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Dividers

Lecture 10. Dividers. Required Reading. Behrooz Parhami, Computer Arithmetic: Algorithms and Hardware Design. Chapter 13, Basic Division Schemes 13.1, Shift/Subtract Division Algorithms 13.3, Restoring Hardware Dividers 13.4, Non-Restoring and Signed Division.

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Dividers

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  1. Lecture 10 Dividers

  2. Required Reading Behrooz Parhami, Computer Arithmetic: Algorithms and Hardware Design Chapter 13, Basic Division Schemes 13.1, Shift/Subtract Division Algorithms 13.3, Restoring Hardware Dividers 13.4, Non-Restoring and Signed Division

  3. Notation and Basic Equations

  4. Notation z Dividend z2k-1z2k-2 . . . z2 z1 z0 d Divisor dk-1dk-2 . . . d1 d0 q Quotient qk-1qk-2 . . . q1 q0 s Remainder sk-1sk-2 . . . s1 s0 (s = z - dq)

  5. Basic Equations of Division z = d q + s | s | < | d | sign(s) = sign(z) z > 0 0  s < | d | z < 0 - | d | < s  0

  6. Unsigned Integer Division Overflow • Must check overflow because obviously the quotient q can also be 2k bits. • For example, if the divisor d is 1, then the quotient q is the dividend z, which is 2k bits Condition for no overflow (i.e. q fits in k bits): z = q d + s < (2k-1) d + d = d 2k z = zH 2k + zL < d 2k zH < d

  7. Sequential Integer Division Basic Equations s(0) = z s(j) = 2 s(j-1) - qk-j (2k d) for j=1..k s(k) = 2k s

  8. Sequential Integer Division Justification s(1) = 2 z - qk-1 (2k d) s(2) = 2(2 z - qk-1 (2k d)) - qk-2 (2k d) s(3) = 2(2(2 z - qk-1 (2k d)) - qk-2 (2k d)) - qk-3 (2k d) . . . . . . s(k) = 2(. . . 2(2(2 z - qk-1 (2k d)) - qk-2 (2k d)) - qk-3 (2k d) . . . - q0 (2k d) = = 2k z - (2k d) (qk-1 2k-1 + qk-2 2k-2 + qk-3 2k-3 + … + q020) = = 2k z - (2k d) q = 2k (z - d q) = 2k s

  9. Fig. 13.2 Examples of sequential division with integer and fractional operands.

  10. Fractional Division

  11. Unsigned Fractional Division zfrac Dividend .z-1z-2 . . . z-(2k-1)z-2k dfrac Divisor .d-1d-2 . . . d-(k-1) d-k qfrac Quotient .q-1q-2 . . . q-(k-1) q-k sfrac Remainder .000…0s-(k+1) . . . s-(2k-1) s-2k k bits

  12. Integer vs. Fractional Division For Integers: z = q d + s  2-2k z 2-2k = (q 2-k) (d 2-k) + s (2-2k) For Fractions: zfrac = qfrac dfrac + sfrac where zfrac = z 2-2k dfrac = d 2-k qfrac = q 2-k sfrac = s 2-2k

  13. Unsigned Fractional Division Overflow Condition for no overflow: zfrac < dfrac

  14. Sequential Fractional Division Basic Equations s(0) = zfrac s(j) = 2 s(j-1) - q-j dfrac for j=1..k 2k · sfrac = s(k) sfrac = 2-k · s(k)

  15. Sequential Fractional Division Justification s(1) = 2 zfrac - q-1 dfrac s(2) = 2(2 zfrac - q-1 dfrac) - q-2 dfrac s(3) = 2(2(2 zfrac - q-1 dfrac) - q-2 dfrac) - q-3 dfrac . . . . . . s(k) = 2(. . . 2(2(2 zfrac - q-1 dfrac) - q-2 dfrac) - q-3 dfrac . . . - q-k dfrac = = 2k zfrac - dfrac (q-1 2k-1 + q-2 2k-2 + q-3 2k-3 + … + q-k20) = = 2k zfrac - dfrac 2k (q-1 2-1 + q-2 2-2 + q-3 2-3 + … + q-k2-k) = = 2k zfrac - (2k dfrac) qfrac = 2k (zfrac - dfrac qfrac) = 2k sfrac

  16. Restoring Unsigned Integer Division

  17. Restoring Unsigned Integer Division s(0) = z for j = 1 to k if 2 s(j-1) - 2k d > 0 qk-j = 1 s(j) = 2 s(j-1) - 2k d else qk-j = 0 s(j) = 2 s(j-1)

