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

Modulo-( 2 𝑜 + 3 ) Parallel Prefix Addition via Diminished-3 Representation of Residues Authors: Ghassem Jaberipur, Sahar Moradi Cherati Arith 26 Kindly presented by: Paulo Sérgio Alves Martins

Contents 2/22

3/22 I NTRODUCTION  Popular τ = 2 𝑜 − 1,2 𝑜 , 2 𝑜 + 1  Residue number systems (RNS)  ℛ = {𝑛 1 , 𝑛 2 … 𝑛 𝑙 } , 𝑛 𝑗 (1 ≤ 𝑗 ≤ 𝑙)  General form: 𝑙  𝑁 = 𝑗=1 𝑛 𝑗 2 𝑜 ± δ 1 ≤ δ < 2 𝑜−1  Applications  Cryptography  digital signal/image processing Parallel prefix modulo- ( 2 𝑜 − δ ) adders: Delay  High Speed  RNS Features 3 + 2 log 𝑜 Δ  δ = 1  Low Power  δ = 3 [ Jaberipur,2015 ] 4 + 2 log 𝑜 Δ  δ = 2 𝑟 + 1 [ Langroudi,2015 ] 5 + 2 log 𝑜 Δ 𝑌, 𝑍 ∈ ℛ 𝑌 = 𝑦 1 , 𝑦 2 … , 𝑦 𝑙 , 𝑍 = (𝑧 1 , 𝑧 2 … , 𝑧 𝑙 ) ,  2 𝑜 + δ = 2 𝑜+1 − δ′ where 𝑦 𝑗 = 𝑌 𝑛 𝑗 , 𝑧 𝑗 = 𝑍 𝑛 𝑗 , where 1 < δ ′ = 2 𝑜 − δ < 2 𝑜 𝑎 = 𝑌 ⊛ 𝑍 , 𝑨 𝑗 = 𝑦 𝑗 ⊛ 𝑧 𝑗 , where ⊛∈ {+, −,×} No direct fast solution

D IMINISHED -1 ADDERS 4/22 Diminished-1 encoding Diminished-1 addition 𝑇 = 𝐵 + 𝐶 2 𝑜 +1 = 𝑇 ′ + 𝑨 𝑇 , 𝐵 = 𝐵 ′ + 𝑨 𝐵 , 𝐶 = 𝐶 ′ + 𝑨 𝐶 𝑌 : A modulo- (2 𝑜 + 1) residue ∈ 0,2 𝑜 𝑇 ′ = 𝑇 − 1 , 𝐵 ′ = 𝐵 − 1 , 𝐶 ′ = 𝐶 − 1 for 𝑇, 𝐵, 𝐶 > 0 𝑌 = 𝑌 ′ + 𝑨 𝑌 , where 𝑌 ′ = 𝑌 − 1 ∈ 0,2 𝑜 − 1 for 𝑌 > 0 𝑨 𝑇 , 𝑨 𝐵 , 𝑨 𝐶 : zero-indicator bits 𝑨 𝑌 = 0(1) , if and only if 𝑌 = 0(> 0) 𝐵 ′ + 𝐶 ′ = 2 𝑜 𝑥 𝑜 ′ + 𝑋 ′ = 𝑥 𝑜−1 𝑨 𝑇 = (𝑨 𝐵 ∨ 𝑨 𝐶 ) ∧ 𝑨 𝐵 𝑨 𝐶 ∧ 𝜊 𝑋 ′ , where ′ ′ … 𝑥 0 𝑇 ′ = 𝐵 ′ + 𝐶 ′ + 𝑨 𝐵 + 𝑨 𝐶 − 𝑨 𝑇 2 𝑜 +1 𝑇 ′ = 𝑇 − 1 = 𝐵 + 𝐶 2 𝑜 +1 − 1 𝑋 ′ + 𝑨 𝐵 𝑨 𝐶 𝑥 𝑜 = ′ 2 𝑜 = 𝐵 ′ + 1 + 𝐶 ′ + 1 − 1 2 𝑜 +1 = 2 𝑜 𝑥 𝑜 ′ + 𝑋 ′ + 1 2 𝑜 +1 ′ = 𝐻 𝑜−1:0 𝑥 𝑜 𝑋 ′ + 1 − 𝑥 𝑜 𝑋 ′ + 𝑥 ′𝑜 = 2 𝑜 +1 = ′ 𝜊 = 𝑄 𝑜−1:0 𝐻 𝑜−1:0 = 1 , Iff 𝐵 ′ + 𝐶 ′ = 2 𝑜 − 1

