How To Create A Signature Scheme For A Smart Card With A Signature On A Twisted Edwards Curve (Eclipse) On A Powerbook (Powerbook) On An Iphone Or Ipad (Powerwolf) On The Powerbook 2 (

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1 Smart card implementation of a digital signature scheme for Twisted Edwards curves By Niels Duif

2 Remark Ask questions PAGE 1

3 Structure Signature schemes Smart cards Elliptic curves My signature scheme Side channel attacks Improvements New findings PAGE 2

4 Smart card Tamper resistant Cheap Limited capacity 64 kb EEPROM 1536 bytes RAM PAGE 3

5 Smart card Tamper resistant Cheap Limited capacity 64 kb EEPROM 1536 bytes RAM PAGE 4

6 Smart card Tamper resistant Cheap Limited capacity 64 kb EEPROM 1536 bytes RAM PAGE 5

7 Signature schemes PAGE 6

8 Signature schemes Authentication PAGE 7

9 Analog signatures Authentication PAGE 8

10 Analog signatures Authentication PAGE 9

11 Analog signatures Authentication PAGE 10

12 Analog signatures Authentication PAGE 11

13 Analog signatures Authentication Integrity PAGE 12

14 Digital signatures PAGE 13

15 Digital signatures Input: message (m) Output: signature (S) PAGE 14

16 Digital signatures Input: message (m) Output: signature (S) Authentication Only Niels Duif can sign the message Integrity Changing the message is detected PAGE 15

17 Digital signatures Niels Duif PAGE 16

18 Digital signatures Niels Duif Secret key k Niels PAGE 17

19 Digital signatures Niels Duif Secret key k Niels = 37 PAGE 18

20 Digital signatures Niels Duif Secret key k Niels = 37 Mathematics Public key Q niels = PAGE 19

21 Digital signatures Niels Duif Secret key k Niels = 37 Mathematics Public key Q niels = PAGE 20

22 Digital signatures Niels Duif Secret key k Niels = 37 Mathematics Public key Q niels = PAGE 21

23 Sign Niels Duif Eindhoven, Dear all, Today there will be free drinks for everyone at hours at Schootsestraat 65, Eindhoven. Kind regards, Niels Duif PAGE 22

24 Sign Niels Duif Secret key k Niels = 37 Eindhoven, Dear all, Today there will be free drinks for everyone at hours at Schootsestraat 65, Eindhoven. Kind regards, Niels Duif PAGE 23

25 Sign Niels Duif Secret key k Niels = 37 Eindhoven, Dear all, Today there will be free drinks for everyone at hours at Schootsestraat 65, Eindhoven. Kind regards, Niels Duif PAGE 24

26 Sign Niels Duif Secret key k Niels = 37 Eindhoven, Dear all, Today there will be free drinks for everyone at hours at Schootsestraat 65, Eindhoven. Kind regards, Niels Duif Signature: E1 B8 AD 2F 12 BF 3D F8 C3 B F A A 9E 0D 68 2F 8B BF 1A 49 F6 15 AE 14 EC 9F 5A F F 23 4E F EA F 0B CD 7F 31 CE AC D4 PAGE 25

27 Verify Someone Eindhoven, Dear all, Today there will be free drinks for everyone at hours at Schootsestraat 65, Eindhoven. Kind regards, Niels Duif Signature: E1 B8 AD 2F 12 BF 3D F8 C3 B F A A 9E 0D 68 2F 8B BF 1A 49 F6 15 AE 14 EC 9F 5A F F 23 4E F EA F 0B CD 7F 31 CE AC D4 PAGE 26

28 Verify Someone Public key Q niels = Eindhoven, Dear all, Today there will be free drinks for everyone at hours at Schootsestraat 65, Eindhoven. Kind regards, Niels Duif Signature: E1 B8 AD 2F 12 BF 3D F8 C3 B F A A 9E 0D 68 2F 8B BF 1A 49 F6 15 AE 14 EC 9F 5A F F 23 4E F EA F 0B CD 7F 31 CE AC D4 PAGE 27

29 Verify Someone Public key Q niels = Eindhoven, Dear all, Today there will be free drinks for everyone at hours at Schootsestraat 65, Eindhoven. Kind regards, Niels Duif Signature: E1 B8 AD 2F 12 BF 3D F8 C3 B F A A 9E 0D 68 2F 8B BF 1A 49 F6 15 AE 14 EC 9F 5A F F 23 4E F EA F 0B CD 7F 31 CE AC D4 PAGE 28

30 Verify Someone Public key Q niels = Eindhoven, Dear all, Today there will be free drinks for everyone at hours at Schootsestraat 65, Eindhoven. Kind regards, Niels Duif Signature: E1 B8 AD 2F 12 BF 3D F8 C3 B F A A 9E 0D 68 2F 8B BF 1A 49 F6 15 AE 14 EC 9F 5A F F 23 4E F EA F 0B CD 7F 31 CE AC D4 Correct PAGE 29

31 Verify Someone Public key Q niels = Eindhoven, Dear all, Today there will be free drinks for everyone at hours at Schootsestraat 65, Eindhoven. Also, I m donating to Compumatica. Kind regards, Niels Duif Signature: E1 B8 AD 2F 12 BF 3D F8 C3 B F A A 9E 0D 68 2F 8B BF 1A 49 F6 15 AE 14 EC 9F 5A F F 23 4E F EA F 0B CD 7F 31 CE AC D4 PAGE 30

