var ches prez 1 · 2010. 8. 26. · title: var_ches_prez_1.cdr author: michal created date:...

New High Entropy Element for FPGA Based

True Random Number Generators

Michal Varchola and Milo Drutarovskš ý

Department of Electronics and Multimedia CommunicationsFaculty of Electrical Engineering and Informatics

šé š

Technical University of Ko icePark Komensk ho 13, 04120 Ko ice

Slovak Republic

The 12 Workshop on Cryptographic Hardwareand Embedded Systems (CHES 2010)UC Santa BarbaraAugust 20, 2010

th

Agenda

CHES 2010 - UC Santa Barbara, CA, USA, August 20, 2010

- Introduction

- New Entropy Element Design Goals

- Transition Effect Ring Oscillator (TERO)

- Mathematical Model of TERO

- Experimental Results

- Conclusions

2 / 26

IntroductionRandom Numbers are Essential...


Session Keys

Signature ParametersTemporary Keys

Challenges for AuthenticationZero Knowledge Protocols

NoncesGeneration of Primes

The TRNG of insufficient quality can weaken an

otherwise strong cryptographic system (see e.g. [1]).

...and therefore random numbers must be independent,

unpredictable, and must fulfill strict statistical properties.

True Random Number Generators (TRNGs) translate

a physical phenomena (e.g. thermal noise) to the random digits.

[1] Markettos, Moore:

, CHES 2009

The Frequency Injection Attack on Ring-Oscillator

Based True Random Number Generators

3 / 26



Session Keys











, CHES 2009

The Frequency Injection Attack on Ring-Oscillator


3 / 26



Session Keys











, CHES 2009

The Frequency Injection


Attack on Ring-Oscillator

3 / 26

IntroductionWhy Random Number Generators?True


“Any one who considers arithmetical methods

of producing random digits is, of course, in a state of sin.

For, as has been pointed out several times, there is no

such thing as a random number –

there are only methods to produce random numbers,

and a strict arithmetic procedure of course

is not such a method.”

John Von Neumann

The Research Challenge:

To discover such a random source and extraction method,

which can be reliably synthesized in modern electronic devices

such as Field Programmable Gate Arrays (FPGAs),

where cryptographic systems are usually implemeted.

4 / 26

IntroductionFPGA-based TRNG Randomness Sources & Designs


Timing Jitter:

ROs- of Ring Oscillators ( ):

- Two accompanied with LFSRs (Tkacik; CHES 2002; by Dichtl)

- Fibonacci and Galois Combined by XOR (Golic; Tran. on Comp. 2006)

- 114 combined by XOR (Sunar; Tran. on Comp.; by Dichtl)

- 20 combined by XOR - the modification of Sunar’s 114 ROs design

(Wold, Tan; Int. J. of Reconfig. Comp. 2009; by Fischer)

ROs compromised

ROs

ROs

ROs

compromised

compromised

RO attack- The general frequency injection (Markettos, Moore; CHES 2009)

(Fischer & Drutarovsky, CHES 2002)- of PLLs’ output

Metastability:

- Metastable Ring Oscillator ( et.al.; CHES 2008)

- Authors mentioned difficulties realated to implementation in FPGA hardware

Vasyltsov

the

5 / 26

IntroductionRing Oscillator Randomness Extraction Method


&

restart

output

Basic RO circuitry

6 / 26



Basic RO operation t1

output t1

restart t1

&

restart

output

Basic RO circuitry

6 / 26



&

restart

output

Basic RO-based TRNG

Basic RO operation t1

DFF rnd. bit

sample

output t1

restart t1

sample t1

rnd. bit t1Randomness

Extractor

sampling

6 / 26



&

restart

output

Basic RO operation t1, t2

DFF rnd. bit

sample

output t2

rnd. bit t2

output t1

restart t1,t2

sample t1,t2

rnd. bit t1Randomness

Extractor

Basic RO-based TRNG

sampling

6 / 26



Pros ROsof the Popular :

- Employs standard logic cells

Easy synthesis in FPGAs and ASICs-

&

restart

output


DFF rnd. bit

sample

output t2

rnd. bit t2

output t1

restart t1,t2

sample t1,t2

rnd. bit t1

Basic RO-based TRNG

sampling

6 / 26



Pros ROs

Cons ROs

of the Popular :

of the Popular :

