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, 3:00-4:00 PM,
Wednesday, 3:00-4:00 PM,
7:30-8:30 PM
and by appointment
ECE 645
Part of:
MS in Computer Engineering
Fundamental course for the specialization areas:
Digital Systems Design
Digital Signal Processing
Elective course in the remaining specialization areas
MS in Electrical Engineering
Elective
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
2 PhD ECE
students
5 MS CpE
students
5 MS EE
students
A few words about You
4 students
who did
not take
ECE 545
8 students
who took
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
Homework
15 %
Midterm exam (in class)
20 %
Final Exam (in class)
30 %
Project
Project
35%
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
2.Bob
…
12. Charlie
Small points
40
36
…
8
Big points
5
4.5
…
1
Digital circuit design course covering
Efficient
• addition and subtraction
• multiplication
• division and modular reduction
• exponentiation
Integers
unsigned and signed
Real numbers
Elements
of the Galois
field GF(2n)
• fixed point
• single and double precision
floating point
• polynomial base
Lecture topics
INTRODUCTION
1. Applications of computer arithmetic algorithms.
ADDITION AND SUBTRACTION
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
MULTIOPERAND ADDITION
1. Sequential multi-operand adders
2. Carry Save Adders
3. Wallace and Dadda Trees
NUMBER REPRESENTATIONS
• Unsigned Integers
• Signed Integers
• Fixed-point real numbers
• Floating-point real numbers
• Elements of the Galois Field GF(2n)
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)
Supplemantory books:
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.
Literature (3)
Digital System Design textbooks:
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)
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
Tentative days of exams:
Midterm Exam - March 26
Final Exam
- Wednesday, May 7, 4:30-7:15 PM
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
Software
Analytical
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
AN
20%
HW
80%
AN
20%
SW
40%
HW
40%
AN
60%
SW
40%
Primary applications (1)
Execution units of general purpose microprocessors
Integer units
Integers
(8, 16, 32, 64, 128 bits)
Floating point units
Real numbers
(32, 64, 128 bits)
Primary applications (2)
Digital signal and digital image processing
e.g., digital filters
Discrete Fourier Transform
Discrete Hilbert Transform
Edge detection
General purpose
DSP processors
Specialized circuits
Real or complex numbers
(fixed-point or floating point)
Primary applications (3)
Coding
Error detection codes
Error correcting codes
Elements of
the Galois fields GF(2n)
(4-64 bits)
Secret-key (Symmetric) Cryptosystems
key of Alice and Bob - KAB
key of Alice and Bob - KAB
Network
Encryption
Alice
Decryption
Bob
Hash Function
arbitrary length
m
message
h
It is computationally
infeasible to find such
m and m’ that
h(m)=h(m’)
h(m)
fixed length
hash
function
hash value
Primary applications (4)
Cryptography
IDEA, RC6, Mars,
SHA-3 candidates:
SIMD, Shabal,
Skein, BLAKE
Integers
(16, 32, 64 bits)
Twofish, Rijndael,
SHA-3 candidates
Elements of
the Galois field GF(2n)
(4, 8 bits)
Main
operations
RC6
2 x SQR32,
2 x ROL32
MARS
MUL32,
2 x ROL32,
S-box 9x32
Twofish
96 S-box 4x4,
24 MUL GF(28)
Auxiliary
operations
XOR,
ADD/SUB32
XOR,
ADD/SUB32
XOR
ADD32
Rijndael
16 S-box 8x8
24 MUL GF(28)
XOR
Serpent
8 x 32
S-box 4x4
XOR
Basic Operations of 14 SHA-3 Candidates
NTT – Number Theoretic Transform, GF MUL – Galois Field multiplication,
42
MUL – integer multiplication, mADDn – multioperand addition with n operands
42
Public Key (Asymmetric) Cryptosystems
Private key of Bob - kB
Public key of Bob - KB
Network
Encryption
Alice
Decryption
Bob
RSA as a trap-door one-way function
PUBLIC KEY
M
C = f(M) = Me mod N
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))
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))
Primary applications (5)
Cryptography
Public key cryptography
RSA, DSA,
Diffie-Hellman
Long integers
(1k-16k bits)
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))