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

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

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

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

5. Prerequisite knowledge This class assumes proficiency with the FPGA CAD tools from ECE 545 You are expected to be proficient with: –Synthesizable VHDL coding –Advanced VHDL testbenches, including file input/output –Xilinx FPGA synthesis and post-synthesis simulation –Xilinx FPGA place-and-route and post-place and route 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/S10/

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

8. Advanced 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(2 n ) polynomial base

9. At the end of this course you should be able to: Understand mathematical and gate-level algorithms for computer addition, subtraction, multiplication, division, and exponentiation Understand tradeoffs involved with different arithmetic architectures between performance, area, latency, scalability, etc. Synthesize and implement computer arithmetic blocks on FPGAs Be comfortable with different number systems, and have familiarity with floating-point and Galois field arithmetic for future study Understand sources of error in computer arithmetic and basics of error analysis This knowledge will come about through homework, projects and practice exams. Course Objectives

10. Lecture topics (1) 1. Applications of computer arithmetic algorithms 2. Number representation Unsigned Integers Signed Integers Fixed-point real numbers Floating-point real numbers Elements of the Galois Field GF(2 n ) INTRODUCTION

11. 1. Basic addition, subtraction, and counting 2. Carry-lookahead, carry-select, and hybrid adders 3. Adders based on Parallel Prefix Networks ADDITION AND SUBTRACTION

12. MULTIOPERAND ADDITION 1. Carry-save adders 2. Wallace and Dadda Trees 3. Adding multiple unsigned and signed numbers

13. TECHNOLOGY 1. Internal Structure of Xilinx and Altera FPGAs 2. ASIC standard cell libraries and synthesis tools for ASICs 3. Two-operand and multi-operand addition in FPGAs

14. MULTIPLICATION 1. Tree and array multipliers 2. Sequential multipliers 3. Multiplication of signed numbers and squaring

15. TECHNOLOGY 1. Pipelining 2. Multi-cycle paths 3. Multiplication in Xilinx and Altera FPGAs - using distributed logic - using embedded multipliers - using DSP blocks

16. LONG INTEGER ARITHMETIC 1.Modular Exponentiation 2.Montgomery Multipliers and Exponentiation Units

17. DIVISION 1.Basic restoring and non-restoring sequential dividers 2. SRT and high-radix dividers 3. Array dividers

18. FLOATING POINT AND GALOIS FIELD ARITHMETIC 1.Floating-point units 2. Galois Field GF(2 n ) units

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

20. Literature (2) 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. Recommended textbooks:

21. Literature (2) 1.Pong P. Chu, RTL Hardware Design Using VHDL: Coding for Efficiency, Portability, and Scalability, Wiley-IEEE Press, 2006. 2. Volnei A. Pedroni, Circuit Design with VHDL, The MIT Press, 2004. 3. Sundar Rajan, Essential VHDL: RTL Synthesis Done Right, S & G Publishing, 1998. VHDL books:

22. Literature (3) 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.

23. 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: 8/70, 6/73, 7/77, 4/83, 8/90, 8/92, 8/94, 7/00, 3/05. IEEE Transactions on Circuits and Systems IEEE Transactions on Very Large Scale Integration IEE Proceedings: Computer and Digital Techniques Journal of VLSI Signal Processing

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

25. 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 Midterm Exam - Monday, March 23 Final Exam - Tuesday, May 11, 7:30-10:15 PM Tentative days of exams:

26. Project (1) Project I (individual, 20% of grade) Comprehensive analysis of basic operations of SHA-3 candidates Final report due Tuesday, March 16 Optimization criteria: minimum latency minimum area minimum product latency · area use of embedded FPGA resources (BRAMs, embedded multipliers, DSP units, Different for all students Done individually

27. Limitations of the Current Approach Time and effort Accuracy of comparison One designer = too long time to implement all candidates Multiple designers = significant inaccuracies associated with different skills and coding styles

28. Problem How to predict ranking and relative performance of candidate algorithms without the actual time-consuming hardware implementation at the Register Transfer Level (RTL)? Applications: Ranking of candidate algorithms submitted to the contests (large number of candidates, time limit) Ranking of candidate algorithms during the design process by designers themselves (no experience in hardware design, short response time needed)

29. Features of our Problem to Exploit No need to obtain the functioning netlist or HDL description (performance numbers sufficient) Limited accuracy required (less than 20% differences in performance considered insignificant) Limited number of basic operations Limited number of architectures used in practice

30. The proposed approach

31. 1.Determine the minimum set of basic operations required to implement a given class of cryptographic transformations 2.Determine the required range of parameters of these operations (e.g., operand sizes in arithmetic operations) 3.Implement basic operations in RTL VHDL (or Verilog) in a parametric fashion (using constants and generics) 4. Characterize all operations, for all required parameter values using Xilinx and/or Altera development environments -Area and latency -Low cost FPGAs and high-performance FPGAs Steps of Our Methodology (1)

32. Mars Twofish Serpent RC6 Rijndael Major operations of AES finalists S-boxes Integer multiplication Variable rotation Multiplication in GF(2 m )

33. Mars Twofish Serpent RC6 Rijndael Auxiliary operations of AES finalists Boolean Addition/ subtraction Permutation Fixed rotation

