Download presentation

Presentation is loading. Please wait.

Published byPaulina Spier Modified about 1 year ago

1
Unified Approach to the Design of Modulo-(2 n ± 1) Adders Based on Signed-LSB Representation of Residues Ghassem Jaberipur Dept. Electrical & Computer Engr. Shahid Beheshti Univ., Tehran, Iran Behrooz Parhami Dept. Electrical & Computer Engr. Univ. of California, Santa Barbara, USA 19th IEEE International Symposium on Computer Arithmetic Portland, Oregon, USA, June 8-10, 2009

2
19th IEEE International Symposium on Computer Arithmetic Portland, Oregon, USA, June 8-10, 2009 Unified Approach to the Design of Modulo-(2 n ± 1) Adders Based on Signed-LSB Representation of Residues Outline Introduction Background Signed-LSB Representation New Modulo-(2 n ± 1) Adders – Mod-(2 n + 1) Adder – Mod-(2 n – 1) Adder Conversion from/to Binary Comparisons & Applications Conclusion 2

3
19th IEEE International Symposium on Computer Arithmetic Portland, Oregon, USA, June 8-10, 2009 Unified Approach to the Design of Modulo-(2 n ± 1) Adders Based on Signed-LSB Representation of Residues Introduction Renewed interest in RNS arithmetic Separate designs for mod-(2 n ± 1) and mod-2 n Error-prone and labor-intensive optimizations New signed-LSB representation of residues Sole use of standard arithmetic building blocks Greater confidence in correctness Configurable RNS processor for fault tolerance 3

4
19th IEEE International Symposium on Computer Arithmetic Portland, Oregon, USA, June 8-10, 2009 Unified Approach to the Design of Modulo-(2 n ± 1) Adders Based on Signed-LSB Representation of Residues Background: Mod-(2 n – 1) Addition Mod-m: Mod-(2 n –1): 4

5
19th IEEE International Symposium on Computer Arithmetic Portland, Oregon, USA, June 8-10, 2009 Unified Approach to the Design of Modulo-(2 n ± 1) Adders Based on Signed-LSB Representation of Residues Background: Symbols Used 5

6
19th IEEE International Symposium on Computer Arithmetic Portland, Oregon, USA, June 8-10, 2009 Unified Approach to the Design of Modulo-(2 n ± 1) Adders Based on Signed-LSB Representation of Residues Background: Mod-2 n Adder 6

7
19th IEEE International Symposium on Computer Arithmetic Portland, Oregon, USA, June 8-10, 2009 Unified Approach to the Design of Modulo-(2 n ± 1) Adders Based on Signed-LSB Representation of Residues Kalamboukas et al., RPP modulo 255 adderTPP modulo 255 adder Background: Mod-(2 n – 1) Adders

8
19th IEEE International Symposium on Computer Arithmetic Portland, Oregon, USA, June 8-10, 2009 Unified Approach to the Design of Modulo-(2 n ± 1) Adders Based on Signed-LSB Representation of Residues Background: Mod-(2 n + 1) Addition Mod-(2 n +1): W' is difficult to compute, therefore, let 8

9
19th IEEE International Symposium on Computer Arithmetic Portland, Oregon, USA, June 8-10, 2009 Unified Approach to the Design of Modulo-(2 n ± 1) Adders Based on Signed-LSB Representation of Residues Background: Mod-(2 n + 1) Adders Efstathiou, et al., Flaw: S n is wrong

10
19th IEEE International Symposium on Computer Arithmetic Portland, Oregon, USA, June 8-10, 2009 Unified Approach to the Design of Modulo-(2 n ± 1) Adders Based on Signed-LSB Representation of Residues Background: Mod-(2 n + 1) Adders The corrected mod-257 TPP adder 10 Same Latency More area

11
19th IEEE International Symposium on Computer Arithmetic Portland, Oregon, USA, June 8-10, 2009 Unified Approach to the Design of Modulo-(2 n ± 1) Adders Based on Signed-LSB Representation of Residues Background: Dim-1 Representation Diminshed-1 mod-(2 n + 1) 11

12
19th IEEE International Symposium on Computer Arithmetic Portland, Oregon, USA, June 8-10, 2009 Unified Approach to the Design of Modulo-(2 n ± 1) Adders Based on Signed-LSB Representation of Residues Signed-LSB Representation Faithful representation of [–1, 2 n – 1] Problem: Mixed posibits and negabits: A + B 12

13
19th IEEE International Symposium on Computer Arithmetic Portland, Oregon, USA, June 8-10, 2009 Unified Approach to the Design of Modulo-(2 n ± 1) Adders Based on Signed-LSB Representation of Residues Universal Full Adders Full adder can compress mixed posibits and negabits 13 ||X 1 + X 2 + x 3 || = X 1 – 1 + X 2 – 1 + x 3 = 2c + s – 2 = ||2C + s||

