FPGA Implementations of Pairing using Residue Number ... · Question : Can we bridge the gap to...

MotivationsBackgrounds

Pairing Coprocessor DesignImplementation Results

Conclusions

FPGA Implementations of Pairing usingResidue Number System and Lazy Reduction

Ray Cheung 1 Sylvain Duquesne 2 Junfeng Fan 4

Nicolas Guillermin 2,3 Ingrid Verbauwhede 4 Gavin Yao1

1 Department of Electronic EngineeringCity University of Hong Kong, Hong Kong SAR,

2 IRMAR, UMR CNRS 6625, Université Rennes 1Campus de Beaulieu, 35042 Rennes cedex, France

3 DGA.IS, La Roche Marguerite - 35170 - Bruz, France4 ESAT/SCD-COSIC, Katholieke Universiteit Leuven

Kasteelpark Arenberg 10, B-3001 Leuven-Heverlee, Belgium

Oct, 2011CHES 2011, Nara FPGA Implementation of Pairing using RNS 1/24

Conclusions

Agenda

1 Motivations

2 Backgrounds

3 Pairing Coprocessor Design

4 Implementation Results

5 Conclusions

CHES 2011, Nara FPGA Implementation of Pairing using RNS 2/24

Conclusions

Motivations

Getting the fastest hardware implementationfor a 128 bit security Pairing

supersingular curves/small char. ordinary curves/big char.

faster hardware architecture faster software implementationsfrobenius embedding degreeparallelism smaller curves

Question : Can we bridge the gap to make ordinary curveswin?

Conclusions

Motivations

Conclusions

Motivations

faster hardware architecture

faster software implementations

frobenius

embedding degree

parallelism

smaller curves

Conclusions

Motivations

Conclusions

Motivations

Conclusions

Pairings and Barreto-Naehrig CurvesResidue number system (RNS)RNS Montgomery

Agenda

1 Motivations

2 Backgrounds

5 Conclusions

Conclusions

What is Pairing and how do we use it?A pairing is a bilinear map e : G1 �G2 Ñ GT with G1 �G2 and GT groupswith hard Discrete Logarithm

Useful for :

Three-party one-roundDiffie-Hellman keyagreement [Joux’00]

Identity-based encryption[Boneh�01]

Short signature[Boneh�01]

Blind signature[Boldyreva’03]

(a, ga)

(b, gb) (c, gc)

(a, ga)

(b, gb) (c, gc)

ga ↔

gb g a ↔

gb ↔ gc

(a, ga)

(b, gb) (c, gc)

ga ↔

gb g a ↔

gb ↔ gc

to B: gab

to C: gac

to A: gab

to C: gbcto A: gac

to B: gbc

(a, ga)

(b, gb) (c, gc)gb gc

(a, ga)

(b, gb) (c, gc)gb

e(gb, gc)a = e(g, g)abc

e(ga, gc)b = e(g, g)abc e(ga, gb)c = e(g, g)abc

Conclusions

Useful for :

(a, ga)

(b, gb) (c, gc)

(a, ga)

(b, gb) (c, gc)

ga ↔

gb g a ↔

gb ↔ gc

(a, ga)

(b, gb) (c, gc)

ga ↔

gb g a ↔

gb ↔ gc

to B: gab

to C: gac

to A: gab

to C: gbcto A: gac

to B: gbc

(a, ga)

(b, gb) (c, gc)gb

Conclusions

Useful for :

(a, ga)

(b, gb) (c, gc)

(a, ga)

(b, gb) (c, gc)

ga ↔

gb g a ↔

gb ↔ gc

(a, ga)

(b, gb) (c, gc)

ga ↔

gb g a ↔

gb ↔ gc

to B: gab

to C: gac

to A: gab

to C: gbcto A: gac

to B: gbc

(a, ga)

(b, gb) (c, gc)gb

Conclusions

Useful for :

(a, ga)

(b, gb) (c, gc)

(a, ga)

(b, gb) (c, gc)

ga ↔

gb g a ↔

gb ↔ gc

(a, ga)

(b, gb) (c, gc)

ga ↔

gb g a ↔

gb ↔ gc

to B: gab

to C: gac

to A: gab

to C: gbcto A: gac

to B: gbc

(a, ga)

(b, gb) (c, gc)gb

Conclusions

Useful for :

(a, ga)

(b, gb) (c, gc)

(a, ga)

(b, gb) (c, gc)

ga ↔

gb g a ↔

gb ↔ gc

(a, ga)