Related Work: Modulo-( 𝟑 𝒐 + 𝟐 ) D1 adder 5/22 RPP TPP

D IMINISHED -3 REPRESENTATION AND ADDITION

7/22 D IMINISHED -3 REPRESENTATION {0, 1, 2} 𝑌: Modulo− 2 𝑜 + 3 residue ∈ [0, 2 𝑜 + 2] [3, 2 𝑜 + 2] 𝑌 ∈ { 0, 1, 2} -indicator 𝑈 𝑌 = 𝑢 1 𝑢 0 𝑈 D3 𝑌 = 𝑌′ + 𝑈 Representation 𝑌 𝑌 ′ ∈ [0, 2 𝑜 − 1] 𝑌 ′ ∶ 𝑜 bit 𝑌 = 0 ⟺ 𝑌 = 0, 𝑌 ′ = 0 𝑈 𝑌 = 1 ⟺ 𝑌 = 1, 𝑌 ′ = 0 𝑈 𝑌 = 2 ⟺ 𝑌 = 2, 𝑌 ′ = 0 𝑈 𝑌 = 3 ⟺ 3 ≤ 𝑌 ≤ 2 𝑜 + 2, 𝑌 ′ ∈ [0,2 𝑜 − 1] 𝑈

Modulo-( 𝟑 𝒐 + 𝟒 ) D3 ADDITION 8/22 𝐵 ∈ [0, 2 𝑜 + 2] 𝐵 = 𝐵 ′ + 𝑈 … 𝐵′ = 𝒃 𝒐−𝟐 𝒃 𝟑 𝒃 𝟐 𝒃 𝟏 𝐵 𝐶 ′ = … 𝒄 𝒐−𝟐 𝒄 𝟑 𝒄 𝟐 𝒄 𝟏 𝐶 ∈ [0, 2 𝑜 + 2] 𝐶 = 𝐶 ′ + 𝑈 𝐶 ′ … ′ ′ ′ ′ 𝑥 𝒐 𝑥 𝒐−𝟐 𝑥 2 𝑥 1 𝑥 0 𝐵, 𝐶, 𝑇 ≥ 3 𝑇 ′ = 𝑇 − 3 = 𝐵 + 𝐶 2 𝑜 +3 − 3 = 𝐵 ′ + 3 + 𝐶 ′ + 3 − 3 2 𝑜 +3 = 2 𝑜 𝑥 𝑜 ′ + 𝑋 ′ + 3 2 𝑜 +3 𝑋 ′ + 3 1 − 𝑥 𝑜 𝑋 ′ + 3𝑥 ′𝑜 = 2 𝑜 +3 = ′

Comparison D1 and D3 9/22 𝟔 + 𝟑 𝒎𝒑𝒉 𝒐 𝜠 𝟒 + 𝟑 𝒎𝒑𝒉 𝒐 𝜠

Detect Special Cases 10/22 𝑻 = 𝑩 + 𝑪 𝟑 𝒐 +𝟒 ∈ {𝟏 ,1,2} 𝑬𝟐 IF 𝐵 ′ + 𝐶 ′ = 2 𝑜 − 1 THEN ξ 1 = 1 ELSE ξ 1 = 0 𝑻 = 𝑩 + 𝑪 𝟑 𝒐 +𝟐 = 𝟏 IF 𝐵 ′ + 𝐶 ′ = 2 𝑜 − 2 THEN ξ 2 = 1 ELSE ξ 2 = 0 IF 𝐵 ′ + 𝐶 ′ = 2 𝑜 − 1 THEN ξ = 1 ELSE ξ = 0 IF 𝐵 ′ + 𝐶 ′ = 2 𝑜 − 3 THEN ξ 3 = 1 ELSE ξ 3 = 0 ξ 1 = 𝑄 𝑜−1:2 𝐻 𝑜−1:2 ℎ 1 𝑣 0 ξ 2 = 𝑄 𝑜−1:2 𝐻 𝑜−1:2 ℎ 1 𝑣 0 ξ 3 = 𝑄 𝑜−1:2 𝐻 𝑜−1:2 𝑣 1 𝑣 0

