Kris Gaj

Office hours: Monday, 3:00-4:00 PM, Wednesday, 3:00-4:00 PM,

7:30-8:30 PM and by appointment

Research and teaching interests: •  cryptography •  computer arithmetic •  FPGA design and verification

Contact: Engineering Bldg., room 3225

kgaj@gmu.edu (703) 993-1575

ECE 645

Part of:

MS in Electrical Engineering

MS in Computer Engineering

Digital Systems Design Digital Signal Processing

Fundamental course for the specialization areas:

Elective

Elective course in the remaining specialization areas

PhD in ECE, Computer Science, and IT

Elective

DIGITAL SYSTEMS DESIGN Concentration advisors: Kris Gaj, Ken Hintz, Houman Homayoun 1.  ECE 545 Digital System Design with VHDL

– K. Gaj, project, FPGA design with VHDL, 2. ECE 645 Computer Arithmetic

– K. Gaj, project, FPGA design with VHDL or Verilog, software, or analytical 3. ECE 681 VLSI Design for ASICs

– H. Homayoun, project/lab, front-end and back-end ASIC design with Synopsys tools

4. ECE 586 Digital Integrated Circuits – D. Ioannou, R. Mulpuri,

5a. ECE 682 VLSI Test Concepts – T. Storey 5b. ECE 699 Digital Signals Processing Hardware Architectures

– A. Cohen, project, FPGA design with VHDL and Matlab/Simulink

DIGITAL SIGNAL PROCESSING Concentration advisors: Aaron Cohen, Kris Gaj, Ken Hintz, Jill Nelson,

Kathleen Wage 1.  ECE 535 Digital Signal Processing

– L. Griffiths, J. Nelson, Matlab 2.  ECE 545 Digital System Design with VHDL

– K. Gaj, project, FPGA design with VHDL

3.  ECE 645 Computer Arithmetic – K. Gaj, project, FPGA design with VHDL

4.  ECE 699 Digital Signals Processing Hardware Architectures – A. Cohen, project, FPGA design with VHDL and Matlab/Simulink

5a. ECE 537 Introduction to Digital Image Processing – K. Hintz

5b. ECE 738 Advanced Digital Signal Processing – K. Wage

A few words about You

5 MS CpE students

5 MS EE students

2 PhD ECE students

A few words about You

8 students who took ECE 545

4 students who did not take ECE 545

ECE 545 not enforced as a prerequisite

Useful Knowledge (which can be used as a background for a project)

•  RTL design with VHDL or Verilog

•  FPGA Devices and Tools

•  High level programming language (e.g., C, Java, Matlab)

•  Basics of digital design and computer organization

Course web page

ECE web page → Courses → ECE 645

http://ece.gmu.edu/coursewebpages/ECE/ECE645/S14

Computer Arithmetic

Lecture Project

Project 35%

Homework 15 % Midterm exam (in class) 20 % Final Exam (in class) 30 %

Bonus Points for Class Activity

•  Based on class exercises during lecture

•  “Small” points earned each week posted on BlackBoard

•  Up to 5 “big” bonus points

•  Scaled based on the performance of the best student

For example:

1. Alice 40 5 2.  Bob 36 4.5 … … … 12. Charlie 8 1

Small points Big points

Digital circuit design course covering

•  addition and subtraction •  multiplication •  division and modular reduction •  exponentiation

Efficient

Integers unsigned and signed

Real numbers

•  fixed point •  single and double precision floating point

Elements of the Galois field GF(2n) •  polynomial base

1.  Applications of computer arithmetic algorithms.

INTRODUCTION

Lecture topics

1.  Basic addition, subtraction, and counting

2.  Addition in Xilinx and Altera FPGAs 3. Carry-lookahead, carry-select, and hybrid adders 4. Adders based on Parallel Prefix Networks 5.  Pipelined Adders

6.  Modular Adders/Subtractors

ADDITION AND SUBTRACTION

MULTIOPERAND ADDITION

1.  Sequential multi-operand adders

2.  Carry Save Adders 3.  Wallace and Dadda Trees

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

NUMBER REPRESENTATIONS

MULTIPLICATION

1.  Tree and array multipliers

2.  Unsigned vs. signed multipliers

3.  Optimizations for squaring 4. Sequential multipliers

- radix-2 multiplier - multipliers based on carry-save adders - radix-4 & radix-8 multipliers - Booth multipliers - serial multipliers

TECHNOLOGY

1.  Embedded resources of Xilinx and Altera FPGAs - block memories - multipliers - DSP units

2. Multiplication in Xilinx and Altera FPGAs - using distributed logic - using embedded multipliers - using DSP blocks 3. Pipelined multipliers

DIVISION

1.  Basic restoring and non-restoring sequential dividers

2.  Array dividers

3.  Dividers by Convergence

4.  SRT and high-radix dividers

LONG INTEGER ARITHMETIC (PROJECTS)

1.  Modular Multiplication

2.  Modular Exponentiation

3.  Montgomery Multipliers and Exponentiation Units

FLOATING POINT AND

GALOIS FIELD ARITHMETIC (PROJECTS)

1.  Floating-point units •  Binary formats •  Decimal formats

2. Galois Field GF(2n) units

Literature (1)

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

Literature (2)

Jean-Pierre Deschamps, Gery Jean Antoine Bioul, Gustavo D. Sutter, Synthesis of Arithmetic Circuits: FPGA, ASIC and Embedded Systems, Wiley-Interscience, 2006. Joseph Cavanagh, Computer Arithmetic and Verilog HDL Fundamentals, CRC Press, 2009. Milos D. Ercegovac and Tomas Lang Digital Arithmetic, Morgan Kaufmann Publishers, 2004.

