Modulo-( 2 + 3 ) Parallel Prefix Addition via Diminished-3 - - PowerPoint PPT Presentation

β–Ά
modulo 2 3 parallel prefix addition via
SMART_READER_LITE
LIVE PREVIEW

Modulo-( 2 + 3 ) Parallel Prefix Addition via Diminished-3 - - PowerPoint PPT Presentation

Dec 09, 2022 273 likes β€’517 views

Modulo-( 2 + 3 ) Parallel Prefix Addition via Diminished-3 Representation of Residues Authors: Ghassem Jaberipur, Sahar Moradi Cherati Arith 26 Kindly presented by: Paulo Srgio Alves Martins Contents 2/22 3/22 I NTRODUCTION Popular

slide-1
SLIDE 1

Modulo-(2π‘œ + 3) Parallel Prefix Addition via Diminished-3 Representation of Residues

Authors: Ghassem Jaberipur, Sahar Moradi Cherati

Arith 26

Kindly presented by: Paulo SΓ©rgio Alves Martins

slide-2
SLIDE 2

Contents

2/22

slide-3
SLIDE 3

INTRODUCTION

ο‚΄ Residue number systems (RNS)

ο‚΄ β„› = {𝑛1, 𝑛2 … 𝑛𝑙}, 𝑛𝑗(1 ≀ 𝑗 ≀ 𝑙) ο‚΄ 𝑁 = 𝑗=1

𝑙

𝑛𝑗

ο‚΄ Applications ο‚΄ RNS Features 3/22

π‘Œ, 𝑍 ∈ β„› π‘Œ = 𝑦1, 𝑦2 … , 𝑦𝑙 , 𝑍 = (𝑧1, 𝑧2 … , 𝑧𝑙), where 𝑦𝑗 = π‘Œ 𝑛𝑗, 𝑧𝑗 = 𝑍 𝑛𝑗 π‘Ž = π‘Œ βŠ› 𝑍, 𝑨𝑗 = 𝑦𝑗 βŠ› 𝑧𝑗, where βŠ›βˆˆ {+, βˆ’,Γ—}

  • Popular Ο„ = 2π‘œ βˆ’ 1,2π‘œ, 2π‘œ + 1
  • General form:

2π‘œ Β± Ξ΄ 1 ≀ Ξ΄ < 2π‘œβˆ’1

Parallel prefix modulo-(2π‘œ βˆ’ Ξ΄) adders:

  • Ξ΄ = 1
  • Ξ΄ = 3 [Jaberipur,2015]
  • Ξ΄ = 2π‘Ÿ + 1 [Langroudi,2015]
  • 2π‘œ + Ξ΄ = 2π‘œ+1 βˆ’ Ξ΄β€²

, where 1 < Ξ΄β€² = 2π‘œ βˆ’ Ξ΄ < 2π‘œ No direct fast solution Delay

3 + 2 log π‘œ Ξ” 4 + 2 log π‘œ Ξ” 5 + 2 log π‘œ Ξ”

  • Cryptography
  • digital signal/image processing
  • High Speed
  • Low Power
slide-4
SLIDE 4

DIMINISHED-1 ADDERS

4/22

π‘Œ: A modulo-(2π‘œ + 1) residue ∈ 0,2π‘œ π‘Œ = π‘Œβ€² + π‘¨π‘Œ, where π‘Œβ€² = π‘Œ βˆ’ 1 ∈ 0,2π‘œ βˆ’ 1 for π‘Œ > 0 π‘¨π‘Œ = 0(1), if and only if π‘Œ = 0(> 0) 𝑨𝑇 = (π‘¨π΅βˆ¨ 𝑨𝐢) ∧ 𝑨𝐡𝑨𝐢 ∧ 𝜊 𝑇′ = 𝐡′ + 𝐢′ + 𝑨𝐡 + 𝑨𝐢 βˆ’ 𝑨𝑇 2π‘œ+1 = 𝑋′ + 𝑨𝐡𝑨𝐢π‘₯π‘œ