(b, gb) (c, gc)

ga ↔

gb g a ↔

gb ↔ gc

to B: gab

to C: gac

to A: gab

to C: gbcto A: gac

to B: gbc

(a, ga)

(b, gb) (c, gc)gb

Conclusions

Useful for :

(a, ga)

(b, gb) (c, gc)

(a, ga)

(b, gb) (c, gc)

ga ↔

gb g a ↔

gb ↔ gc

(a, ga)

(b, gb) (c, gc)

ga ↔

gb g a ↔

gb ↔ gc

to B: gab

to C: gac

to A: gab

to C: gbcto A: gac

to B: gbc

(a, ga)

(b, gb) (c, gc)gb

Conclusions

Useful for :

(a, ga)

(b, gb) (c, gc)

(a, ga)

(b, gb) (c, gc)

ga ↔

gb g a ↔

gb ↔ gc

(a, ga)

(b, gb) (c, gc)

ga ↔

gb g a ↔

gb ↔ gc

to B: gab

to C: gac

to A: gab

to C: gbcto A: gac

to B: gbc

(a, ga)

(b, gb) (c, gc)gb

Conclusions

Useful for :

(a, ga)

(b, gb) (c, gc)

(a, ga)

(b, gb) (c, gc)

ga ↔

gb g a ↔

gb ↔ gc

(a, ga)

(b, gb) (c, gc)

ga ↔

gb g a ↔

gb ↔ gc

to B: gab

to C: gac

to A: gab

to C: gbcto A: gac

to B: gbc

(a, ga)

(b, gb) (c, gc)gb

Conclusions

Barreto-Naehrig Curves

BN curve over Fp:y 2� x 3

� bwhere b � 0 such that #E � `

p � 36u4 � 36u3 � 24u2 � 6u � 1

` � 36u4 � 36u3 � 18u2 � 6u � 1

for u P Z and p, ` primes.

Very adapted for 128 bits security

Conclusions

Barreto-Naehrig Curves

BN curve over Fp:y 2� x 3

� bwhere b � 0 such that #E � `

p � 36u4 � 36u3 � 24u2 � 6u � 1

` � 36u4 � 36u3 � 18u2 � 6u � 1

for u P Z and p, ` primes.

Very adapted for 128 bits security

Conclusions

Pairing Parameter Selection

Chosen curves

Security u rlog2 ps

126-bit �p262 � 255 � 1q 254128-bit �p263 � 222 � 218 � 27 � 1q 258192-bit �p2160 � 274 � 212 � 1q 646

Conclusions

Optimal Ate Pairing on BN Curves

pairing

Tate [Frey+94]Ate [Hess+06]Optimal [Lee+08,Vercauteren'10]pairing

Miller's loopFinal exponentiation

Tate [Frey+94]Ate [Hess+06]Optimal [Lee+08,Vercauteren'10]

[Miller'04]

pairing

BN, Fp12, Fp2, Fp arithmetic

[Miller'04]

KaratsubaLazy reduction[Scott'08, Aranha+11]

pairing

Fp arithmetic (modular operations)

[Miller'04]

Barrett [Barrett'86]Montgomery [Montgomery'85]FVV [Fan+11]Blakley [Ghosh+10]RNS [Kawamura+00]

Conclusions

pairing

[Miller'04]

pairing

[Miller'04]

pairing

[Miller'04]

Conclusions

pairing

[Miller'04]

pairing

[Miller'04]

pairing

[Miller'04]

Conclusions

pairing

[Miller'04]

pairing

[Miller'04]

pairing

[Miller'04]

Conclusions

pairing

[Miller'04]

pairing

[Miller'04]

pairing

[Miller'04]

Conclusions

Residue Number System (RNS)

RNS is defined by n pairwise coprime integer constants:B � tb1,b2, � � � ,bnu.

MB :�±n

i�1 bi ,bi P B

Any integer X ,0 ¤ X MB, X is uniquely represented by:X � tX mod b1,X mod b2, � � � ,X mod bnu,

Arithmetic operations on RNS (Z{MBZ)

Normal RNSR � X � Y mod MB ri � xi � yi mod biR � X � Y mod MB ri � xi � yi mod biR � X {Y mod MB ri � xi y�1

i mod bi

only if gcdpY ,MBq � 1

Conclusions

MB :�±n

i�1 bi ,bi P B

i mod bi

Conclusions

MB :�±n

i�1 bi ,bi P B

i mod bi

Conclusions

RNS Montgomery reduction [Bajard�98]

RNS Montgomery – MB

Input: A � aMB mod p andB � bMB mod p

Output: T � abMB mod p

in B1: TB Ð ABBB

2: QB Ð TB � p�pq�1

3: QBBase Extension

ÝÝÝÝÝÝÝÝÝÝÝÝÝÑQC

3: SB Ð pT �QBpq{MB

5: SBBase Extension

ÐÝÝÝÝÝÝÝÝÝÝÝÝÝSC

gcdpMB,MBq � MB � 1M�1

B does not exist in B.