- Employs standard logic cells

Easy synthesis in FPGAs and ASICs

- Low entropy rate (acquired from

internal noise processes in RO electronic components)

- Strong dependence on working conditions and external perturbation

- RO can synchronize on parallel operating ROs or on perturbation

-

&

restart

output


DFF rnd. bit

sample

output t2

rnd. bit t2

output t1

restart t1,t2

sample t1,t2

rnd. bit t1

Basic RO-based TRNG

sampling

6 / 26

Agenda


- Introduction




- Conclusions


7 / 26

New Entropy Element Design Goals


New

Entropy

Element

entropy rate

higher than RO

8 / 26



entropy rate

higher than RO sensitivity on

global interference

and working cond.

lower than RO

New

Entropy

Element

8 / 26



entropy rate


global interference

and working cond.

lower than RO

provable

ability to extract

the intrinsic noise

of logic elements

New

Entropy

Element

8 / 26



entropy rate


global interference

and working cond.

lower than RO

provable

ability to extract

the intrinsic noise

of logic elements

New

Entropy

Element

acceptable

math. model for

wide scientific

community

8 / 26



entropy rate


global interference

and working cond.

lower than RO

provable

ability to extract

the intrinsic noise

of logic elements

acceptable

math. model for

wide scientific

communityfeature of

inner testability

for the instant

evaluation

low number

of logic elements

in its structurecombination of

multiple known

rnd. extraction

methods

New

Entropy

Element

8 / 26



entropy rate


global interference

and working cond.

lower than RO

provable

ability to extract

the intrinsic noise

of logic elements

acceptable

math. model for

wide scientific

communityfeature of

inner testability

for the instant

evaluation

stateless

entropy

concept

parallel

operation

low number

of logic elements

in its structure

straightforward

place & route

strategies

combination of

multiple known

rnd. extraction

methods

New

Entropy

Element

8 / 26

Agenda


- Introduction




- Conclusions


9 / 26

&

restart (rst)

output

Transition Effect Ring Oscillator (TERO)TERO Circuitry and Operation in FPGA Hardware


Starting from simple RO...

10 / 26



AND1

output

rst

...two inverters are replaced by buffers, still the same circuit behavior...

10 / 26



AND

XOR AND22

1XOR1

‘0’

‘1’

output

Normal Ring

Oscillator Operation

(RO Mode)

rst

...buffer and inverter are replaced by XORs, which act as andbuffer inverter,

second buffer is replaced by AND (in order to have the loop symmetric)...

10 / 26



rst

ctrlAND

XOR AND22

1XOR1 AND

XOR AND22

1XOR1

output

Transition Effect Ring

Oscillator Operation

(TERO Mode)

...both XORs can be switched from inverter to buffer logic function simultaneously

what can cause oscillations in the circuit if the feedback path is long enough ...

10 / 26



Isolated TERO loop

Internal

terout

Single CLB Implementation

clr

tffout

tffout2

rst

ctrl INVAND

XOR AND22

1 1XOR1

INV2

INV3

INV4

INV5

TFF

TFF2

Isolated TERO loop

AND

XOR AND22

1XOR1

Neighboring Logic Isolation(in order to prevent routing of

internal TERO loop signals

outside the CLB )

Randomness Extractor(TFF resolves whether TERO

made even or odd number

of oscillations)

10 / 26



ctrl

rst

terout

tffout

nrstT = 3200 ns

ctrlT = 4000 nsrnd. bit gener.start

rnd. bit gener. finish

TEROoscillation

randombit

TFF is clearedVariation

of s# oscillation

Isolated TERO loop

Internal

terout

Single CLB Implementation

clr

tffout

tffout2

rst

ctrl INVAND

XOR AND22

1 1XOR1

INV2

INV3

INV4

INV5

TFF1

TFF2

Single CLB Implementation in Xilinx Spartan 3E FPGA and the Oscilloscope Screenshot