34. Major cipher operations (1) - S-box S-box n x m ROM Software Hardware C ASM WORD S[1<<n]= { 0x23, 0x34, 0x56.............. } S DW 23H, 34H, 56H ….. direct logic n m 2 n words n-bit address m-bit output... x1x1 x2x2 xnxn y1y1 y2y2 ymym S 2 n  m bits

35. variable rotation ROL32 Mux-based rotation High-speed clock C ASM Major cipher operations (2) – Variable Rotation A <<< B ROL A, B C = (A > (32-B)); min (B, 32-B) CLK’ cycles Hardware Software fast clock CLK’ A A<<<B A<<<0A<<<16 32 B[4] B[3] B[2] B[1] B[0]

36. Permutation C order of wires Auxiliary cipher operations (1) - Permutation P Hardware Software ASM complex sequence of instructions <<, |, & complex sequence of instructions ROL, OR, AND n n x1x1 x2x2 x3x3 xnxn x n-1... y1y1 y2y2 y3y3 ynyn y n-1...

37. C=A+B mod 2 n Adder/subtractor ASM C Auxiliary cipher operations (4) Addition/subtraction Hardware Software C = A+B; ADD n n n nn n unsigned long A, B, C; A B C n=32, 16

38. Delay Area Multiple designs for hardware adders Ripple carry adder (RC) Carry-Skip adder (CS) Carry-LookAhead adder (CLA) Carry-Select adder Parallel-Prefix Network adder (Kogge-Stone, Brent-Kung)

39. Delay Area modular multiplication Boolean permutation variable rotation GF(2 n ) multiplication fixed rotation Delay and area in HARDWARE Basic operations addition (CLA) addition (RC) S-box 4x4 S-box 8x8 S-box 9x32 modular inverse

40. addition multiplication Boolean permutation fixed rotation GF(2 n ) multiplication variable rotation Delay and area in SOFTWARE Basic operations Delay Memory S-box 4x4 S-box 8x8 S-box 9x32 modular inverse

41. 5.Develop a simple and human-friendly notation to describe cryptographic algorithms (or their repetitive parts [rounds]), which reveals the parallelism present in the algorithm  Graphical representation more human friendly  Textual representation easier to process by computer programs Steps of Our Methodology (2) Possible Approach: start from a textual description adopt one of the existing graphical editors

43. 6.Develop a tool capable of estimating algorithm performance in terms of area and throughput using  High-level description  Library of basic components  Choice of architecture  Optimization criteria (minimum area, maximum throughput, maximum throughput to area ratio, etc.)  Other constraints, such as required clock frequency, etc. 7.Calibration of the developed tools using existing RTL designs for a limited subset of the algorithms Steps of Our Methodology (2)

44. Possible Problems Routing (interconnect) delays Optimizations on the boundary between two operations Combining multiple operations into one (e.g., using look-up table approach) Inter-round optimizations Resource sharing techniques, in particular resource sharing between encryption and decryption circuits Dependence of results on selected FPGA devices Others…

45. Summary Main project goals: Provide cryptographic community and in particular standardization organizations/groups with a reliable and fast way of comparing large number of candidates for a cryptographic standard Save designers of cryptographic algorithms from design blunders (such as that of IBM team in case of MARS) Project in progress… Feedback and collaboration is very welcome

46. addition multiplication Boolean permutation fixed rotation GF(2 n ) multiplication variable rotation Delay and area in SOFTWARE MARS – IBM team Delay Memory S-box 4x4 S-box 8x8 S-box 9x32 modular inverse

47. Delay Area modular multiplication Boolean permutation variable rotation GF(2 n ) multiplication fixed rotation Delay and area in HARDWARE MARS – IBM team addition (CLA) addition (RC) S-box 4x4 S-box 8x8 S-box 9x32 modular inverse

48. Project II (30% of grade) Project (2) Real life application Requirements derived from the analysis of an application Software implementation (typically public domain) used as a source of test vectors and to determine HW/SW speed ratio Several project topics proposed on the web You can suggest project topic by yourself New Design in the area of Public Key Cryptography, Cryptanalysis, Digital Signal Processing, etc.

49. Cooperation (but not exchange of codes) between teams is encouraged Every team works on a slightly different problem Project topics should be more complex for larger teams Project II (rules) Can be done in a group of 1-3 students Oral presentation and written report: Tuesday, May 4

50. Degrees of freedom and possible trade-offs speedarea power testability ECE 645 ECE 682 ECE 586, 681

51. speed area latency throughput Degrees of freedom and possible trade-offs

52. 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)

53. 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 General purpose DSP processors Specialized circuits

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

55. Secret-key (Symmetric) Cryptosystems key of Alice and Bob - K AB Alice Bob Network Encryption Decryption

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

57. Primary applications (4) Cryptography Integers (16, 32 bits) IDEA, RC6, MarsTwofish, Rijndael, SHA-3 candidates Elements of the Galois field GF(2 n ) (4, 8 bits)

58. 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(2 8 ) XOR ADD32 Rijndael Serpent 8 x 32 S-box 4x4 XOR 16 S-box 8x8 24 MUL GF(2 8 ) XOR

59. Public Key (Asymmetric) Cryptosystems Public key of Bob - K B Private key of Bob - k B Alice Bob Network Encryption Decryption

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

61. 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

62. Primary applications (5) Cryptography Long integers (1000-16,000 bits) Public key cryptography RSA, DSA, Diffie-Hellman Elliptic Curve Cryptosystems Elements of the Galois field GF(2 n ) (150-500 bits)

63. 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))