  18. Fig. 13.6 Example of restoring unsigned division.

  19. Fig. 13.5 Shift/subtract sequential restoring divider.

  20. Signed Integer Division

  21. Signed Integer Division z d | z | | d | sign(z) sign(d) Unsigned division sign(s) = sign(z) sign(z) = sign(d) + | q | | s | sign(q) = - sign(z)  sign(d) q s

  22. Examples of Signed Integer Division Examples of division with signed operands z = 5 d = 3 q = 1 s = 2 z = 5 d = –3 q = –1 s = 2 z = –5 d = 3 q = –1 s = –2 z = –5 d = –3 q = 1 s = –2 Magnitudes of q and s are unaffected by input signs Signs of q and s are derivable from signs of z and d

  23. Fast Review of Fast Dividers

  24. Classification of Dividers Array Dividers Dividers by Convergence Sequential Radix-2 High-radix • Restoring • Non-restoring • regular • SRT • regular using carry save adders • SRT using carry save adders

  25. Array Dividers

  26. Fig. 15.7 Restoring array divider composed of controlledsubtractor cells.

  27. Sequential Dividers with Carry-Save Adders

  28. Fig. 14.8 Block diagram of a radix-2 divider with partialremainder in stored-carry form.

  29. Pentium bug (1) October 1994 Thomas Nicely, Lynchburg Collage, Virginia finds an error in his computer calculations, and traces it back to the Pentium processor November 7, 1994 First press announcement, Electronic Engineering Times Late 1994 Tim Coe, Vitesse Semiconductor presents an example with the worst-case error c = 4 195 835/3 145 727 Pentium = 1.333 739 06... Correct result = 1.333 820 44...

  30. Pentium bug (2) Intel admits “subtle flaw” November 30, 1994 Intel’s white paper about the bug and its possible consequences Intel - average spreadsheet user affected once in 27,000 years IBM - average spreadsheet user affected once every 24 days Replacements based on customer needs December 20, 1994 Announcement of no-question-asked replacements

  31. Pentium bug (3) Error traced back to the look-up table used by the radix-4 SRT division algorithm 2048 cells, 1066 non-zero values {-2, -1, 1, 2} 5 non-zero values not downloaded correctly to the lookup table due to an error in the C script

  32. Multiply/Divide Unit

  33. Multiply-Divide Unit The control unit proceeds through necessary steps for multiplication or division (including using the appropriate shift direction) The slight speed penalty owing to a more complex control unit is insignificant Fig. 15.9 Sequential radix-2 multiply/divide unit.

  34. DIGITAL SYSTEMS DESIGN • Concentration advisors:Kris Gaj • ECE 545 Digital System Design with VHDL (Fall semesters)– K. Gaj, project, FPGA design with VHDL, Aldec/Synplicity/Xilinx/Altera • 2. ECE 645 Computer Arithmetic (Spring semesters)– K. Gaj, project, FPGA design with VHDL or Verilog, • Aldec/Synplicity/Xilinx/Altera • 3. ECE 586 Digital Integrated Circuits(Spring semesters) – D. Ioannou • 4. ECE 681 VLSI Design for ASICs (Fall semesters)– TBD, project/lab, front-end and back-end ASIC design with Synopsys tools • 5. ECE 682 VLSI Test Concepts (Spring semesters)– T. Storey, homework

  35. Follow-up courses Computer Arithmetic ECE 645 ECE 646 Cryptography and Computer Network Security (Fall 2012) ECE 746 Advanced Applied Cryptography (Spring2013) ECE 899 Cryptographic Engineering (Spring 2012/Spring 2014)

  36. Cryptography and Computer Network Security Advanced Applied Cryptography Operations in the Galois Fields GF(2n) Modular integer arithmetic • AES • Stream ciphers • Elliptic curve cryptosystems • Random number generators • Smart cards • Attacks against implementations • (timing, power, fault analysis) • Efficient and secure • implementations of cryptography • Security in various kinds of • networks (IPSec, wireless) • Zero-knowledge identification • schemes • Historical ciphers • Classical encryption • (DES, Triple DES, RC5, IDEA) • Modes of operation of block • ciphers • Public key encryption (RSA) • Hash functions and MACs • Digital signatures • Public key certificates • PGP • Secure Internet Protocols • Cryptographic standards

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