1. ECE 545 – Introduction to VHDL ECE 645—Project 2 Project Options

2. 2 Project 2 Overview Project 2 will involve the FPGA implementation of a complex digital arithmetic function The project will have an application in either cryptography or signal processing Due to the scope of the project, students should be in groups of 3 The specification and scope of the project will be an interactive process between groups and the instructor

3. 3 Project Options Each group will involve the FPGA implementation of one of the following projects: Cryptography related 1.Trial division sieve 2.Elliptic curve method of factoring 3.RSA encryption & decryption with Montgomery multipliers based on carry save adders Signal processing related 4.Iterative and pipeline CORDIC (coordinate rotation digital computer) processors 5.Finite impulse response filter architectures for FPGA implementations 6.Direct digital frequency synthesis

4. Cryptography Projects Background ECE 645 – Computer Arithmetic

5. 5 RSA Public Key Cryptosystem 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))

6. 6 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

7. 7 Polynomial Selection Linear Algebra Square Root RelationCollection Sieving Cofactoring 200 bit numbers & 350 bit Trial division ECM method Factoring 1024-bit RSA keys using Number Field Sieve (NFS)

8. Topic 1: Trial Division Sieve ECE 645 – Computer Arithmetic

9. 9 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

10. 10 Topic 1: Trial Division Sieve (1) Given: Inputs: Variables: 1.Integers N 1, N 2, N 3,.... each of the size of k-bits Constants: 2. Factor base = set of all primes smaller smaller than a certain bound B = { p 1 =2, p 2 =3, p 3 =5,..., p t ≤ B } Parameters of interest: 4 ≤ k ≤ 512 3 ≤ B ≤ 10 5

11. 11 Topic 1: Trial Division Sieve (2) Required: Outputs: For each integer N i : A list of primes from the factor base that divides N i, and the number of times each prime divides N i. For example if N i = p 1 e1 · p 2 e2 · p 3 e3 · M i, where M i is not divisible by any prime belonging to a factor base, then the output is {p 1, e1}, {p 2, e2}, {p 3, e3}

12. 12 Topic 1: Trial Division Sieve (3) Example: Constants: k=10, B=5 Factor base = {2, 3, 5} Variables: N 1 = 408 = 2 3 · 3 · 17 N 2 = 630 = 2 · 3 2 · 5 · 7 Outputs: {2, 3}, {3, 1} {2, 1}, {3, 2}, {5, 1}

13. Topic 2: Elliptic Curve Method of Factoring ECE 645 – Computer Arithmetic

14. 14 P=(6,19) Q=(7,12) R=P+Q=(13,7) A Addition P=(3,13) 2P=P+P=(7,11) D Doubling  P: Elliptic Curves

15. 15 Inputs : N – number to be factored E – elliptic curve P 0 – point of the curve E : initial point B 1 – smoothness bound for Phase1 B 2 – smoothness bound for Phase2 Outputs: q - factor of N, 1 < q ≤ N or FAIL ECM Algorithm

16. 16 precomputations postcomputations main computations ECM Algorithm Phase 1

17. 17 postcomputations main computations ECM Algorithm Phase 2

18. 18 ECM k·P P+Q2P x·y mod Nx+y mod Nx-y mod N Top level Medium level Point addition Low level Moduar multiplication Modular addition Modular subtraction Scalar multiplication Point doubling Elliptic curve point operations Modular arithmetic (ring operations) Functional units Control unit Host computer Hierarchy of Elliptic Curve Operations

19. Topic 3: RSA Encryption & Decryption with Montgomery Multipliers based on Carry Save Adders ECE 645 – Computer Arithmetic

20. 20 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)) RSA as a Trap-Door One-Way Function

21. 21 Right-to-left binary exponentiation Left-to-right binary exponentiation E = (e L-1, e L-2, …, e 1, e 0 ) 2 Y = 1; S = X; for i=0 to L-1 { if (e i == 1) Y = Y  S mod N; S = S 2 mod N; } Y = 1; for i=L-1 downto 0 { Y = Y 2 mod N; if (e i == 1) Y = Y  X mod N; } Exponentiation: Y = X E mod N

22. 22 C = A  B mod M A Integer domain Montgomery domain A’ = A  2 k mod M B B’ = B  2 k mod M C’ = MP(A’, B’, M) = = A’  B’  2 -k mod M = = (A  2 k )  (B  2 k )  2 -k mod M = = A  B  2 k mod M C’ = C  2 k mod M C = A  B A, B, M – k-bit numbers Montgomery Modular Multiplication

23. 23 A’ = MP(A, 2 2k mod M, M) C = MP(C’, 1, M) A A’ C C’ Montgomery Modular Multiplication

24. 24 = MPMP CPCP P dPdP mod = MQMQ CQCQ Q dQdQ C P = C mod P d P = d mod (P-1) C Q = C mod Q d Q = d mod (Q-1) = mod C M d N M = M P ·R Q + M Q ·R P mod N where R P = (P -1 mod Q) ·P = P Q-1 mod N R Q = (Q -1 mod P) ·Q= Q P-1 mod N Fast Modular Exponentiation using Chinese Remainder Theorem