Isolated TERO loop

AND

XOR AND22

1XOR1

tffout2INV2 TFF2

10 / 26

Transition Effect Ring Oscillator (TERO)TERO Circuitry Simulation ResultLT Spice


1 sμ 3 μs

3.02 μs 3.10 μs0 V

0 V

2.5 V

3.06 μs

2 μs

ST MT

2.5 V

T - width of pulse

when the oscillations

are started

S T - width of pulse


disappear

M

terout

ctrl edge ctrl edge

TT ≈ 5 ns

T - period of a

single TERO

oscillation

T

11 / 26

Transition Effect Ring Oscillator (TERO)TERO Circuitry Simulation ResultLT Spice


The raised pulse plays a crucial role...shortening of

1 sμ 3 μs

3.02 μs 3.10 μs0 V

0 V

2.5 V

3.06 μs

2 μs

ST TT ≈ 5 ns MT

2.5 V

T - width of pulse


are started

S T - width of pulse


disappear

MT - period of a

single TERO

oscillation

T

terout

ctrl edge ctrl edge

1T 2T

T - each oscillation

(average) pulse

shortening factor

D

DT T T= -1 2

DT ≈ 10 ps T ≈ 80 psD

11 / 26

Transition Effect Ring Oscillator (TERO)TERO vs. RO comparison using VHDL Macro-Model


VHDL Macro-Model of TERO (and RO - when ctrl = ‘1’ for XOR and ctrl = ‘0’ for XOR )1 2

wire CNT

results file

stim

uli

noise file 1 noise file 2 noise file 3 noise file 4

clr

rst

ctrl

smp

wire wire wireXOR AND XOR AND2211

12 / 26


VHDL Macro-Model of TERO (and RO - when ctrl = ‘1’ for XOR and ctrl = ‘0’ for XOR )1 2

0 50 100143

148

153

TE

RO

#oscs.

σ= 1ps, b= 0ps

0 50 100

110

150

190

σ= 10ps, b= 50ps

0 50 100146

152

158

σ= 0.5ps, b= 50ps

0 100

319320321

ctrl period

RO

#oscs.

σ= 1ps, b= 0ps

0 100

310320330

ctrl period

σ= 10ps, b= 50ps

0 100

310320330

ctrl period

σ= 0.5ps, b= 50ps

wire CNT

results file

stim

uli

noise file 1 noise file 2 noise file 3 noise file 4

clr

rst

ctrl

smp

wire wire wireXOR AND XOR AND2211

The ModelSim simulation results of the VHDL structure for both TERO and RO modes

12 / 26

Transition Effect Ring Oscillator (TERO)TERO vs. RO comparison using VHDL Macro-Model

Agenda


- Introduction




- Conclusions


13 / 26

Mathematical Model of TERO


“Remember that all models are wrong;

the practical question is how wrong do they

have to be to not be useful.”

George E. P. Box

14 / 26

Mathematical Model of TERO


“Remember that all models are wrong;

the practical question is how wrong do they

have to be to not be useful.”

George E. P. Box

We need the useful mathematical model in order to:

- show that “randomness” relies on the physical phenomena

- determine how much “randomness” is available

- fulfill TRNG evaluation criteria

- persuade a community on the reliability of a random number

generation method

14 / 26

Mathematical Model of TEROTERO Mathematical Model Definition


Σ ΣYTj

i =1

( )Δ +T = T Y + ΔTij DT T =-S M

i =1

YTj

D Tj Tij

N(0, )σ 2

TD T T-S M ctrl

stopcondition

start &reset

noise model

ΣYTj

i =1

sum the inputuntil reaching the stopcondition: = -

each periodΣ T T

ctrl(comb filter)

Y ,Y Y,..., ,...T1 T2 Tj

ΔT2( -1)j

sumationinput

TT TM

ΔTij

TS

TS

TD

Tctrl

ΔTΣ ij

lastpulse

firstpulse

ctrl

terout

S M

Δ +TTij D

Tctrl j -1Tctrl j

TT1TT2

TTiTTi +1

T - width of pulse


are started

S

T - width of pulse


disappear

M


(average) pulse

shortening factor

D

Δ period jitter or

instability of

shortening factor

-Tij

Y - number of

oscillations done

each -th ctrl periodj

Tj

T - period of a

single TERO

oscillation

T

15 / 26



Σ ΣYTj

i =1

( )Δ + = T Y +T ΔTij DT T =-S M

i =1

YTj

D Tj Tij

TD T T-S M ctrl

stopcondition

start &reset

ΣYTj

i =1


each periodΣ T T

ctrl(comb filter)

