Fun with Certificates part II - Institute for Advanced Study · Fun with Certificates part II Elliptic Curve Cryptography May 13, 2019 Network Security Institute for Advanced Study

Fun with Certificates part IIElliptic Curve Cryptography

May 13, 2019

Network SecurityInstitute for Advanced Study

Fun with Certificates part IIa Deep Dive into Elliptic Curve Cryptography for all ages

Brian Epstein(he/him/his)

Institute for Advanced Study

Computer Manager, Network and Security

Information Security Officer

[email protected] - @epepepep

mailto:[email protected]

https://security.ias.edu 3


May 13, 2019


Topics● Explain why ECC came about● ECC deep dive● Safe Curves and Trust● Certs● Demo



May 13, 2019


So I was browsing the Interwebs...

5

6



May 13, 2019


26%

1%

64%

7%1%

no SSL

RSA 1024 bit

RSA 2048 bit

RSA 4096 bit

ECC 256 bit

.edu's taken from Majestic's top 1 million websites (3096 total)

2017 TLS Certificate Breakdown for Edu's



May 13, 2019


13%

0%

78%

6%

3%

0%

no SSL

RSA 1024 bit

RSA 2048 bit

RSA 4096 bit

ECC 256 bit

ECC 384 bit

.edu's taken from Majestic's top 1 million websites (4008 total)

2018 TLS Certificate Breakdown for Edu's



May 13, 2019


Why create ECC, we have RSA?

● If RSA breaks, what then?● Faster computers force increased key size● Speed is faster with ECC (for most things)



May 13, 2019


Key Length Comparison



May 13, 2019


Elliptic Curve Cryptography

Neal Koblitz

Victor Miller

1985



May 13, 2019


Elliptic Curve Cryptography (ECC)

● Explain the end goal for ECC● Review a little math● Show how to get to our end goal

So, let's begin at the end...



May 13, 2019


Secret Exchange

Where can wetalk privately??I have an

idea . . .



May 13, 2019




May 13, 2019


dingo (2,38)

gazelle (21,31)

stallion (17,15)

donkey (30,35)

iguana (19,18)

jackal (15,40)

orangutan (33,14)

goat (38,15)

mongoose (32,29)

rat (28,19)

deer (27,26)

cow (10,38)



May 13, 2019


dingo(2,38)

gazelle

(21,31)

stallio

n

(17,15)

donke

y

(30,

35)

chipmunk(2,3)

mule(21,10)alligator(17,26)

jack

al(1

5,4

0)

igua

na(1

9,18

)

ferret

(19,2

3)

boar

(30,6)

ora

ng

uta

n(3

3,1

4)

goat

(38,1

5)

mong

oose

(32,2

9)

rat

(28

,19)

deer

(27,2

6)

cow

(10

,38)

hip

po

(29

,3)

meeka

t(6

,34

)

musk

rat

(34,3

5)

cou

gar

(37

,5)

wom

bat

(22

,16)

newt(35,11)

gibbon

(25,33)

opossum

(18,35)

panda

(26,28)

llama(5,0)

sloth (35,30) pa

rake

et(2

5,8)

reindee

r(3

7,3

6)

dormouse

(22,25)

hed

geh

og

(26,1

3)

chim

panz

ee

(18,

6)

cat

(34,6

)

an

teate

r(6

,7)

oce

lot

(29,3

8)

beaver

(10,3

)

pon

y(2

7,1

5)

porcu

pin

e(2

8,2

2)

squ

irrel

(32,1

2)

koala

(38

,26)

an

telo

pe

(33,2

7)

chicke

n(1

5,1

)