β€² 2π‘œ

Diminished-1 encoding π‘₯π‘œ

β€² = π»π‘œβˆ’1:0

𝜊 = π‘„π‘œβˆ’1:0π»π‘œβˆ’1:0 = 1, Iff 𝐡′ + 𝐢′ = 2π‘œ βˆ’ 1 𝑇 = 𝐡 + 𝐢 2π‘œ+1 = 𝑇′ + 𝑨𝑇, 𝐡 = 𝐡′ + 𝑨𝐡, 𝐢 = 𝐢′ + 𝑨𝐢 𝑇′ = 𝑇 βˆ’ 1, 𝐡′ = 𝐡 βˆ’ 1, 𝐢′ = 𝐢 βˆ’ 1 for 𝑇, 𝐡, 𝐢 > 0 𝑨𝑇, 𝑨𝐡, 𝑨𝐢: zero-indicator bits 𝐡′ + 𝐢′ = 2π‘œπ‘₯π‘œ

β€² +

𝑋′, where 𝑋′ = π‘₯π‘œβˆ’1

β€²

… π‘₯0

β€²

𝑇′ = 𝑇 βˆ’ 1 = 𝐡 + 𝐢 2π‘œ+1 βˆ’ 1 = 𝐡′ + 1 + 𝐢′ + 1 βˆ’ 1 2π‘œ+1 = 2π‘œπ‘₯π‘œ

β€² +

𝑋′ + 1 2π‘œ+1 = 𝑋′ + 1 βˆ’ π‘₯π‘œ

β€² 2π‘œ+1 =

𝑋′ + π‘₯β€²π‘œ Diminished-1 addition

slide-5
SLIDE 5

Related Work: Modulo-(πŸ‘π’ + 𝟐) D1 adder

5/22

RPP TPP

slide-6
SLIDE 6

DIMINISHED-3 REPRESENTATION

AND ADDITION

slide-7
SLIDE 7

DIMINISHED-3 REPRESENTATION

7/22

π‘Œ: Moduloβˆ’ 2π‘œ + 3 residue ∈ [0, 2π‘œ + 2] π‘Œβ€² ∈ [0, 2π‘œ βˆ’ 1]

D3 Representation

π‘Œ = π‘Œβ€² + π‘ˆ

π‘Œ

π‘ˆ

π‘Œ = 𝑒1𝑒0

{0, 1, 2} [3, 2π‘œ + 2] π‘ˆ

π‘Œ ∈ {0, 1, 2}-indicator

π‘Œβ€² ∢ π‘œ bit

π‘ˆ

π‘Œ = 0 ⟺ π‘Œ = 0, π‘Œβ€² = 0

π‘ˆ

π‘Œ = 1 ⟺ π‘Œ = 1, π‘Œβ€² = 0

π‘ˆ

π‘Œ = 2 ⟺ π‘Œ = 2, π‘Œβ€² = 0

π‘ˆ

π‘Œ = 3 ⟺ 3 ≀ π‘Œ ≀ 2π‘œ + 2, π‘Œβ€² ∈ [0,2π‘œ βˆ’ 1]

slide-8
SLIDE 8

Modulo-(πŸ‘π’ + πŸ’) D3 ADDITION

8/22

𝐡 ∈ [0, 2π‘œ + 2] 𝐡 = 𝐡′ + π‘ˆ

𝐡

𝐡′= π’ƒπ’βˆ’πŸ … π’ƒπŸ‘ π’ƒπŸ π’ƒπŸ 𝐢′ = π’„π’βˆ’πŸ … π’„πŸ‘ π’„πŸ π’„πŸ π‘₯𝒐

β€²

π‘₯π’βˆ’πŸ

β€²

… π‘₯2

β€²

π‘₯1

β€²

π‘₯0

β€²

𝐡, 𝐢, 𝑇 β‰₯ 3 𝑇′ = 𝑇 βˆ’ 3 = 𝐡 + 𝐢 2π‘œ+3 βˆ’ 3 = 𝐡′ + 3 + 𝐢′ + 3 βˆ’ 3 2π‘œ+3 = 2π‘œπ‘₯π‘œ