Y ,Y Y,..., ,...T1 T2 Tj

ΔT2( -1)j

sumationinput

TT TM TS

TS

TD

Tctrl

ΔTΣ ij

lastpulse

firstpulse

ctrl

terout

S M

Δ +TTij D

Tctrl j -1Tctrl j

TT1TT2

TTiTTi +1

T - width of pulse


are started

S

T - width of pulse


disappear

M

D

Δ period jitter or

instability of

shortening factor

-Tij

Y - number of

oscillations done


Tj

T - period of a

single TERO

oscillation

T


(average) pulse

shortening factor

N(0, )σ 2

noise model

ΔTij

15 / 26


Σ ΣYTj

i =1

( )Δ + = T Y +T ΔTij DT T =-S M

i =1

YTj

D Tj Tij

TD T T-S M ctrl

stopcondition

start &reset

ΣYTj

i =1


each periodΣ T T

ctrl(comb filter)

Y ,Y Y,..., ,...T1 T2 Tj

ΔT2( -1)j

sumationinput

TT TM TS

TS

TD

Tctrl

ΔTΣ ij

lastpulse

firstpulse

ctrl

terout

S M

Δ +TTij D

Tctrl j -1Tctrl j

TT1TT2

TTiTTi +1

T - width of pulse


are started

S

T - width of pulse


disappear

M

D

Δ period jitter or

instability of

shortening factor

-Tij

Y - number of

oscillations done


Tj

T - period of a

single TERO

oscillation

T



(average) pulse

shortening factor

N(0, )σ 2

noise model

ΔTij

15 / 26


oscillationsY oscillationsYTj-1 Tj

Σ ΣYTj

i =1

( )Δ +T = T Y + ΔTij DT T =-S M

i =1

YTj

D Tj Tij

TD T T-S M ctrl

stopcondition

start &reset

ΣYTj

i =1


each periodΣ T T

ctrl(comb filter)

Y ,Y Y,..., ,...T1 T2 Tj

ΔT2( -1)j

sumationinput

TT TM TS

TS

TD

Tctrl

ΔTΣ ij

lastpulse

firstpulse

ctrl

terout

S M

Δ +TTij D

Tctrl j -1Tctrl j

TT1TT2

TTiTTi +1

T - width of pulse


are started

S

T - width of pulse


disappear

M

D

Δ period jitter or

instability of

shortening factor

-Tij

Y - number of

oscillations done


Tj

T - period of a

single TERO

oscillation

T



(average) pulse

shortening factor

N(0, )σ 2

noise model

ΔTij

=Y YTj-1 Tj

15 / 26


Y ≈T

T T-S M

TD

mean value of

number of oscs.

in TERO mode

std. deviation of

number of oscs.

in TERO mode

where aσ is std. deviation of period jitter

σ ≈Y

σ

TDT

YT

Mathematical Model of TEROTERO Mathematical Model Practical Impact

16 / 26


Y ≈T

T T-S M

TD

Y ≈R

Tnrst

2TT

mean value of

number of oscs.

in RO mode

where time when RO is oscillatingT isnrst


std. deviation of

number of oscs.

in RO mode

σ ≈Y

σ

TDT

YT

σ ≈Y

σ

2TTR

YR


mean value of

number of oscs.

in TERO mode

std. deviation of

number of oscs.

in TERO mode

16 / 26


Y ≈T

T T-S M

TD

Y ≈R

Tnrst

2TT



σ ≈Y

σ

TDT

YT

σ ≈Y

σ

2TTR

YR

~ ps ~ ns


mean value of

number of oscs.

in RO mode

std. deviation of

number of oscs.

in RO mode

mean value of

number of oscs.

in TERO mode

std. deviation of

number of oscs.

in TERO mode

16 / 26


Y ≈T

T T-S M

TD

Y ≈R

Tnrst

2TT


σYT

σYR

≈2TT

TD

2TT ( )T T-S M

TD Tnrst

≈ 100 500 !!!~


σ ≈Y

σ

TDT

YT

σ ≈Y

σ

2TTR

YR

~ ps ~ ns

≈


mean value of

number of oscs.

in RO mode

std. deviation of

number of oscs.

in RO mode

mean value of

number of oscs.

in TERO mode

std. deviation of

number of oscs.