May 13, 2019


dingo(2,38)

gazelle

(21,31)

stallio

n

(17,15)

donke

y

(30,

35)

chipmunk(2,3)


jack

al(1

5,4

0)

igua

na(1

9,18

)

ferret

(19,2

3)

boar

(30,6)

ora

ng

uta

n(3

3,1

4)

goat

(38,1

5)

mong

oose

(32,2

9)

rat

(28

,19)

deer

(27,2

6)

cow

(10

,38)

hip

po

(29

,3)

meeka

t(6

,34

)

musk

rat

(34,3

5)

cou

gar

(37

,5)

wom

bat

(22

,16)

newt(35,11)

gibbon

(25,33)

opossum

(18,35)

panda

(26,28)

llama(5,0)

sloth (35,30) pa

rake

et(2

5,8)

reindee

r(3

7,3

6)

dormouse

(22,25)

hed

geh

og

(26,1

3)

chim

panz

ee

(18,

6)

cat

(34,6

)

an

teate

r(6

,7)

oce

lot

(29

,38)

beaver

(10,3

)

pon

y(2

7,1

5)

porcu

pin

e(2

8,2

2)

squ

irrel

(32,1

2)

koala

(38

,26)

an

telo

pe

(33,2

7)

chicke

n(1

5,1

)

G = dingo (2,38)

d = ?

Q = wombat (22,16)

r = ?

R = panda (26,28)



May 13, 2019


dingo(2,38)

gazelle

(21,31)

stallio

n

(17,15)

donke

y

(30,

35)

chipmunk(2,3)


jack

al(1

5,4

0)

igua

na(1

9,18

)

ferret

(19,2

3)

boar

(30,6)

ora

ng

uta

n(3

3,1

4)

goat

(38,1

5)

mong

oose

(32,2

9)

rat

(28

,19)

deer

(27,2

6)

cow

(10

,38)

hip

po

(29

,3)

meeka

t(6

,34

)

musk

rat

(34,3

5)

cou

gar

(37

,5)

wom

bat

(22

,16)

newt(35,11)

gibbon

(25,33)

opossum

(18,35)

panda

(26,28)

llama(5,0)

sloth (35,30) pa

rake

et(2

5,8)

reindee

r(3

7,3

6)

dormouse

(22,25)

hed

geh

og

(26,1

3)

chim

panz

ee

(18,

6)

cat

(34,6

)

an

teate

r(6

,7)

oce

lot

(29

,38)

beaver

(10,3

)

pon

y(2

7,1

5)

porcu

pin

e(2

8,2

2)

squ

irrel

(32,1

2)

koala

(38

,26)

an

telo

pe

(33,2

7)

chicke

n(1

5,1

)

G = dingo (2,38)

d = ?

Q = wombat (22,16)

r = ?

R = panda (26,28)



May 13, 2019


dingo(2,38)

gazelle

(21,31)

stallio

n

(17,15)

donke

y

(30,

35)

chipmunk(2,3)


jack

al(1

5,4

0)

igua

na(1

9,18

)

ferret

(19,2

3)

boar

(30,6)

ora

ng

uta

n(3

3,1

4)

goat

(38,1

5)

mong

oose

(32,2

9)

rat

(28

,19)

deer

(27,2

6)

cow

(10

,38)

hip

po

(29

,3)

meeka

t(6

,34

)

musk

rat

(34,3

5)

cou

gar

(37

,5)

wom

bat

(22

,16)

newt(35,11)

gibbon

(25,33)

opossum

(18,35)

panda

(26,28)

llama(5,0)

sloth (35,30) pa

rake

et(2

5,8)

reindee

r(3

7,3

6)

dormouse

(22,25)

hed

geh

og

(26,1

3)

chim

panz

ee

(18,

6)

cat

(34,6

)

an

teate

r(6

,7)

oce

lot

(29

,38)

beaver

(10,3

)

pon

y(2

7,1

5)

porcu

pin

e(2

8,2

2)

squ

irrel

(32,1

2)

koala

(38

,26)

an

telo

pe

(33,2

7)

chicke

n(1

5,1

)

G = dingo (2,38)

d = ?

Q = wombat (22,16)

r = ?