β€² +

𝑋′ + 3 2π‘œ+3 = 𝑋′ + 3 1 βˆ’ π‘₯π‘œ

β€² 2π‘œ+3 =

𝑋′ + 3π‘₯β€²π‘œ 𝐢 ∈ [0, 2π‘œ + 2] 𝐢 = 𝐢′ + π‘ˆπΆ

slide-9
SLIDE 9

Comparison D1 and D3

9/22

πŸ’ + πŸ‘ π’Žπ’‘π’‰ 𝒐 𝜠 πŸ” + πŸ‘ π’Žπ’‘π’‰ 𝒐 𝜠

slide-10
SLIDE 10

Detect Special Cases 𝑻 = 𝑩 + π‘ͺ πŸ‘π’+πŸ’ ∈ {𝟏,1,2}

10/22

IF 𝐡′ + 𝐢′ = 2π‘œ βˆ’ 1 THEN ΞΎ1 = 1 ELSE ΞΎ1 = 0 IF 𝐡′ + 𝐢′ = 2π‘œ βˆ’ 2 THEN ΞΎ2 = 1 ELSE ΞΎ2 = 0 IF 𝐡′ + 𝐢′ = 2π‘œ βˆ’ 3 THEN ΞΎ3 = 1 ELSE ΞΎ3 = 0 ΞΎ1 = π‘„π‘œβˆ’1:2π»π‘œβˆ’1:2β„Ž1𝑣0 ΞΎ2 = π‘„π‘œβˆ’1:2π»π‘œβˆ’1:2β„Ž1𝑣0 ΞΎ3 = π‘„π‘œβˆ’1:2π»π‘œβˆ’1:2𝑣1𝑣0 𝑻 = 𝑩 + π‘ͺ πŸ‘π’+𝟐 = 𝟏 IF 𝐡′ + 𝐢′ = 2π‘œ βˆ’ 1 THEN ΞΎ = 1 ELSE ΞΎ = 0 π‘¬πŸ

slide-11
SLIDE 11

11/22

ο‚΄ Οƒ1 = Ξ±1Ξ²1 ∨ Ξ±0Ξ²0 𝒧 ∨ 𝑔

1(Ξ±1, Ξ²1, Ξ±0, Ξ²0, 𝑣1, 𝑀1, 𝑣0)

ο‚΄ Οƒ0 = Ξ±1Ξ²0 ∨ Ξ±0Ξ²1 𝒧 ∨ 𝑔

0 Ξ±1, Ξ²1, Ξ±0, Ξ²0, 𝑣1, 𝑀1, 𝑣0

𝒧 = π‘„π‘œβˆ’1:2π»π‘œβˆ’1:2,

DERIVATION OF π‘ˆ

𝑇

ΞΎ1 = π’§β„Ž1𝑣0, ΞΎ2 = π’§β„Ž1𝑣0, ΞΎ3 = 𝒧𝑣1𝑣0

slide-12
SLIDE 12

Impact of ΞΎ-dependent noise terms on 𝑼𝑻

12/22

𝑼𝑩 𝑼π‘ͺ π›πŸ π›πŸ‘ π›πŸ’ 𝑼𝑻 𝑻 Justification 3 1 X X 3 β‰₯ 4 𝐡′ + 𝐢′ = 𝐡′ < 2π‘œ βˆ’ 1 ⟹ 𝑇 = 𝐡 + 𝐢 = 𝐡′ + 3 + 1 < 2π‘œ + 3 3 1 1 X X 𝐡′ + 𝐢′ = 2π‘œ βˆ’ 1 ⟹ 𝐡 + 𝐢 = 2π‘œ + 3, 𝑇 = 𝐡 + 𝐢 2π‘œ+3 = 0 π‘ˆ

𝑇 = 3𝜊1

𝑼𝑩 𝑼π‘ͺ π›πŸ π›πŸ‘ π›πŸ’ 𝑼𝑻 𝑻 Justification 3 2 X 3 β‰₯ 5 𝐡′ + 𝐢′ = 𝐡′ < 2π‘œ βˆ’ 2 ⟹ 𝑇 = 𝐡 + 𝐢 = 𝐡′ + 3 + 2 < 2π‘œ + 3 3 2 1 X 𝐡′ + 𝐢′ = 2π‘œ βˆ’ 2 ⟹ 𝐡 + 𝐢 = 2π‘œ + 3, 𝑇 = 𝐡 + 𝐢 2π‘œ+3 = 0 3 2 1 X 1 1 𝐡′ + 𝐢′ = 2π‘œ βˆ’ 1 ⟹ 𝐡 + 𝐢 = 2π‘œ + 4, 𝑇 = 𝐡 + 𝐢 2π‘œ+3 = 1 π‘ˆ

