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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