R = panda (26,28)



May 13, 2019


dingo(2,38)

gazelle

(21,31)

stallio

n

(17,15)

donke

y

(30,

35)

chipmunk(2,3)


jack

al(1

5,4

0)

igua

na(1

9,18

)

ferret

(19,2

3)

boar

(30,6)

ora

ng

uta

n(3

3,1

4)

goat

(38,1

5)

mong

oose

(32,2

9)

rat

(28

,19)

deer

(27,2

6)

cow

(10

,38)

hip

po

(29

,3)

meeka

t(6

,34

)

musk

rat

(34,3

5)

cou

gar

(37

,5)

wom

bat

(22

,16)

newt(35,11)

gibbon

(25,33)

opossum

(18,35)

panda

(26,28)

llama(5,0)

sloth (35,30) pa

rake

et(2

5,8)

reindee

r(3

7,3

6)

dormouse

(22,25)

hed

geh

og

(26,1

3)

chim

panz

ee

(18,

6)

cat

(34,6

)

an

teate

r(6

,7)

oce

lot

(29

,38)

beaver

(10,3

)

pon

y(2

7,1

5)

porcu

pin

e(2

8,2

2)

squ

irrel

(32,1

2)

koala

(38

,26)

an

telo

pe

(33,2

7)

chicke

n(1

5,1

)

G = dingo (2,38)

d = ?

Q = wombat (22,16)

r = ?

R = panda (26,28)

S = mule (21,10)



May 13, 2019


G = dingo (2,38)

d = ?

Q = wombat (22,16)

r = ?

R = panda (26,28)

S = mule (21,10)

d r

+ r+ d = = SWhy does this work?

d = 16 r = 25

25 + 16 = 16 + 25 = 41



May 13, 2019


Humpf,how romantic...

I should'vepicked a better

number.



May 13, 2019


Math● Square and Square root● Graphing● Elliptic Curves with point math● Finite Fields



May 13, 2019


Square and Square Root

√9=3

√9=−3√9=±3

32=3⋅3=9

(−3)2=−3⋅−3=9

32=3⋅332

(−3)2=−3⋅−3(−3)2



May 13, 2019


Graphing

0 1 2 3 40

2

4

6

8

10

12

14

16

18

1

x

y

y=x2+1

x yx2+1

0

1

2

34

5

02+1

1017

26

0•0+10+11

12+11•1+11+12

22+12•2+14+15

0 1 2 3 40

2

4

6

8

10

12

14

16

18

12

x

y

0 1 2 3 40

2

4

6

8

10

12

14

16

18

12

5

x

y

0 1 2 3 40

2

4

6

8

10

12

14

16

18

12

5

10

x

y

0 1 2 3 40

2

4

6

8

10

12

14

16

18

12

5

10

17

x

y

0 1 2 3 4 50

5

10

15

20

25

30

12

5

10

17

26

x

y



May 13, 2019


Elliptic Curves


May 13, 2019


An Elliptical Machine



May 13, 2019


Elliptic Curvesy2= x3−x+3{(x , y )∈ℝ2∣y2= x3+ax+b ,4 a3+27 b2≠0}∪{0}



May 13, 2019


Elliptic Curve Math● Create “point addition” ⊕

P⊕Q⊕R=0

P⊕Q=-R● Create “point multiplication” ⊙

2⊙P = P⊕P

5⊙P = P⊕P⊕P⊕P⊕P● Demo D

32

33

34

35

36

37

38

39

40

41

42

43

44

45



May 13, 2019


0 1 2 3 40

2

4

6

8

10

12

14

16

18

12

5

10

17

x

y

x x2+1

0

1

2

34

5

1017

26

0

2

5

Graphing



May 13, 2019


Finite Fields● Finite

– There is an end● Field

– Football– Soccer

● Demo A



May 13, 2019


Benefits from Finite Fields● computers are terrible at irrational

numbers● get to use whole numbers (integers)● reduce the size of the problem● Field is “closed”



May 13, 2019


Example Finite Field● Field size is 41● x axis goes from 0 to 40● y axis goes from 0 to 40