Conclusions

in B1: TB Ð ABBB

3: QBBase Extension

3: SB Ð pT �QBpq{MB

5: SBBase Extension

gcdpMB,MBq � MB � 1M�1

B does not exist in B.

Conclusions

in B in C1: TB Ð ABBB TC Ð ACBC

3: QBBase Extension

4: SC Ð pTC �QCpqpM�1B qC

5: SBBase Extension

Introduce a new base C to perform division by MB.

Needs the computation of Base Extension [Kawamura�00]

Conclusions

in B in C1: TB Ð ABBB TC Ð ACBC

3: QBBase Extension

4: SC Ð pTC �QCpqpM�1B qC

5: SBBase Extension

Introduce a new base C to perform division by MB.Needs the computation of Base Extension [Kawamura�00]

Conclusions

RNS complexity and Lazy reduction

RNS Montgomery

Multiplication :2n MUL

Reduction (RED):2n2 � 3n MUL

Conventional Montgomery

Multiplication:n2 MUL

Reduction:n2 � n MUL

AB mod p:2n2 � 5n MUL

AB mod p:2n2 � n MUL

AB � CD mod p:2n2 � 7n MUL

AB � CD mod p:3n2 � n MUL

°ki�1 Ai Bi mod p:

2n2 � p3 � 2k qn MUL

p1 � k qn2 � n MUL

Conclusions

RNS Montgomery

Conclusions

RNS Montgomery

Conclusions

Theoretical conclusions

Lazy reduction reduces the complexity of Pairings[Aranha�11]

RNS reduces the complexity of lazy reductionRNS involves easy parallelism

Remains to verify it "in real world"

Conclusions

RNS reduces the complexity of lazy reduction

RNS involves easy parallelism

Conclusions

Architectural Design IFurther OptimizationArchitectural Design II

Agenda

1 Motivations

2 Backgrounds

5 Conclusions

Conclusions

Cox-Rower Architecture [Kawamura�00]

Main featuren rowers

MUL: 2 cyclesRED: 2n+3 cycles

One rower,one channelMicrocodedsequencer

ControllerROM

(BRAM)

DATACTRL

ξ register

Conclusions

Rower Design

2 accumulatorsSmall-constant multiplier3-port RAMsSame data path

Multiplier

Constant Multiplier{2,3,4,6}

Acc0 Acc1

Accumulators

Channel Reduction

Conclusions

Rower Design

Underlying Field

Fp2 � Fpris{pi2 � 1q

Fp12 � Fp2rγs{pγ6 � p1 � iqq

px0 � x1iqpy0 � y1iqp1� iq�px0y0 � x1y1 � x0y1 � x1y0q�

px0y0 � x1y1 � x0y1 � x1y0qi

Multiplier

Acc0 Acc1

Accumulators

Channel Reduction

Conclusions

Rower Design

Underlying Field

px0 � x1iqpy0 � y1iqp1� iq

�px0y0 � x1y1 � x0y1 � x1y0q�

px0y0 � x1y1 � x0y1 � x1y0qi

Multiplier

Acc0 Acc1

Accumulators

Channel Reduction

Conclusions

Rower Design

Underlying Field

px0 � x1iqpy0 � y1iqp1� iq�px0y0 � x1y1 � x0y1 � x1y0q�

px0y0 � x1y1 � x0y1 � x1y0qi

Multiplier

Acc0 Acc1

Accumulators

Channel Reduction

Conclusions

Further Optimization

ObservationsComputational intensiveness: base extension.

Base extension: n2 multiplications by constant.Constant is determined by base B and C.

Constant size is 35 bits.With selected bases, bit-length: 35 Ñ 25.Bit-length of the constant is shortened.

Conclusions

ObservationsComputational intensiveness: base extension.Base extension: n2 multiplications by constant.

Constant is determined by base B and C.

Conclusions

ObservationsComputational intensiveness: base extension.Base extension: n2 multiplications by constant.Constant is determined by base B and C.

Conclusions

Constant size is 35 bits.