32 Verify Someone Public key Q niels = Eindhoven, Dear all, Today there will be free drinks for everyone at hours at Schootsestraat 65, Eindhoven. Also, I m donating to Compumatica. Kind regards, Niels Duif Signature: E1 B8 AD 2F 12 BF 3D F8 C3 B F A A 9E 0D 68 2F 8B BF 1A 49 F6 15 AE 14 EC 9F 5A F F 23 4E F EA F 0B CD 7F 31 CE AC D4 PAGE 31

33 Verify Someone Public key Q niels = Eindhoven, Dear all, Today there will be free drinks for everyone at hours at Schootsestraat 65, Eindhoven. Also, I m donating to Compumatica. Kind regards, Niels Duif Signature: E1 B8 AD 2F 12 BF 3D F8 C3 B F A A 9E 0D 68 2F 8B BF 1A 49 F6 15 AE 14 EC 9F 5A F F 23 4E F EA F 0B CD 7F 31 CE AC D4 Wrong PAGE 32

34 Applications Signing bank transactions (ideal) Signing your tax declaration (DigID) Secure And many more PAGE 33

35 Elliptic curves PAGE 34

36 Elliptic curves Elliptic curve over a prime field F p PAGE 35

37 Elliptic curves Elliptic curve over a prime field F p p = is prime PAGE 36

38 Elliptic curves Elliptic curve over a prime field F p p = is prime y 2 x 3 a 2 x 2 a 4 x a 6 (mod p) PAGE 37

39 Elliptic curves Elliptic curve over a prime field F p p = is prime y 2 x 3 a 2 x 2 a 4 x a 6 (mod p) Twisted Edwards curve PAGE 38

40 Elliptic curves Elliptic curve over a prime field F p p = is prime y 2 x 3 a 2 x 2 a 4 x a 6 (mod p) Twisted Edwards curve ax 2 y 2 1 dx 2 y 2 (mod p) PAGE 39

41 Elliptic curves Elliptic curve over a prime field F p p = is prime y 2 x 3 a 2 x 2 a 4 x a 6 (mod p) Twisted Edwards curve ax 2 a = -1 y 2 1 dx 2 y 2 (mod p) PAGE 40

42 Elliptic curves Elliptic curve over a prime field F p p = is prime y 2 x 3 a 2 x 2 a 4 x a 6 (mod p) Twisted Edwards curve ax 2 y 2 1 dx 2 y 2 (mod p) a = -1 Fast arithmetic PAGE 41

43 My signature scheme PAGE 42

44 My signature scheme Similar to Schnorr signatures PAGE 43

45 My signature scheme Similar to Schnorr signatures Inversion-free PAGE 44

46 My signature scheme Similar to Schnorr signatures Inversion-free No point compression (X:Y:Z) (X:Y:Z:T) PAGE 45

47 Side channel attacks PAGE 46

48 Side channel attacks Get secret information by: PAGE 47

49 Side channel attacks Get secret information by: Measuring power consumption, radiation, etc PAGE 48

50 Side channel attacks Get secret information by: Measuring power consumption, radiation, etc Disturbing the computation PAGE 49

51 Improvements PAGE 50

52 Improvements p = % Signed byte representation -11% Twisted Edwards curves -2% Extended coordinates -1% Fast squaring -5% PAGE 51

53 Improvements Schnorr variant instead of ECDSA variant -2% Side channel attack resistance +46% Java Card PAGE 52

54 Improvements Schnorr variant instead of ECDSA variant -2% Side channel attack resistance +46% Java Card % That means: 7000 times slower PAGE 53

55 PAGE 54

56 New findings PAGE 55

57 New findings Precomputed points are not presented as (X:Y:Z:T) but as (Y-X : Y+X : 2Z : kt), where k = 2d PAGE 56

58 New findings Precomputed points are not presented as (X:Y:Z:T) but as (Y-X : Y+X : 2Z : kt), where k = 2d Saves 2 additions, a multiplication by 2 and a multiplication by k PAGE 57

59 New findings Precomputed points are not presented as (X:Y:Z:T) but as (Y-X : Y+X : 2Z : kt), where k = 2d Saves 2 additions, a multiplication by 2 and a multiplication by k Helps to prevent side channel attacks PAGE 58

60 New findings Precomputed points are not presented as (X:Y:Z:T) but as (Y-X : Y+X : 2Z : kt), where k = 2d Saves 2 additions, a multiplication by 2 and a multiplication by k Helps to prevent side channel attacks The scalar r is represented as r = r 0 + r 1 * r 63 *2 252, where r 0, r 1,..., r 63 are taken at random in {1,2,...,16} This avoids the neutral element (0:1:1:0) PAGE 59

61 Acknowledgements Thanks Tanja! PAGE 60

62 Acknowledgements Thanks Tanja! Thanks Cees! PAGE 61

63 Acknowledgements Thanks Tanja! Thanks Cees! Thanks Henk and Aart! PAGE 62

64 Acknowledgements Thanks Tanja! Thanks Cees! Thanks Henk and Aart! Thanks Eline! PAGE 63

65 Acknowledgements Thanks Tanja! Thanks Cees! Thanks Henk and Aart! Thanks Eline! Thanks to all my friends and family! PAGE 64

66 End of presentation Twisted Edwards curves ax 2 y 2 1 dx 2 y 2 (mod p) Java Card Precomputing (Y-X : Y+X : 2Z : kt) Smart cards Signatures Java Card PAGE 65

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