May 13, 2019


Graphing an EC on a Finite Field



May 13, 2019


0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930313233343536373839400

5

10

15

20

25

30

35

40

x

y

y2(mod 41)≡ x3−x+3 (mod 41)x



May 13, 2019


0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930313233343536373839400

5

10

15

20

25

30

35

40

x

y

2 3;

y2(mod 41)≡ x3−x+3 (mod 41)x



May 13, 2019


0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930313233343536373839400

5

10

15

20

25

30

35

40

x

y

2 3; 38

y2(mod 41)≡ x3−x+3 (mod 41)x



May 13, 2019


0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930313233343536373839400

5

10

15

20

25

30

35

40

x

y

2

5

3; 38

0

y2(mod 41)≡ x3−x+3 (mod 41)x



May 13, 2019


0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930313233343536373839400

5

10

15

20

25

30

35

40

x

y

2

6

3; 38

7; 34

5 0

y2(mod 41)≡ x3−x+3 (mod 41)x



May 13, 2019


0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930313233343536373839400

5

10

15

20

25

30

35

40

x

y

2

610

3; 38

7; 343; 38

5 0

y2(mod 41)≡ x3−x+3 (mod 41)x



May 13, 2019


0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930313233343536373839400

5

10

15

20

25

30

35

40

x

y

2

610

1517

18

1; 4015; 26

6; 35

3; 38

7; 343; 38

5 0

y2(mod 41)≡ x3−x+3 (mod 41)x

... …



May 13, 2019


0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930313233343536373839400

5

10

15

20

25

30

35

40

x

y

2

610

1517

18

1; 4015; 26

6; 35

3; 38

7; 343; 38

5 0; 41

y2(mod 41)≡ x3−x+3 (mod 41)x

... …



May 13, 2019


0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930313233343536373839400

5

10

15

20

25

30

35

40

x

y

2

610

1517

18

1; 4015; 26

6; 35

3; 38

7; 343; 38

5 0

y2(mod 41)≡ x3−x+3 (mod 41)x

... …



May 13, 2019


0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930313233343536373839400

5

10

15

20

25

30

35

40

x

y

2

610

1517

18

1; 4015; 26

6; 35

3; 38

7; 343; 38

5 0

y2(mod 41)≡ x3−x+3 (mod 41)x

... …



May 13, 2019


Point Addition ⊕● Draw a line between points P and Q● Flip over at the sides, keep your slope● When you hit the next point, flip to

opposite side of the graph



May 13, 2019


0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930313233343536373839400

5

10

15

20

25

30

35

40

x

y

2

610

1517

18

1; 4015; 26

6; 35

3; 38

7; 343; 38

5 0

y2(mod 41)≡ x3−x+3 (mod 41)x

... …

P

Q

R

-R

P ⊕ Q = -RP ⊕ Q ⊕ R = 0



May 13, 2019


One way function

X



May 13, 2019


One way function● Point addition ⊕ and multiplication ⊙ are easy● Point subtraction ⊖ and division ⊘ are hard● Given R, what are P & Q?



May 13, 2019


0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930313233343536373839400

5

10

15

20

25

30

35

40

x

y

2

610

1517

18

1; 4015; 26

6; 35

3; 38

7; 343; 38

5 0

y2(mod 41)≡ x3−x+3 (mod 41)x

... …R

P + Q = -R

P + Q + R = 0

-R

P



May 13, 2019


Point Multiplication ⊙



May 13, 2019


2 ⊙ (2,38) =(2,38) ⊕ (2,38) =(21,31)



May 13, 2019


3 ⊙ (2,38) =(2,38) ⊕ ((2,38) ⊕ (2,38)) =(2,38) ⊕ (21,31) =(17,15)



May 13, 2019


4 ⊙ (2,38) =(2,38) ⊕ ((2,38) ⊕ ((2,38) ⊕ (2,38))) =(2,38) ⊕ ((2,38) ⊕ (21,31)) =(2,38) ⊕ (17,15) =(30,35)