Supplemantory books:

Literature (3)

1.  Pong P. Chu, RTL Hardware Design Using VHDL: Coding for Efficiency, Portability, and Scalability, Wiley-IEEE Press, 2006 (ECE 545 textbook) 2.  Hubert Kaeslin, Digital Integrated Circuit Design: From VLSI Architectures to CMOS Fabrication, Cambridge University Press; 1st Edition, 2008. (ECE 681 textbook) 3. Stephen Brown and Zvonko Vranesic, Fundamentals of Digital Logic with VHDL Design, 3rd Edition, McGraw-Hill, 2008. (ECE 331 textbook)

Digital System Design textbooks:

Literature (4)

Supplementary books:

1.  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.

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

Homework

•  reading assignments

•  analysis of computer arithmetic algorithms and implementations

•  design of small arithmetic units

Getting Help Outside of Office Hours

•  System for asking questions 24/7

•  Answers can be given by students and instructors

•  Student answers endorsed (or corrected) by instructors

•  Average response time in ECE 545 = 2.1 hour

•  You can submit your questions anonymously

•  You can ask private questions visible only to

the instructors

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

Midterm Exam - March 26 Final Exam - Wednesday, May 7, 4:30-7:15 PM

Tentative days of exams:

Project

•  Can be done individually or in groups of two students

•  Suggested project topics posted early in the semester

•  You can propose your own project topic •  Regular meetings with the instructor •  Presentations at the end of the semester

•  Contest for the best project

ECE 645 Project

Hardware Analytical Software

Hardware Projects

•  Real-life circuit requiring the use of arithmetic operations •  FPGA implementation using embedded resources, such as DSP units and Block Memories •  Options of FPGA tools, initial placement point, and optimum target clock frequency selected using ATHENa •  Possible experimental testing using PLDA boards with PCI Express interface based on Virtex 6, Virtex 7, and Stratix V FPGAs (optional)

Software Projects

•  Real-life application requiring the use of arithmetic operations •  Software implementation in a high-level programming language of your choice •  Possible use of arithmetic libraries, such as GMP, MIRACL, RELIC, NTL

•  Optimization of software tool options

Analytical Projects

•  Review of literature concerning algorithms and hardware architectures for a specific class of arithmetic operations •  Qualitative comparison of competing designs

•  Quantitative comparison based on published results

Mixed Projects

HW 80%

AN 20% HW

40% SW 40%

AN 20%

AN 60%

SW 40%

Primary applications (1)

Execution units of general purpose microprocessors

Integer units Floating point units

Integers (8, 16, 32, 64, 128 bits)

Real numbers (32, 64, 128 bits)

Primary applications (2)

Digital signal and digital image processing

Real or complex numbers (fixed-point or floating point)

e.g., digital filters Discrete Fourier Transform Discrete Hilbert Transform Edge detection

General purpose DSP processors Specialized circuits

Primary applications (3)

Coding

Elements of the Galois fields GF(2n) (4-64 bits)

Error detection codes Error correcting codes

Secret-key (Symmetric) Cryptosystems

key of Alice and Bob - KAB key of Alice and Bob - KAB

Alice Bob

Network

Encryption Decryption

Hash Function

arbitrary length

message

hash function

hash value h(m)

h

m

fixed length

It is computationally infeasible to find such

m and m’ that h(m)=h(m’)

Primary applications (4)

Cryptography

Integers (16, 32, 64 bits)

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)

RC6

MARS

Twofish

MUL32, 2 x ROL32, S-box 9x32

Main operations

Auxiliary operations

XOR, ADD/SUB32

2 x SQR32, 2 x ROL32

XOR, ADD/SUB32

96 S-box 4x4, 24 MUL GF(28)

XOR ADD32

Rijndael

Serpent 8 x 32 S-box 4x4

XOR

16 S-box 8x8 24 MUL GF(28)

XOR

42

Basic Operations of 14 SHA-3 Candidates

42 NTT – Number Theoretic Transform, GF MUL – Galois Field multiplication,

MUL – integer multiplication, mADDn – multioperand addition with n operands

Public Key (Asymmetric) Cryptosystems

Public key of Bob - KB Private key of Bob - kB

Alice Bob

Network

Encryption Decryption

RSA as a trap-door one-way function

M C = f(M) = Me mod N C

M = f-1(C) = Cd mod N

PUBLIC KEY

PRIVATE KEY

N = P ⋅ Q P, Q - large prime numbers

e ⋅ d ≡ 1 mod ((P-1)(Q-1))

RSA keys

PUBLIC KEY PRIVATE KEY

{ e, N } { d, P, Q }

N = P ⋅ Q

e ⋅ d ≡ 1 mod ((P-1)(Q-1))

P, Q - large prime numbers

Primary applications (5)

Cryptography

Long integers (1k-16k bits)

Public key cryptography

RSA, DSA, Diffie-Hellman

Elliptic Curve Cryptosystems, Pairing Based Cryptosystems

Elements of the Galois field GF(2n) (160-512 bits)

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))