS • Unsigned Integers • Signed Integers • Fixed-point real numbers • Floating-point real numbers • Elements of the Galois Field GF(2n)

  13. LONG INTEGER ARITHMETIC • Modular Multiplication • Modular Exponentiation • Montgomery Multipliers and Exponentiation Units

  14. MULTIPLICATION • Tree and array multipliers • Unsigned vs. signed multipliers • Optimizations for squaring • 4. Sequential multipliers • - radix-2 multiplier • - multipliers based on carry-save adders • - radix-4 & radix-8 multipliers • - Booth multipliers • - serial multipliers

  15. TECHNOLOGY • Embedded resources of Xilinx and AlteraFPGAs • - block memories • - multipliers • - DSP units • 2. Multiplication in Xilinx and AlteraFPGAs • - using distributed logic • - using embedded multipliers • - using DSP blocks • 3. Pipelined multipliers

  16. DIVISION • Basic restoring and non-restoring • sequential dividers • 2. SRTand high-radix dividers • 3. Array dividers

  17. FLOATING POINT AND GALOIS FIELD ARITHMETIC • Floating-point units • 2. Galois Field GF(2n) units

  18. Literature (1) Required textbook: Behrooz Parhami, Computer Arithmetic: Algorithms and Hardware Design, 2nd edition, Oxford University Press, 2010.

  19. Literature (2) Recommended textbooks: Jean-Pierre Deschamps, Gery Jean Antoine Bioul, Gustavo D. Sutter, Synthesis of Arithmetic Circuits: FPGA, ASIC and Embedded Systems, Wiley-Interscience, 2006. Milos D. Ercegovac and Tomas Lang Digital Arithmetic, Morgan Kaufmann Publishers, 2004. Isreal Koren, Computer Arithmetic Algorithms, 2nd edition, A. K. Peters, Natick, MA, 2002.

  20. Literature (2) VHDL books: • Pong P. Chu, RTL Hardware Design Using VHDL: • Coding for Efficiency, Portability, and Scalability, • Wiley-IEEE Press, 2006. • Volnei A. Pedroni, Circuit Design and Simulation • with VHDL, 2nd edition, The MIT Press, 2010. • 3. Sundar Rajan, Essential VHDL: RTL Synthesis Done Right, • S & G Publishing, 1998.

  21. Literature (3) Supplementary books: • E. E. Swartzlander, Jr., Computer Arithmetic, • vols. I and II, IEEE Computer Society Press, 1990. • 2. Alfred J. Menezes, Paul C. van Oorschot, • and Scott A. Vanstone, • Handbook of Applied Cryptology, • Chapter 14, Efficient Implementation, • CRC Press, Inc.,1998.

  22. Literature (3) Proceedings of conferences ARITH - International Symposium on Computer Arithmetic ASIL - Asilomar Conference on Signals, Systems, and Computers ICCD - International Conference on Computer Design CHES - Workshop on Cryptographic Hardware and Embedded Systems Journals and periodicals IEEE Transactions on Computers, in particular special issues on computer arithmetic. IEEE Transactions on Circuits and Systems IEEE Transactions on Very Large Scale Integration IEE Proceedings: Computer and Digital Techniques Journal of Signal Processing Systems

  23. Homework • reading assignments • analysis of computer arithmetic algorithms • and implementations • design of small hardware units using VHDL

  24. Midterm exams Midterm Exam - 2 hrs 30 minutes, in class multiple choice + short problems Final Exam – 2 hrs 45 minutes comprehensive conceptual questions analysis and design of arithmetic units Practice exams on the web Tentative days of exams: Midterm Exam - Monday, March 26 Final Exam - Monday, May 14, 7:30-10:15 PM

  25. Project 1 Project I (individual, 20% of grade) Adders in Xilinx and Altera FPGAs • Choosing optimal architecture for • combinational adder • pipelined adder • in • Xilinx FPGAs (Virtex 5 & Virtex 6) • Altera FPGAs (Stratix III & Stratix IV) • ASICs (bonus) Done individually Final report & deliverables due Monday, March 19

  26. Project 2 Project II (in groups of two or individually, 30% of grade) Modular Exponentiation of Large Integers or Floating Point Operations • Investigation of alternative architectures for • the best performance in terms of • Latency • Latency x Area product • in • Xilinx FPGAs (Virtex 5 & Virtex 6) • Altera FPGAs (Stratix III & Stratix IV) • ASICs (bonus) Final report & deliverables due Monday, May 7

  27. Primary applications (1) Execution units of general purpose microprocessors Integer units Floating point units Integers (8, 16, 32, 64 bits) Real numbers (32, 64 bits)

  28. Primary applications (2) Digital signal and digital image processing e.g., digital filters Discrete Fourier Transform Discrete Hilbert Transform General purpose DSP processors Specialized circuits Real or complex numbers (fixed-point or floating point)

  29. Primary applications (3) Coding Error detection codes Error correcting codes Elements of the Galois fields GF(2n) (4-64 bits)

  30. Secret-key (Symmetric) Cryptosystems key of Alice and Bob - KAB key of Alice and Bob - KAB Network Decryption Encryption Bob Alice

  31. Hash Function arbitrary length m message hash function h It is computationally infeasible to find such m and m’ that h(m)=h(m’) h(m) hash value fixed length

  32. Primary applications (4) Cryptography IDEA, RC6, Mars, SHA-3 candidates: SIMD, Shabal, Skein, BLAKE Twofish, Rijndael, SHA-3 candidates Elements of the Galois field GF(2n) (4, 8 bits) Integers (16, 32, 64 bits)

  33. Main operations Auxiliary operations 2 x SQR32, 2 x ROL32 XOR, ADD/SUB32 RC6 MARS XOR, ADD/SUB32 MUL32, 2 x ROL32, S-box 9x32 XOR ADD32 Twofish 96 S-box 4x4, 24 MUL GF(28) Rijndael 16 S-box 8x8 24 MUL GF(28) XOR 8 x 32 S-box 4x4 Serpent XOR

  34. Basic Operations of 14 SHA-3 Candidates NTT – Number Theoretic Transform, GF MUL – Galois Field multiplication, MUL – integer multiplication, mADDn – multioperand addition with n operands

  35. Public Key (Asymmetric) Cryptosystems Private key of Bob - kB Public key of Bob - KB Network Decryption Encryption Bob Alice

  36. RSA as a trap-door one-way function PUBLIC KEY C = f(M) = Me mod N M C M = f-1(C) = Cd mod N PRIVATE KEY N = P  Q P, Q - large prime numbers e  d  1 mod ((P-1)(Q-1))

  37. RSA keys PUBLIC KEY PRIVATE KEY { e, N } { d, P, Q } N = P  Q P, Q - large prime numbers e  d  1 mod ((P-1)(Q-1))

  38. Primary applications (5) Cryptography Public key cryptography Elliptic Curve Cryptosystems, Pairing Based Cryptosystems RSA, DSA, Diffie-Hellman Long integers (1k-16k bits) Elements of the Galois field GF(2n) (160-512 bits)

  39. Primary applications (5) Cipher Breaking Public key cryptography RSA PUBLIC KEY RSA PRIVATE KEY { e, N } { d, P, Q } N = P  Q P, Q e  d  1 mod ((P-1)(Q-1))

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