May 13, 2019




May 13, 2019




May 13, 2019




May 13, 2019


dingo(2,38)

gazelle

(21,31)

stallio

n

(17,15)

donke

y

(30,

35)

chipmunk(2,3)


jack

al(1

5,4

0)

igua

na(1

9,18

)

ferret

(19,2

3)

boar

(30,6)

ora

ng

uta

n(3

3,1

4)

goat

(38,1

5)

mong

oose

(32,2

9)

rat

(28

,19)

deer

(27,2

6)

cow

(10

,38)

hip

po

(29

,3)

meeka

t(6

,34

)

musk

rat

(34,3

5)

cou

gar

(37

,5)

wom

bat

(22

,16)

newt(35,11)

gibbon

(25,33)

opossum

(18,35)

panda

(26,28)

llama(5,0)

sloth (35,30) pa

rake

et(2

5,8)

reindee

r(3

7,3

6)

dormouse

(22,25)

hed

geh

og

(26,1

3)

chim

panz

ee

(18,

6)

cat

(34,6

)

an

teate

r(6

,7)

oce

lot

(29

,38)

beaver

(10,3

)

pon

y(2

7,1

5)

porcu

pin

e(2

8,2

2)

squ

irrel

(32,1

2)

koala

(38

,26)

an

telo

pe

(33,2

7)

chicke

n(1

5,1

)



May 13, 2019


And they lived happily ever after...

… until Dual_EC_DRBG



May 13, 2019


Dual Elliptic Curve Deterministic Random Bit Generator (Dual_EC_DRBG)● Developed prior to 2004 by NSA

● Approved by NIST

● RSA Security used as default

● Bruce Schneier concluded it weak

● Edward Snowden leaks included documents revealing plot by NSA



May 13, 2019


Is ECC compromised then?● No, but we have some trust issues.● ANSI X9.62 (1999), IEEE P1363 (2000)?● SEC 2 (2000), NIST FIPS 186-2 (2000)?● ANSI X9.63 (2001), Brainpool (2005)?● NSA Suite B (2005)?● ANSSI FRP256V1 (2011)?



May 13, 2019


SafeCurves

● Choosing safe curves for elliptic-curve cryptography

● https://safecurves.cr.yp.to/



May 13, 2019


Million Dollar ECC curve

● Publicly verifiable randomness produced in February 2016 by many national lotteries

● http://cryptoexperts.github.io/million-dollar-curve/



May 13, 2019


And they lived happily ever after...

Hello? It's Eve here.Did you forget about me?

I heard everything!



May 13, 2019




May 13, 2019


https://safecurves.cr.yp.to/



May 13, 2019


(Jimmy)nkwwm

http://cryptoexperts.github.io/million-dollar-curve/

http://cryptoexperts.github.io/million-dollar-curve/



May 13, 2019


nkwwm



May 13, 2019


nkwwmJimmy (Jimmy)orqql



May 13, 2019


Jimmy orqql



May 13, 2019


Jimmy orqqlJimmy



May 13, 2019


Jimmy Jimmy

(got it)ldg jg



May 13, 2019


Jimmy Jimmy

ldg jg



May 13, 2019


Jimmy Jimmy

ldg jggot it(got it)tes fs



May 13, 2019


Jimmy Jimmy

got ittes fs



May 13, 2019


Jimmy Jimmy

got ittes fsgot it



May 13, 2019


RSA Certificates● Subject (FQDN)● Issuer (CA)● Public Key

● Modulus (n) product of two prime numbers● Public Exponent (e)

● x509 extensions● Certificate Authority Signature

2008-05-29 Fun with Certificates




May 13, 2019


ECC Certificates● Subject (FQDN)● Issuer (CA)● Public Key

● Curve● Generator (start)● Public x,y coordinate

● x509 extensions● Certificate Authority Signature

98

RSA ECC



May 13, 2019


RSA Private Keys● Private Key

● Modulus (n) product of two prime numbers (p*q)● Public Exponent (e)● Private Exponent (d)● Prime1 (p)● Prime2 (q)




May 13, 2019


ECC Private Keys● Private Key

● Private number (how many steps)● Public x,y coordinate● Public Generator (starting point)● Curve

102

RSA ECC



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May 13, 2019


Trust● Public Key Infrastructure (PKI)

● Certificate Authority (CA) i.e. notary● Intermediate Certificate● Client Certificate

● Web of Trust



May 13, 2019


Public Key Infrastructure(PKI)

Web of Trust



May 13, 2019


PKI● Why do we trust CAs?