D ERIVATION OF 𝑈 11/22 𝑇 𝒧 = 𝑄 𝑜−1:2 𝐻 𝑜−1:2 , ξ 1 = 𝒧ℎ 1 𝑣 0 , ξ 2 = 𝒧ℎ 1 𝑣 0 , ξ 3 = 𝒧𝑣 1 𝑣 0  σ 1 = α 1 β 1 ∨ α 0 β 0 𝒧 ∨ 𝑔 1 (α 1 , β 1 , α 0 , β 0 , 𝑣 1 , 𝑤 1 , 𝑣 0 )  σ 0 = α 1 β 0 ∨ α 0 β 1 𝒧 ∨ 𝑔 0 α 1 , β 1 , α 0 , β 0 , 𝑣 1 , 𝑤 1 , 𝑣 0

Impact of ξ -dependent noise terms on 𝑼 𝑻 12/22 𝑼 𝑩 𝑼 𝑪 𝛐 𝟐 𝛐 𝟑 𝛐 𝟒 𝑼 𝑻 𝑻 Justification 𝑇 = 𝐵 + 𝐶 = 𝐵 ≤ 2 𝑜 + 2 ≥ 3 3 0 X X X 3 𝑈 𝑇 = 3 𝑼 𝑩 𝑼 𝑪 𝛐 𝟐 𝛐 𝟑 𝛐 𝟒 𝑼 𝑻 𝑻 Justification 𝐵 ′ + 𝐶 ′ = 𝐵 ′ < 2 𝑜 − 1 ⟹ 𝑇 = 𝐵 + 𝐶 = 𝐵 ′ + 3 + 1 < 2 𝑜 + 3 ≥ 4 3 1 0 X X 3 𝐵 ′ + 𝐶 ′ = 2 𝑜 − 1 ⟹ 𝐵 + 𝐶 = 2 𝑜 + 3, 𝑇 = 𝐵 + 𝐶 2 𝑜 +3 = 0 3 1 1 X X 0 0 𝑈 𝑇 = 3 𝜊 1 𝑼 𝑩 𝑼 𝑪 𝛐 𝟐 𝛐 𝟑 𝛐 𝟒 𝑼 𝑻 𝑻 Justification 𝐵 ′ + 𝐶 ′ = 𝐵 ′ < 2 𝑜 − 2 ⟹ 𝑇 = 𝐵 + 𝐶 = 𝐵 ′ + 3 + 2 < 2 𝑜 + 3 ≥ 5 3 2 0 0 X 3 𝐵 ′ + 𝐶 ′ = 2 𝑜 − 2 ⟹ 𝐵 + 𝐶 = 2 𝑜 + 3, 𝑇 = 𝐵 + 𝐶 2 𝑜 +3 = 0 3 2 0 1 X 0 0 𝐵 ′ + 𝐶 ′ = 2 𝑜 − 1 ⟹ 𝐵 + 𝐶 = 2 𝑜 + 4, 𝑇 = 𝐵 + 𝐶 2 𝑜 +3 = 1 3 2 1 0 X 1 1 𝑈 𝑇 = 𝜊 1 + 3 𝜊 1 𝜊 2 𝑼 𝑩 𝑼 𝑪 𝛐 𝟐 𝛐 𝟑 𝛐 𝟒 𝑼 𝑻 𝑻 Justification 𝐵 ′ + 𝐶 ′ < 2 𝑜 − 3 ⟹ 𝑇 = 𝐵 + 𝐶 = 𝐵 ′ + 𝐶′ + 6 < 2 𝑜 + 3 ≥ 6 3 3 0 0 0 3 2 𝑜 ≤ 𝐵 ′ + 𝐶 ′ ≤ 2 𝑜 + 2 𝑜 − 2 ⟹ 3 ≤ 𝑇 = 𝐵 + 𝐶 2 𝑜 +3 ≤ 2 𝑜 + 1 3 3 0 0 0 3 ≥ 3 𝐵 ′ + 𝐶 ′ = 2 𝑜 − 3 ⟹ 𝐵 + 𝐶 = 2 𝑜 + 3, 𝑇 = 𝐵 + 𝐶 2 𝑜 +3 = 0 3 3 0 0 1 0 0 𝐵 ′ + 𝐶 ′ = 2 𝑜 − 2 ⟹ 𝐵 + 𝐶 = 2 𝑜 + 4, 𝑇 = 𝐵 + 𝐶 2 𝑜 +3 = 1 3 3 0 1 0 1 1 𝐵 ′ + 𝐶 ′ = 2 𝑜 − 1 ⟹ 𝐵 + 𝐶 = 2 𝑜 + 5, 𝑇 = 𝐵 + 𝐶 2 𝑜 +3 = 2 3 3 1 0 0 2 2 𝑈 𝑇 = 2 𝜊 1 + 𝜊 2 + 3 𝜊 1 𝜊 2 𝜊 3