25. Topic 4: Iterative and Pipeline CORDIC (Coordinate Rotation Digital Computer) Processors ECE 645 – Computer Arithmetic

26. 26 - If we have a computationally efficient way of rotating a vector, we can evaluate cos, sin, and tan –1 functions Rotation by an arbitrary angle is difficult, so we: Perform psuedorotations that require simpler operations Use special angles to synthesize the desired angle z z =   (1) +   (2) +... +   (m) Key ideas in CORDIC COordinate Rotation DIgital Computer used this method in 1950s; modern electronic calculators also use it Rotations and Pseudo-Rotations in CORDIC

27. 27 Fig. 22.1 A pseudorotation step in CORDIC Our strategy: Eliminate the terms (1 + tan 2  (i) ) 1/2 and choose the angles  (i) ) so that tan  (i) is a power of 2; need two shift-adds x (i+1) = x (i) cos  (i) – y (i) sin  (i) = (x (i) – y (i) tan  (i) ) / (1 + tan 2  (i) ) 1/2 y (i+1) = y (i) cos  (i) + x (i) sin  (i) = (y (i) + x (i) tan  (i) ) / (1 + tan 2  (i) ) 1/2 z (i+1) = z (i) –  (i) Recall that cos  = 1 / (1 + tan 2  ) 1/2 Rotating a Vector by an Angle

28. 28 Fig. 22.1 A pseudorotation step in CORDIC Pseudorotation: Whereas a real rotation does not change the length R (i) of the vector, a pseudorotation step increases its length to: R (i+1) = R (i) / cos  (i) = R (i) (1 + tan 2  (i) ) 1/2 x (i+1) = x (i) – y (i) tan  (i) y (i+1) = y (i) + x (i) tan  (i) z (i+1) = z (i) –  (i) Pseudorotating a Vector by an Angle

29. 29 CORDIC iteration: In step i, we pseudorotate by an angle whose tangent is d i 2 –i (the angle e (i) is fixed, only direction d i is to be picked) x (i+1) = x (i) – d i y (i) 2 –i y (i+1) = y (i) + d i x (i) 2 –i z (i+1) = z (i) – d i tan –1 2 –i = z (i) – d i e (i) –––––––––––––––––––––––––––––––– i –––––––––––––––––––––––––––––––– 0 45.00.785 398 163 1 26.60.463 647 609 2 14.00.244 978 663 3 7.10.124 354 994 4 3.60.062 418 810 5 1.80.031 239 833 6 0.90.015 623 728 7 0.40.007 812 341 8 0.20.003 906 230 9 0.10.001 953 123 –––––––––––––––––––––––––––––––– e (i) in degrees (approximate) e (i) in radians (precise) Table 22.1 Value of the function e (i) = tan –1 2 –i, in degrees and radians, for 0  i  9 Example: 30  angle 30.0  45.0 – 26.6 + 14.0 – 7.1 + 3.6 + 1.8 – 0.9 + 0.4 – 0.2 + 0.1 = 30.1 Basic CORDIC Iterations

30. 30 Project Task Implement iterative and pipeline solutions to CORDIC in various modes

31. Topic 5: Finite Impulse Response Filter Architectures for FPGA Implementations ECE 645 – Computer Arithmetic

32. 32 Digital filters are widely used in digital communications and audio/video processing. In particular, finite impulse response (FIR) filters are used for their ease of implementation and stability. FIR Filters

33. 33 As seen above digital filters, boxed in blue, play a crucial role in digital communication chips such as Ethernet transceivers, cable modems, DSL modems, satellite receivers, mobile phones, etc. Example: Gigabit Ethernet

34. 34 x(n) Z -1 h0h0 h1h1 h2h2 h N-1 An FIR filter implements a convolution in the time- domain Critical path of N-tap filter: N-1 adds + 1 multiply Arithmetic complexity of N-tap filter modeled as: N multiplications/sample + N-1 adds/sample y(n) Direct Form Filter

35. 35 Project Task: FIR Architecture Explorations and Optimizations Transpose form Parallel subexpression sharing Canonic signed digit representations using carry- save addition Parallel, word-serial, bit-serial implementation Xilinx DSP multipliers and multiply-accumulate structures

36. Topic 6: Direct Digital Frequency Synthesis ECE 645 – Computer Arithmetic

37. 37 Direct Digital Frequency Synthesis Direct digital frequency synthesis is used to generate sin and cosine functions for digital communication applications Used in many applications: cell phones, cable modems, satellite receivers, etc.

38. 38 DDFS: Basic Understanding and Architecture Output of DDFS is a sine and cosine waveform k = frequency control word L = accumulator bit width N=2 L =number of slots in ROM D=number of output bits phi(n) = (nk) mod N 1/T = clock frequency f 0 = 1/ (NT) = lowest frequency output (i.e. resolution) f c = kf 0 = k/(NT) = desired frequency, output will be cos(2π f c nT) and sin(2π f c nT) f max = greatest frequency achievable = 1/(2T) = ½ f clk +N slots of ROM k D D Lcos(2π/N * phi(n)) sin(2π/N * phi(n))

39. 39 DDFS: Example Output

40. 40 Project task The ROM-based architecture is simplistic; new architectures which are superior exist Investigate various architectures of DDFS and implement in FPGA