14
19th IEEE International Symposium on Computer Arithmetic Portland, Oregon, USA, June 8-10, 2009 Unified Approach to the Design of Modulo-(2 n ± 1) Adders Based on Signed-LSB Representation of Residues New Modulo-(2 n + 1) Adder 14

15
19th IEEE International Symposium on Computer Arithmetic Portland, Oregon, USA, June 8-10, 2009 Unified Approach to the Design of Modulo-(2 n ± 1) Adders Based on Signed-LSB Representation of Residues Mod-(2 n + 1) Signed-LSB Addition 15

16
19th IEEE International Symposium on Computer Arithmetic Portland, Oregon, USA, June 8-10, 2009 Unified Approach to the Design of Modulo-(2 n ± 1) Adders Based on Signed-LSB Representation of Residues New Mod-(2 n – 1) Adder 16

17
19th IEEE International Symposium on Computer Arithmetic Portland, Oregon, USA, June 8-10, 2009 Unified Approach to the Design of Modulo-(2 n ± 1) Adders Based on Signed-LSB Representation of Residues Mod-(2 n + 1) vs. Mod-(2 n – 1) 17

18
19th IEEE International Symposium on Computer Arithmetic Portland, Oregon, USA, June 8-10, 2009 Unified Approach to the Design of Modulo-(2 n ± 1) Adders Based on Signed-LSB Representation of Residues Conversion from/to Binary 18 Weighted representation Conversion of input to residue representation is very simple Fast residue-to-binary converters implement the Chinese remainder theorem via CSAs Weighted representation Signed-LSB representation

19
19th IEEE International Symposium on Computer Arithmetic Portland, Oregon, USA, June 8-10, 2009 Unified Approach to the Design of Modulo-(2 n ± 1) Adders Based on Signed-LSB Representation of Residues Applications Fault-tolerant RNS processor 19

20
19th IEEE International Symposium on Computer Arithmetic Portland, Oregon, USA, June 8-10, 2009 Unified Approach to the Design of Modulo-(2 n ± 1) Adders Based on Signed-LSB Representation of Residues Comparison: Gate-Level Analysis 20 RepresentationModulusRPPTPPRef. Weighted2 n + 12 log n + 92 log n + 6[1] Diminished-12 n + 12 log n + 72 log n + 5[9] Signed-LSB2 n + 12 log n + 72 log n + 5New Weighted2 n – 12 log n + 52 log n + 3[5] Signed-LSB2 n – 12 log n + 72 log n + 5New

21
19th IEEE International Symposium on Computer Arithmetic Portland, Oregon, USA, June 8-10, 2009 Unified Approach to the Design of Modulo-(2 n ± 1) Adders Based on Signed-LSB Representation of Residues Comparison: Synthesis Results 21 RepresentationModulus Delay (ns)Critical pathArea ( m 2 ) Weighted a4 s1a4 s Diminished a0 s0a0 s Signed-LSB a0 s3a0 s Weighted 2 4 – a2 s2a2 s Signed-LSB 2 4 – a0 s0a0 s Weighted; b5 s6b5 s Diminished b3 s2b3 s Signed-LSB b1 s1b1 s Weighted 2 8 – a0 s5a0 s Signed-LSB 2 8 – a0 s6a0 s Weighted a 16 s Diminished a 13 s Signed-LSB b4 s5b4 s Weighted 2 16 – a 13 s Signed-LSB 2 16 – a9 s8a9 s8 6243

22
19th IEEE International Symposium on Computer Arithmetic Portland, Oregon, USA, June 8-10, 2009 Unified Approach to the Design of Modulo-(2 n ± 1) Adders Based on Signed-LSB Representation of Residues Conclusion Implementing mod-(2 n – 1) and mod-(2 n + 1) addition using generic CSA and binary adders Easier/faster exploration of the design space Simpler testing and verification Greater confidence in design correctness Configurable modular adders (fault tolerance) Potential for less complex modular subtractors and modular multipliers 22

23
19th IEEE International Symposium on Computer Arithmetic Portland, Oregon, USA, June 8-10, 2009 Unified Approach to the Design of Modulo-(2 n ± 1) Adders Based on Signed-LSB Representation of Residues Questions? The authors gratefully acknowledge the assistance of Mr. Saeed Nejati and Ms. Hanieh Alavi. G. Jaberipur also acknowledges support from IPM School of Computer Science and from Shahid Beheshti University. Supplement at: 23

Similar presentations

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google