13/22 D ERIVATION OF 𝑇′ 𝑇 ′ = 𝐵 ′ + 𝑈 𝐵 + 𝐶 ′ + 𝑈 𝐶 − 𝑈 𝑇 2 𝑜 +3 = 𝐵 ′ + 𝐶 ′ + 𝑈 𝐵 + 𝑈 𝐶 − 𝑈 𝑇 2 𝑜 +3 ′ + 𝑋 ′ + 𝛆′ 2 𝑜 +3 , 𝛆 ′ = 𝑈 − 3𝑥 𝑜 𝑋 ′ + 𝑈 2 𝑜 +3 = 𝑋 ′ + 𝑈 − 3𝑥 𝑜 = 2 𝑜 𝑥 𝑜 ′ 2 𝑜 +3 = ′ 𝑼 𝑪 0 1 2 3 𝑼 𝑩 0 0 0 0 0 0 0 0 3 𝜊 1 + 1 1 0 0 1 2 𝜊 1 + 3 𝜊 2 + 2 2 0 3 𝜊 1 + 1 2 𝜊 1 + 3 𝜊 2 + 2 𝜊 1 + 2 𝜊 2 + 3 𝜊 3 + 3 3 T HE NOISE TERM 𝑈 = 𝑈 𝐵 + 𝑈 𝐶 − 𝑈 𝑇 IN TERMS OF 𝑈 𝐵 , 𝑈 𝐶 , AND ξ BITS

14/22 C OMPOUND RPP REALIZATION OF 𝑻′ ′ δ 0 ′ ∈ {0,1,2} δ ′ = δ 1 ′ + 𝛆′ 𝑿 ′ + 𝒜𝒙 𝒐 𝑻′ = 𝟑 𝒐 ′ = ξ 1 α 1 β 1 α 0 ⨁β 0 ∨ ξ 1 ′ , δ 1 ξ 2 𝑨𝑥 𝑜 ′ = ξ 1 𝑦 ∨ ξ 2 𝑨 ∨ α 0 β 0 α 1 ⨁β 1 ∨ α 1 β 1 α 0 ∨ β 0 δ 0 𝑨 = α 1 β 1 α 0 β 0