With selected bases, bit-length: 35 Ñ 25.Bit-length of the constant is shortened.

Conclusions

Constant size is 35 bits.With selected bases, bit-length: 35 Ñ 25.

Bit-length of the constant is shortened.

Conclusions

Dual mode 4-stage pipelined MUL

One 35�35multiplier

Two 35�25multipliers

@250MHzon Virtex-6

MUL: 2 cycles

RED: n+4 cycles(v.s. 2n+3 cycles)

35x25 35x2535x35

Conclusions

Dual mode 4-stage pipelined MUL

One 35�35multiplier

Two 35�25multipliers

@250MHzon Virtex-6

MUL: 2 cycles

RED: n+4 cycles(v.s. 2n+3 cycles)

35x25 35x2535x35

Conclusions

Comparison

Agenda

1 Motivations

2 Backgrounds

5 Conclusions

Conclusions

Comparison

Pairing Parameter Selection

Design I

Security u rlog2 ps126-bit �p262 � 255 � 1q 254128-bit �p263 � 222 � 218 � 27 � 1q 258192-bit �p2160 � 274 � 212 � 1q 646

Design II

126-bit �p262 � 255 � 1q 254

Conclusions

Comparison

Logic Utilization and Cycle Count

Logic utilizationDesign n Device Multipliers Logic Embedded

Elements Memory8 Cyclone II 35 18multipliers 14274 LC 67 M4k

I 8 Stratix III 72 DSP18el 4233 ALMs 1 M144k + 18 M9k(Altera) 19 Stratix III 171 DSP18el 9910 ALMs 1 M144k + 40 M9k

II 8 Virtex-6 32 DSP48E1s 7032 Slices 45 18Kb BRAMs(Xilinx)

Cycle count and Latency

Curve Cycles Technology Frequency Latency

Design I

BN126 176111 Cyclone II 91 MHz 1.93 msBN126 176111 Stratix III 165 MHz 1.07 msBN128 192502 Stratix III 165 MHz 1.16 msBN192 789849 Stratix III 131 MHz 6.02 ms

Design II BN126 143111 Virtex-6 250 MHz 0.57 ms

Conclusions

Comparison

Logic Utilization and Cycle Count

Logic utilizationDesign n Device Multipliers Logic Embedded

Elements Memory8 Cyclone II 35 18multipliers 14274 LC 67 M4k

I 8 Stratix III 72 DSP18el 4233 ALMs 1 M144k + 18 M9k(Altera) 19 Stratix III 171 DSP18el 9910 ALMs 1 M144k + 40 M9k

II 8 Virtex-6 32 DSP48E1s 7032 Slices 45 18Kb BRAMs(Xilinx)

Cycle count and Latency

Curve Cycles Technology Frequency Latency

Design I

BN126 176111 Cyclone II 91 MHz 1.93 msBN126 176111 Stratix III 165 MHz 1.07 msBN128 192502 Stratix III 165 MHz 1.16 msBN192 789849 Stratix III 131 MHz 6.02 ms

Design II BN126 143111 Virtex-6 250 MHz 0.57 ms

Conclusions

Comparison

Design Pairing/ Platform Algorithm Area Freq. Cycle DelaySecurity[bit] [MHz] [ms]

Design optimal ate Altera RNS 4233 ALMs165 176,111 1.07

I 126 (Stratix III) (Parallel) 72 DSPsDesign optimal ate Xilinx RNS 7032 slices

250 143,111 0.573II 126 (Virtex-6) (Parallel) 32 DSPs

Fan�11ate/128 Xilinx HMM 4014 slices

210336,366 1.60

opt. ate/128 (Virtex-6) (Parallel) 42 DSPs 245,430 1.17

Estibals’10Tate F35�97 Xilinx

-4755 Slices

192 428,853 2.23128 (Virtex-4) 7 BRAMs

Aranha�10opt. Eta F2367 Xilinx

- 4518 Slices 220 773,960� 3.52128 (Virtex-4)

Ghosh�11ηTF21223 Xilinx

- 15167 Slices 250 76,000� 0.19128 (Virtex-6)

Beuchat�10optimal ate Core

Montgomery - 2800 2,330,000 0.83126 i7

Aranha�11optimal ate Phenom

Montgomery - 3000 1,562,000 0.52126 II

Conclusions

ConclusionsQ&A

Conclusions

Conclusions1 Pairing using RNS + lazy reduction.2 Novel base selection specification.3 Hardware architectures.4 New speed record.

Conclusions

ConclusionsQ&A

Thank you!

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