in TERO mode

16 / 26


Y ≈T

T T-S M

TD

Y ≈R

Tnrst

2TT


σYT

σYR

≈2TT

TD

2TT ( )T T-S M

TD Tnrst


σ ≈Y

σ

TDT

YT

σ ≈Y

σ

2TTR

YR

~ ps ~ ns

140 160 1800

5000

occure

nce

TE

RO

320 325 3300

5

10x 10

4

TD

= 13ps

σ= 10ps

# oscs.

occure

nce

RO

20 22 240

5

10x 10

4

320 325 3300

5

10x 10

4

TD

= 100ps

σ= 10ps

# oscs.

15 30 450

5000

320 325 3300

5

10x 10

4

TD

= 100ps

σ= 150ps

# oscs.

≈

Matlab simulation of TERO and RO math. models


mean value of

number of oscs.

in RO mode

std. deviation of

number of oscs.

in RO mode

mean value of

number of oscs.

in TERO mode

std. deviation of

number of oscs.

in TERO mode

≈ 100 500 !!!~

16 / 26

Agenda


- Introduction




- Conclusions


17 / 26

Experimental ResultsSynthesis of the Evaluation Platform #1


TEROch. A

Asynch.counter

ctrl rst clr, ,

TEROch. B

Asynch.counter

1

1

Xilinx Spartan 3E FPGA

12

Contr

ol

sta

tem

achin

e

3

12

27USBchip

Com-puter

USB2.0

Evaluation Platform implemented in Xilinx Spartan 3E FPGA

18 / 26



TEROch. A

Asynch.counter

ctrl rst clr, ,

TEROch. B

Asynch.counter

1

1


12

Contr

ol

sta

tem

achin

e

3

12

27USBchip

Com-puter

USB2.0

TERO ch. A TERO ch. B

CLB


TERO Place & Route detail,

where each channel is implemented

in separated CLB

18 / 26



TEROch. A

Asynch.counter

ctrl rst clr, ,

TEROch. B

Asynch.counter

1

1


12

Contr

ol

sta

tem

achin

e

3

12

27USBchip

Com-puter

USB2.0

TERO ch. A TERO ch. B

CLB


TERO Place & Route detail,

where each channel is implemented

in separated CLB

The FPGA Fabric

A B

A

B

Two mutual TERO compositions:

and used in

practical experiments

Next Diagonal

18 / 26

Experimental ResultsResults of the Evaluation Platform #1 (1 of 2)


0 50 100

220270330 TERO ch.A

0 50 100

115130145

TE

RO

#o

scs.

TERO ch.B

0 100315330345

ctrl periodRO

#o

scs.

RO both ch.

The of number of oscillations

for both and

comparison

modesTERO RO

TERO mode:

- higher variance

- uncorrelated

RO mode:

- lower variance

- correlated !!!

19 / 26



0 50 100

220270330 TERO ch.A

0 50 100

115130145

TE

RO

#o

scs.

TERO ch.B

0 100315330345

ctrl periodRO

#o

scs.

RO both ch.

0 50 100-4

04 LSB TERO ch.A

0 50 100-4

04

TE

RO

no

rm.

au

toco

rr.

LSB TERO ch.B

0 100-4

04

ctrl period shift

LSB(ch.A XOR ch.B)

0 50 100-800

0800 LSB RO ch.A

0 50 100-800

0800

RO

no

rm.

au

toco

rr.

LSB RO ch.B

0 100-800

0800

ctrl period shift

LSB(ch.A XOR ch.B)

The of normalized autocorrelation

of number of oscillations for both and

comparison

modes(LSB) TERO RO

Repetitive

pattern !!!

for RO mode

No repetitive

pattern

for TERO mode

19 / 26

0 50 100

220270330 TERO ch.A

0 50 100

115130145

TE

RO

#o

scs.

TERO ch.B

0 100315330345

ctrl periodRO

#o

scs.

RO both ch.

0 50 100-4

04 LSB TERO ch.A

0 50 100-4

04

TE

RO

no

rm.

au

toco

rr.

LSB TERO ch.B

0 100-4

04

ctrl period shift

LSB(ch.A XOR ch.B)

0 50 100-800

0800 LSB RO ch.A

0 50 100-800

0800

RO

no

rm.

au

toco

rr.

LSB RO ch.B

0 100-800

0800

ctrl period shift

LSB(ch.A XOR ch.B)



The of normalized autocorrelation

of number of oscillations (LSB) for both and

comparison

modesTERO RO

n d- -1

n d-2( )ΣX = s s n d2 + - / -i i d+( )

i=1

X - should approximately follow (0,1)

- should fall into 3 l: <-3,3>

N

σ interva

Normalized autocorrellation is expressed as:X

s

d d

n n

- random bit sequence

- number of shifts (1 < < 100)

- number of bits ( = 1Mbit)

Repetitive

pattern !!!

for RO mode

No repetitive

pattern

for TERO mode

19 / 26



0 50 100

220270330 TERO ch.A

0 50 100

115130145

TE

RO

#o

scs.