15/22 The required RPP circuitry 𝑏 7 𝑐 7 𝑏 6 𝑐 6 𝑏 2 𝑐 2 𝑏 1 𝑐 1 𝑏 0 𝑐 0 𝑏 5 𝑐 5 𝑏 4 𝑐 4 𝑏 3 𝑐 3 ( g , p ) ( g , p ) ( g , p ) ( g , p ) ( g , p ) 𝑣 𝑗 𝑤 𝑗 l l r r l l r r i i HA HA HA HA HA HA HA HA pgh 𝑣 7 𝑤 7 𝑣 6 𝑤 6 𝑣 5 𝑤 5 𝑣 4 𝑤 4 𝑣 3 𝑤 3 𝑣 2 𝑤 2 𝑣 1 𝑤 1 𝑣 0 𝑤 8   h ( g , p ) ( g p g , p p ) ( g p g ) ( g , p ) 𝑕 1 , 𝑞 1 i i i l l r l r l l r i i pgh pgh pgh pgh pgh pgh pgh 𝑸 𝟖 : 𝟑 𝑸 𝟖 : 𝟑 (4 Δ ) 𝑟 12 𝑟 02 (3 Δ ) (6 Δ ) 𝑟 03 𝑸 𝟖 : 𝟑 𝑸 𝟖 : 𝟑 𝑟 13 (4 Δ ) 𝟖 + 𝟑 𝒎𝒑𝒉 𝒐 𝜠 𝑟 01 (4 Δ ) 𝑣 0 (4 Δ ) 𝑧 𝑟 11 (5 Δ ) α 1 β 1 α 1 β 0 α 0 β 1 α 0 β 0 𝑯 𝟖 : 𝟑 𝑯 𝟖 : 𝟑 (3 Δ ) 𝑦 (𝒉 𝟖 : 𝟑 , 𝒒 𝟖 : 𝟑 ) 𝑯 𝟖 : 𝟑 𝑸 𝟖 : 𝟑 𝑧 (4 Δ ) 𝑑 2 𝑑 6 𝑑 5 𝑑 4 𝑑 3 𝑑 7 𝑑 1 𝑑 0 𝑔 0 (7 Δ ) 𝑔 1 (7 Δ ) σ 1 σ 0 ′ ′ ′ ′ ′ ′ ′ ′ 𝑡 1 𝑡 0 𝑡 7 𝑡 4 𝑡 2 𝑡 6 𝑡 5 𝑡 3

16/22 Carry Bits for RPP Architecture ′ ∨ 𝑞 1 ′ ∨ 𝑞 1 ′ 𝐻 𝑜−1:2 = 𝐻 𝑗−1:2 ∨ 𝑄 𝑗−1:2 𝑕 1 ′ 𝐻 𝑜−1:𝑗 𝑑 𝑗 = 𝐻 𝑗−1:2 ∨ 𝑄 𝑗−1:2 𝑑 2 = 𝐻 𝑗−1:2 ∨ 𝑄 𝑗−1:2 𝑕 1 ′ ∨ 𝑞 1 ′ ∘ 𝐻 7:2 , 1 ′ 𝐻 7:2 , 𝑕 1 ′ , 𝑞 1 𝑑 2 = 𝑕 1 ′ ∨ 𝑞 1 ′ ∘ 𝐻 7:3 , 1 ′ 𝐻 7:3 , 𝑕 2 , 𝑞 2 ∘ 𝑕 1 ′ , 𝑞 1 𝑑 3 = 𝑕 2 ∨ 𝑞 2 𝑕 1 ′ ∨ 𝑞 1 ′ 𝐻 7:4 , (𝐻, 𝑄) 3:2 ∘ (𝑕 1 ′ , 𝑞 1 ′ ) ∘ 𝐻 7:4 , 1 𝑑 4 = 𝐻 3:2 ∨ 𝑄 3:2 𝑕 1 ′ ∨ 𝑞 1 ′ 𝐻 7:5 , 𝐻, 𝑄 4:2 ∘ (𝑕 1 ′ , 𝑞 1 ′ ) ∘ 𝐻 7:5 , 1 𝑑 5 = 𝐻 4:2 ∨ 𝑄 4:2 𝑕 1 ′ ∨ 𝑞 1 ′ 𝐻 7:6 , 𝐻, 𝑄 5:2 ∘ (𝑕 1 ′ , 𝑞 1 ′ ) ∘ 𝐻 7:6 , 1 𝑑 6 = 𝐻 5:2 ∨ 𝑄 5:2 𝑕 1 ′ ∨ 𝑞 1 ′ 𝑕 7 , 𝐻, 𝑄 6:2 ∘ (𝑕 1 ′ , 𝑞 1 ′ ) ∘ 𝑕 7 , 1 𝑑 7 = 𝐻 6:2 ∨ 𝑄 6:2 𝑕 1