1. Enhanced Doublng Attacks on Signed-All-Bits Set Recoding 1 Graduate School of Information Management and Security, Korea University, Korea http://cist.korea.ac.kr Hee-seok Kim 1, Tae Hyun Kim 1, Jeong Choon Ryoo 1, Dong-Guk Han 2, Ho Won Kim 2, and Jongin Lim 1 2 Electronics and Telecommunications Research Institute(ETRI), Korea http://www.etri.re.kr/ WISTP 2007

2. Side Channel attacks-Power analysis Side Channel attacks-Power analysis Scalar multiplication & Simple power analysis to ECC Scalar multiplication & Simple power analysis to ECC Countermeasures & Original Doubling Attack ( DA ) Countermeasures & Original Doubling Attack ( DA )  Countermeasure1 - Coron ’ s dummy method  Countermeasure2 - sABS recoding method  DA & Weakness of Coron ’ s dummy method  Security of sABS recoding against DA Proposed Attacks Proposed Attacks  Recursive attack  Initializing attack Experiments & Statistical approach of noise reduction Experiments & Statistical approach of noise reduction Countermeasures & Conclusion Countermeasures & Conclusion Contents

3. WISTP 2007 Which are Side Channel Attacks 1. Timing Attacks - Kocher (1996) 2. Differential Fault Analysis (DFA) - Biham-Shamir (1997) 3. Simple Power Analysis (SPA) - Kocher, Jaffe, Jun (1998) 4. Differential Power Analysis (DPA) - Kocher, Jaffe, Jun (1998)

4. WISTP 2007 Power attacks  Kocher et al., June 1998: Measure instantaneous power consu mption of a device while it runs a cryptographic algorithm  Different power consumption when operating on logical ones vs. logical zeroes.

5. WISTP 2007 In general, Addition has different power consumption from Doubling. – C. Clavier et al. [3] Simple Power analysis to ECC Point Doubling ( D ) : Execution in all bit values of secret key d : secret exponent Point Addition ( A ) : Execution when bit value is only ‘ 1 ’ DDDAAAD General scalar multiplication algorithm

6. WISTP 2007 Countermeasure against SPA-Coron ’ s method d11101 P2P6P14P28P 3P7P29P Point Doubling ( D ), Point Addition ( A ) : Execution in all bit values of secret key Coron ’ s dummy method d11101 P2P6P14P28P 3P7P29P D A D 15P D A D A

7. WISTP 2007 Countermeasure against SPA-sABS recoding sABS recoding d111 1 P2P D 6P D 14P D 26P D 50P D 102P D 3P A 7P A 13P S 25P S 51P A 101P S D : Doubling, A : Addition, S : Subtraction The power consumption of Addition is similar to that of Subtraction !!  It ’ s secure against original SPA.

8. WISTP 2007 Doubling Attack ( DA ) – Fouque et al. Characteristics Assumption Attacker has an ability to decide whether A=B or not when a smartcard computes ECDBL(A) and ECDBL(B). When input values are P and 2P, Coron ’ s dummy method carries out the same doubling in the vicinity of the bit value ‘ 0 ’. Attack method d101001 P P 2P 3P 4P 5P 10P 11P 2OP 21P 40P 41P 2P 2P 4P 6P 8P 10P 20P 22P 40P 42P 80P 82P

9. WISTP 2007 Doubling Attack ( DA ) – Fouque et al. Key : 1 0 1 0.... = = ≠

10. WISTP 2007 Security of sABS recoding against DA Characteristics Because sABS recoded value has not ‘ 0 ’ bit, it is secure against original DA Example d111 P P 2P 3P 6P 5P 10P 11P 22P 21P 42P 41P 2P 2P 4P 6P 12P 10P 20P 22P 44P 42P 84P 82P

11. WISTP 2007 Characteristics Feasible attack – Supporting a concrete method for experiment ObjectNew power attacks on scalar multiplication using recoding countermeasures (sABS recoding) Proposed ‘ initializing attack ’ - Combination of ‘ doubling attack ’ and ‘ Goubin ’ s attack ’ SPA-based attacks on one-bit of key Proposed attacks

12. WISTP 2007 Proposed attack 1 - Recursive Attack ObjectNew power attack on scalar multiplication using recoding countermeasures (sABS recoding) If an attacker knows upper n bits of secret key, he can find the upper (n+1)-th bit by this attack. By this method, attacker can find all bits of secret key in sequence. Characteristic An attacker that knows upper n bits of secret key ( = d ’ ) selects two inputs A, B for originating same ECDBL in the vicinity of upper (n+1)-th bit ( = t ). A = d ’ P, B = (2d ’ +1) P  if t = 1, (2d ’ +1)A = d ’ B if t = -1, (2d ’ +1)A ≠d ’ B

13. WISTP 2007 Proposed attack 1 - Recursive Attack d11111 A = d ’ P, B = (2d ’ +1) P  if t = 1, (2d ’ +1)A = d ’ B if t = -1, (2d ’ +1)A ≠d ’ B d ’ =11 11P 22P 33P 66P 55P 110P 121P 242P 253P 506P 517P 1034P 1023P 23P 46P 69P 138P 115P 230P 253P 506P 529P 1058P 1081P 2162P 2139P 1

14. WISTP 2007 Proposed attack 2 - Initializing Attack An attacker that knows upper n bits of secret key ( = d ’ ), he selects one input A for originating ECDBL(P) in the upper (n+1)- th bit ( = t ). A = (2d ’ +1) -1 P  if t = 1, (2d ’ +1)A = P if t = -1, (2d ’ +1)A ≠P An attacker acquires the first doubling signal-ECDBL(P) in the signal according to input point ‘ P ’. the first doubling signal-ECDBL(P) in the signal according to input point ‘ P ’ compares with the (n+1)-th doubling signal-ECDBL(P) in the power signal according to input point ‘ (2d ’ +1) -1 P ’

15. WISTP 2007 Proposed attack 2 - Initializing Attack d11111 d ’ =11 54P 35P 16P 32P 51P 29P 10P 20P P 2P 56P 39P 20P 1 A = (2d ’ +1) -1 P  if t = 1, (2d ’ +1)A = P if t = -1, (2d ’ +1)A ≠P The order of curve : 73 (2*11+1) -1 mod 73 = 54

16. WISTP 2007 Experiments & Statistical approach of noise reduction Setting PIC Microcontroller Power supply – 5V Function generator – 1MHz Oscilloscope

17. m 1 =E(X 1 ), m 2 =E(X 2 ), a 1 =max(X 1 ), b 1 =min(X 2 ) k points Experiments & Statistical approach of noise reduction

19. WISTP 2007 INPUT : 3P Key : 1 1 -1.... Key1?? Disc < D 1 INPUT : P 1?? INPUT : 7P Disc > D k points uk points Experiments & Statistical approach of noise reduction

20. WISTP 2007 Countermeasures & Conclusion Characteristics of proposed attacks Characteristics of proposed attacks  These new attacks is applicable to sABS recoding countermeasure.  SPA-based attacks on one-bit of key.  Initializing attack is more powerful than Goubin ’ s attack. Countermeasures Countermeasures  Using a Projective coordinates – affine coordinates is not secure.  BRIP can be applied to our attacks [ 13]  BRIP can be applied to our attacks [ 13].

21. WISTP 2007 Questions and Comments Hee Seok Kim : heeseokkim@cist.korea.ac.kr heeseokkim@cist.korea.ac.kr