TERO ch.B

0 100315330345

ctrl periodRO

#o

scs.

RO both ch.

0 50 100-4

04 LSB TERO ch.A

0 50 100-4

04

TE

RO

no

rm.

au

toco

rr.

LSB TERO ch.B

0 100-4

04

ctrl period shift

LSB(ch.A XOR ch.B)

0 50 100-800

0800 LSB RO ch.A

0 50 100-800

0800

RO

no

rm.

au

toco

rr.

LSB RO ch.B

0 100-800

0800

ctrl period shift

LSB(ch.A XOR ch.B)

XOR combination of A and B

and channelsTERO RO

19 / 26


CHES 2010 - UC Santa Barbara, CA, USA, August 20, 2010 20 / 26



mean value

improved by XOR

combination

20 / 26

mean value

worsened by XOR

combination !!!



TEROs are uncorelated ROs are correlated !!!

The outputs of The outputs of

20 / 26

mean value

improved by XOR

combination

mean value

worsened by XOR

combination !!!



NIST 800-22 and

FIPS 140-2 results

improved by

XOR combination

FIPS 140-2 tests

fail completely !!!



20 / 26

mean value

improved by XOR

combination

mean value

worsened by XOR

combination !!!



FIPS 140-2 tests

fail completely !!!Random sequences

are independent

(with high probability)



20 / 26

mean value

improved by XOR

combination

mean value

worsened by XOR

combination !!!

NIST 800-22 and

FIPS 140-2 results

improved by

XOR combination



NAND1 INV1 INV11

NAND2 INV2 INV22

INV TFF

BUF ctrl

tffout

Isolated TERO loop

terout

Actel Fusion Impl.

TERO 1

TERO 2

TERO 16

ACNT 1

ACNT 2

ACNT 16

12

12

12

ctr

l

ControlFSM

USBChip

PCwork-

stationTarget FPGA

27

USB 2.0

TERO structure adapted for Actel Fusion FPGA

Evaluation Platform implemented in Actel Fusion FPGA

21 / 26



Histograms of number of TERO oscillations of all 16 TERO channels; Placement #1

Histograms of number of TERO oscillations of all 16 TERO channels; Placement #2

22 / 26



V = 1210mVT = +20°C

CORE V = 1240mVT = +20°C

CORE V = 1 0mVT = +20°C

30COREV = 1 0mV

T = +20°C35CORE

V = 1 0mVT = +20°C

40CORE V = 1 0mVT = +20°C

45COREV = 1 0mV

T = +20°C50CORE V = 1 0mV

T = +20°C55CORE

V = 1 0mVT = +20°C

60CORE V = 1 0mVT = +20°C

65CORE V = 12 0mVT = 0°C

5-1

CORE V = 1 0mVT = 0°C

50-1

CORE

V = 1 0mVT = +2°C

50COREV = 1 0mV

T = + °C5035

CORE V = 1 0mVT = + 0°C

507

CORE V = 1 0mVT = + °C

50110

CORE

Histograms of number of TERO oscillations of a single TERO channel

under variously violated working conditions

23 / 26

Agenda


- Introduction





- Conclusions

24 / 26

Conclusion


- New entropy element (TERO) introduced

- High sensitivity to random processes inside

FPGA logic cells, while rejecting global perturbation

- Inner testability - counting of number of oscillations

- Basic mathematical model was introduced

- Two XOR-combined TERO channels can pass

the NIST 800-22 statistical tests

- Speed from 100kpbs to 250 kbps per single TRNG

- Stateless entropy concept

- Performance of TEROs cluster appears to be

independent from the position in FPGA and from

implementation in another identical FPGA board,

and from working conditions

25 / 26

Conclusion















25 / 26

Acknowledgment


“We would like to thank

Actel University Program

for a donation of 10 Actel Fusion

FPGA evaluation boards,

which enabled us to

confirm the TERO principle using

number of identical FPGA boards.”

26 / 26

Thank You

for the Attention

var ches prez 1 · 2010. 8. 26. · title: var_ches_prez_1.cdr author: michal created date:...

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