𝑇 = 𝜊1 + 3 𝜊1 𝜊2

𝑼𝑩 𝑼π‘ͺ π›πŸ π›πŸ‘ π›πŸ’ 𝑼𝑻 𝑻 Justification 3 3 3 β‰₯ 6 𝐡′ + 𝐢′ < 2π‘œ βˆ’ 3 ⟹ 𝑇 = 𝐡 + 𝐢 = 𝐡′ + 𝐢′ + 6 < 2π‘œ + 3 3 3 3 β‰₯ 3 2π‘œ ≀ 𝐡′ + 𝐢′ ≀ 2π‘œ + 2π‘œ βˆ’ 2 ⟹ 3 ≀ 𝑇 = 𝐡 + 𝐢 2π‘œ+3 ≀ 2π‘œ + 1 3 3 1 𝐡′ + 𝐢′ = 2π‘œ βˆ’ 3 ⟹ 𝐡 + 𝐢 = 2π‘œ + 3, 𝑇 = 𝐡 + 𝐢 2π‘œ +3 = 0 3 3 1 1 1 𝐡′ + 𝐢′ = 2π‘œ βˆ’ 2 ⟹ 𝐡 + 𝐢 = 2π‘œ + 4, 𝑇 = 𝐡 + 𝐢 2π‘œ +3 = 1 3 3 1 2 2 𝐡′ + 𝐢′ = 2π‘œ βˆ’ 1 ⟹ 𝐡 + 𝐢 = 2π‘œ + 5, 𝑇 = 𝐡 + 𝐢 2π‘œ +3 = 2 π‘ˆ

𝑇 = 2𝜊1 + 𝜊2 + 3 𝜊1 𝜊2 𝜊3

𝑼𝑩 𝑼π‘ͺ π›πŸ π›πŸ‘ π›πŸ’ 𝑼𝑻 𝑻 Justification 3 X X X 3 β‰₯ 3 𝑇 = 𝐡 + 𝐢 = 𝐡 ≀ 2π‘œ + 2 π‘ˆ

𝑇 = 3

slide-13
SLIDE 13

13/22

THE NOISE TERM π‘ˆ = π‘ˆ

𝐡 + π‘ˆπΆ βˆ’ π‘ˆ 𝑇 IN TERMS OF π‘ˆ 𝐡, π‘ˆπΆ, AND ΞΎ BITS

𝑇′ = 𝐡′ + π‘ˆ

𝐡 + 𝐢′ + π‘ˆπΆ βˆ’ π‘ˆ 𝑇 2π‘œ+3 = 𝐡′ + 𝐢′ + π‘ˆ 𝐡 + π‘ˆπΆ βˆ’ π‘ˆ 𝑇 2π‘œ+3

= 2π‘œπ‘₯π‘œ

β€² +

𝑋′ + π‘ˆ 2π‘œ+3 = 𝑋′ + π‘ˆ βˆ’ 3π‘₯π‘œ

β€² 2π‘œ+3 =

𝑋′ + 𝛆′ 2π‘œ+3, 𝛆′ = π‘ˆ βˆ’ 3π‘₯π‘œ

β€²

DERIVATION OF 𝑇′

𝑼π‘ͺ 𝑼𝑩 1 2 3 1 3𝜊1 + 1 2 1 2𝜊1 + 3𝜊2 + 2 3 3𝜊1 + 1 2𝜊1 + 3𝜊2 + 2 𝜊1 + 2𝜊2 + 3𝜊3 + 3

slide-14
SLIDE 14

COMPOUND RPP REALIZATION OF 𝑻′

14/22

𝑻′ = 𝑿′ + π’œπ’™π’

β€² + 𝛆′ πŸ‘π’