● time consuming vetting process● regularly audited● $$$● bundled with product● certificate revocation



May 13, 2019


Structure● Root CA

● self signed● Intermediate certificate● Server certificate



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May 13, 2019


Who provides the CA certificate, the client or the server?

The client.

...the intermediate certificate?

The server. (or it should)



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May 13, 2019


Getting Your Cert Signed● Internal Certificate Authority● Commercial Certificate Authority

● Be a reseller ($12/yr, $119/yr wildcard)● inCommon for .edu’s ($2k-$20k/yr)

● https://www.incommon.org/certificates/



May 13, 2019


Semi-primes

41 * 43 = easy!1763

1739 = 47 * 37 difficult

1791904897 = 49943 * 35879

170122668341587273458646411386585043888873643113298660753168823105496218048396254258389541689798276387535036676575062116463749217204880781486238521463801806647717753763762209533452596443765433132839199250997874070119227832756249288919712152428105344288137338378592441098310151010596800002333954751873349228763 * 143685366445138003711595402594806625836106895764255994658099545498390517894693472991085893832864915801761970155763201096759761623694012072299292478856561357050062892354466628960025947611851554780658080196114743327960874693198902680721554877864174333388893106637708514607610834750473283277858418617695308935563 = 24444137941285645379511684911299365678423833046448779381238796084162536046797899019234205442218213499926991297229281024701278950648068677702332885730383357978977040184484121175079987603694398742376695650950853277837222494281038135867022877083226479856395867447419772143605903245226717018069307504429199930327344784767917383283267106133917174472280561457908186415882389738067587305825291144415722855157890883871648649466532813832921881732883942736314267482744271752456430649004239402313393638372879487394870568428620598721555293620836002747794896212943069775576590434653324242136440479444891894641015313209968513198569

hard, I need a computer!

Really hard, I need a super computer and a couple of millennia!



May 13, 2019


Breaking semi-primes● Brute force● Sieve methods (slightly better)● Rainbow table

● What if we stored all 174 bit primes on micro-SD cards?



May 13, 2019


100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001001101100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010100101100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010111111100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000110000101100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000110101111100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001100111111100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001110001101100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010001111101100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010011111011100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010100010011100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010100111001100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010111000011100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011000001111100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011011100001100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011101000001100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011110000101.........................................................................

11972621413014756705924586149611790497021399392059337≅ 1.2*1052

23945242826029513411849172299223580994042798784118783≅ 2.4*1052

100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001001101

111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111101

Let's store every 174 bit prime number!



May 13, 2019




May 13, 2019


How many?# primes = π(x) = x/ln(x)

x1 = 11972621413014756705924586149611790497021399392059392

x2 = 23945242826029513411849172299223580994042798784118783

π(x2) – π(x

1) ≅ 9.87*1049 primes

0.005 g 1pb microsd

solar mass 1.9891*1030 kg

174 bitsprime

1 byte8 bits

1 kb 1024 bytes

1 mb 1024 kb

1 gb 1024 mb

1 tb 1024 gb

1 pb 1024 tb

1 kg 1000 gx

x x x x

x x x x =

x9.87*1049 primes

4.8 solar masses ≅ ???



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May 13, 2019




May 13, 2019


Cert Lab



May 13, 2019


Wrap-up● Cryptography● RSA overview● Explain why ECC came about● ECC deep dive● Safe Curves and Trust● Certs



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Fun with Certificates part II - Institute for Advanced Study · Fun with Certificates part II Elliptic Curve Cryptography May 13, 2019 Network Security Institute for Advanced Study

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