𝑨 = Ξ±1Ξ²1Ξ±0Ξ²0 Ξ΄β€² = Ξ΄1

β€² Ξ΄0 β€² ∈ {0,1,2}

Ξ΄1

β€² =

ΞΎ1Ξ±1Ξ²1 Ξ±0⨁β0 ∨ ΞΎ1 ΞΎ2𝑨π‘₯π‘œ

β€²,

Ξ΄0

β€² = ΞΎ1𝑦 ∨ ΞΎ2𝑨 ∨ Ξ±0Ξ²0 Ξ±1⨁β1 ∨ Ξ±1Ξ²1Ξ±0 ∨ Ξ²0

slide-15
SLIDE 15

15/22

The required RPP circuitry

pgh pgh pgh pgh pgh pgh pgh

𝑣7 𝑀7

𝑕 1, π‘ž 1 HA HA HA HA HA HA HA HA

𝑏0 𝑐0 𝑏1 𝑐1 𝑏2 𝑐2 𝑏3 𝑐3 𝑏4 𝑐4 𝑏5 𝑐5 𝑏6 𝑐6 𝑏7 𝑐7 𝑣6 𝑀6 𝑣5 𝑀5 𝑣4 𝑀4 𝑣3 𝑀3 𝑣2 𝑀2 𝑣1 𝑀1 𝑣0 𝑀8 𝑑7

β€²

𝑑6

β€²

𝑑5

β€²

𝑑4

β€²

𝑑3

β€²

𝑑2

β€²

𝑑1

β€²

𝑑0

β€²

𝑑7 𝑑6 𝑑5 𝑑4 𝑑3 𝑑2 𝑑0 𝑑1

(π’‰πŸ–:πŸ‘,π’’πŸ–:πŸ‘)

π‘ΈπŸ–:πŸ‘

π‘―πŸ–:πŸ‘ π‘―πŸ–:πŸ‘ 𝑣0 (6Ξ”)π‘Ÿ03 π‘Ÿ02(3Ξ”) π‘ΈπŸ–:πŸ‘ π‘Ÿ01(4Ξ”) 𝑧(4Ξ”) (4Ξ”)π‘Ÿ12 π‘Ÿ13(4Ξ”) π‘ΈπŸ–:πŸ‘ π‘ΈπŸ–:πŸ‘ π‘Ÿ11(5Ξ”) (3Ξ”)𝑦 (4Ξ”)𝑧

Οƒ1 Οƒ0

Ξ±0Ξ²1 Ξ±1Ξ²1 Ξ±1Ξ²0 𝑔

0(7Ξ”)

Ξ±0Ξ²0 𝑔

1(7Ξ”)

π‘―πŸ–:πŸ‘ π‘ΈπŸ–:πŸ‘

pgh

( , )

i i

g p

i

h ( , )

l l r l r

g p g p p οƒš ( , )

r r

g p ( , )

l l

g p ( )

l l r

g p g οƒš ( , )

r r

g p ( , )

l l

g p ( , )

i i

g p ( , )

i i

g p

𝑣𝑗 𝑀𝑗

πŸ– + πŸ‘ π’Žπ’‘π’‰ 𝒐 𝜠

slide-16
SLIDE 16

16/22

𝑑𝑗 = π»π‘—βˆ’1:2 ∨ π‘„π‘—βˆ’1:2𝑑2=π»π‘—βˆ’1:2 ∨ π‘„π‘—βˆ’1:2 𝑕1

β€² ∨ π‘ž1 β€²π»π‘œβˆ’1:2 = π»π‘—βˆ’1:2 ∨ π‘„π‘—βˆ’1:2 𝑕1 β€² ∨ π‘ž1 β€²π»π‘œβˆ’1:𝑗

𝑑2 = 𝑕1

β€² ∨ π‘ž1 ′𝐻7:2, 𝑕1 β€², π‘ž1 β€² ∘ 𝐻7:2, 1

𝑑3 = 𝑕2 ∨ π‘ž2 𝑕1

β€² ∨ π‘ž1 ′𝐻7:3 , 𝑕2, π‘ž2 ∘ 𝑕1 β€², π‘ž1 β€² ∘ 𝐻7:3, 1

𝑑4 = 𝐻3:2 ∨ 𝑄3:2 𝑕1

β€² ∨ π‘ž1 ′𝐻7:4 , (𝐻, 𝑄)3:2∘ (𝑕1 β€², π‘ž1 β€²) ∘ 𝐻7:4, 1

𝑑5 = 𝐻4:2 ∨ 𝑄

4:2 𝑕1 β€² ∨ π‘ž1 ′𝐻7:5 , 𝐻, 𝑄 4:2 ∘ (𝑕1 β€², π‘ž1 β€²) ∘ 𝐻7:5, 1

𝑑6 = 𝐻5:2 ∨ 𝑄5:2 𝑕1

β€² ∨ π‘ž1 ′𝐻7:6 , 𝐻, 𝑄 5:2 ∘ (𝑕1 β€², π‘ž1 β€²) ∘ 𝐻7:6, 1

𝑑7 = 𝐻6:2 ∨ 𝑄6:2 𝑕1

β€² ∨ π‘ž1 ′𝑕7 , 𝐻, 𝑄 6:2 ∘ (𝑕1 β€², π‘ž1 β€²) ∘ 𝑕7, 1

Carry Bits for RPP Architecture

slide-17
SLIDE 17

17/22

The required TPP circuitry

Ξ±0Ξ²1

Οƒ1

Ξ±1Ξ²0 𝑔

0(7Ξ”)

Ξ±0Ξ²0

Οƒ0

π‘―πŸ–:πŸ‘ π‘ΈπŸ–:πŸ‘ Ξ±1Ξ²1 𝑔

1(7Ξ”)

π‘ΈπŸ–:πŸ‘ π‘―πŸ–:πŸ‘ π‘ΈπŸ–:πŸ‘ π‘ΈπŸ–:πŸ‘ π‘―πŸ–:πŸ‘ π‘―πŸ–:πŸ‘ π‘―πŸ–:πŸ‘ π‘―πŸ–:πŸ‘ π‘Ÿ12

β€² (7Ξ”)

(6Ξ”)π‘Ÿ04

β€²

(7Ξ”)π‘Ÿ02

β€²

π‘Ÿ11

β€² (7Ξ”)

π‘Ÿ03

β€² (5Ξ”)

π‘Ÿ01

β€² (7Ξ”)

(6Ξ”)π‘Ÿ13

β€²

π‘ΈπŸ–:πŸ‘

𝑑0

β€²

𝑑1

β€²

pgh pgh pgh pgh pgh pgh pgh

𝑣1 𝑀1 𝑣2 𝑀2 𝑣3 𝑀3 𝑣4 𝑀4 𝑣5 𝑀5 𝑣6 𝑀6 𝑣7 𝑀7

𝑕 12, π‘ž 12 𝑕 11, π‘ž 11 HA HA HA HA HA HA HA HA

𝑏0 𝑐0 𝑏1 𝑐1 𝑏2 𝑐2 𝑏3 𝑐3 𝑏4 𝑐4 𝑏5 𝑐5 𝑏6 𝑐6 𝑏7 𝑐7 𝑣0 𝑑7

β€²

𝑑6

β€²

𝑑5

β€²

𝑑4

β€²

𝑑3

β€²

𝑑2

β€²

𝑑7 𝑑6 𝑑5 𝑑4 𝑑3 𝑑2

(π’‰πŸ–:πŸ‘, π’’πŸ–:πŸ‘)

πŸ” + πŸ‘ π’Žπ’‘π’‰ 𝒐 𝜠

slide-18
SLIDE 18

Carry Bits for TPP Architecture

𝑑2, (( π‘žβ€², 𝑕′

11 ∘ π‘žβ€², 𝑕′ 12) ∘ ( 𝑕, π‘ž 7 ∘ 𝑕, π‘ž 6)) ∘ (( 𝑕, π‘ž 5 ∘ (𝑕, π‘ž)4) ∘ ((𝑕, π‘ž)3∘ (𝑕, π‘ž)2))

𝑑3, (((π‘ž, 𝑕)2∘ π‘žβ€², 𝑕′

11) ∘ ( π‘žβ€², 𝑕′ 12 ∘ (𝑕, π‘ž)7)) ∘ (((𝑕, π‘ž)6∘ (𝑕, π‘ž)5) ∘ ((𝑕, π‘ž)4∘ (𝑕, π‘ž)3))

𝑑4, 𝑕, π‘ž 3 ∘ 𝑕, π‘ž 2 ∘ 𝑕′, π‘žβ€² 11 ∘ 𝑕′, π‘žβ€² 12 ∘ (((𝑕, π‘ž)7∘ (𝑕, π‘ž)6) ∘ ((𝑕, π‘ž)5∘ (𝑕, π‘ž)4)) 𝑑5, (( π‘ž, 𝑕 4 ∘ π‘ž, 𝑕 3) ∘ ( π‘ž, 𝑕 2 ∘ π‘žβ€², 𝑕′

11)) ∘ (( π‘žβ€², 𝑕′ 12 ∘ (𝑕, π‘ž)7) ∘ ((𝑕, π‘ž)6∘ (𝑕, π‘ž)5))

𝑑6, (( 𝑕, π‘ž 5 ∘ (𝑕, π‘ž)4) ∘ ((𝑕, π‘ž)3∘ (𝑕, π‘ž)2)) ∘ (( 𝑕′, π‘žβ€² 11 ∘ 𝑕′, π‘žβ€² 12) ∘ ( 𝑕, π‘ž 7 ∘ (𝑕, π‘ž)6)) 𝑑7, 𝑕, π‘ž 6 ∘ 𝑕, π‘ž 5 ∘ 𝑕, π‘ž 4 ∘ 𝑕, π‘ž 3 ∘ (((π‘ž, 𝑕)2∘ π‘žβ€², 𝑕′

11) ∘ ( π‘žβ€², 𝑕′ 12 ∘ (𝑕7, 1)))

18/22

slide-19
SLIDE 19

EVALUATION AND COMPARISON

19/22

RPP SYNTHESIS RESULTS 𝒐 = πŸπŸ• 𝒐 = πŸ—

Design(RPP) Delay Area Power 𝒐𝒕 Ratio π‚π’πŸ‘ Ratio 𝒏𝒙 Ratio D3 0.76 1.00 25318 1.00 0.682 1.00 D1 [1] 0.59 0.77 10128 0.40 0.328 0.48 2π‘œ βˆ’ 3 [2] 0.72 0.94 13227 0.52 0.467 0.68 Design(RPP) Delay Area Power 𝒐𝒕 Ratio π‚π’πŸ‘ Ratio 𝒏𝒙 Ratio D3 0.81 1.00 42043 1.00 1.19 1.00 D1 [1] 0.72 0.88 23776 0.56 0.716 0.60 2π‘œ βˆ’ 3 [2] 0.78 0.96 30637 0.73 1.01 0.85

[1] Jaberipur,2011 [2] Jaberipur,2015

slide-20
SLIDE 20

TPP Results

20/22

SYNTHESIS RESULTS FOR 𝒐 = πŸπŸ• SYNTHESIS RESULTS FOR 𝒐 = πŸ— DELAY AND AREA MEASURES

Design(TPP) Delay(𝜠) Area (# of gates) D3

(5 + 2 log π‘œ) 3π‘œ π‘šπ‘π‘• π‘œ + 15π‘œ + 39

D1 [1]

(3 + 2 log π‘œ) 3π‘œ π‘šπ‘π‘• π‘œ + 12π‘œ βˆ’ 1

2π‘œ βˆ’ 3 [2]

(4 + 2 log π‘œ) 3π‘œ π‘šπ‘π‘• π‘œ + 12π‘œ + 4 Design(TPP) Delay Area Power 𝒐𝒕 Ratio π‚π’πŸ‘ Ratio 𝒏𝒙 Ratio D3 0.65 1.00 30686 1.00 0.956 1.00 D1 [1] 0.57 0.88 14227 0.46 0.375 0.39 2π‘œ βˆ’ 3 [2] 0.64 0.98 14152 0.46 0.500 0.52 Design(TPP) Delay Area Power 𝒐𝒕 Ratio π‚π’πŸ‘ Ratio 𝒏𝒙 Ratio D3 0.73 1.00 51121 1.00 1.60 1.00 D1 [1] 0.64 0.88 34441 0.67 0.939 0.59 2π‘œ βˆ’ 3 [2] 0.73 1.00 32712 0.64 1.081 0.67

[1] Jaberipur,2011 [2] Jaberipur,2015

slide-21
SLIDE 21

CONCLUSIONS

ο‚΄ Implemented the required parallel prefix (RPP and TPP architectures) adders based on the novel diminished-3 representation of residues in {3,2π‘œ + 2} and 2-bit {0,1,2} indicator ο‚΄ The adder delay is only 2Ξ” more than the modulo-(2π‘œ + 1) diminished-1 adder, and 1Ξ” more than that of the companion modulo-(2π‘œ βˆ’ 3) adder ο‚΄ Same speed (synthesis result) for the proposed designs and those of the modulo-(2π‘œ βˆ’ 3) adders ο‚΄ Area and Power overhead reduces as π‘œ grows larger

21/22

slide-22
SLIDE 22

Thank You

jaberipur@sbu.ac.ir saha.moradi@mail.sbu.ac.ir