2. Sliding Windows Succumbs to Big Mac Attack Colin D. Walter www.co.umist.ac.uk

3. CHES 2001C.D. Walter, UMIST2 Aims Re-think the power of DPA; Use it on a single exponentiation; Longer keys are more unsafe!

4. CHES 2001C.D. Walter, UMIST3 DPA Attack on RSA Summary: Differential Power Analysis (DPA) is used to determine the secret exponent in an embedded RSA cryptosystem. Assumption: The implementation uses a small multiplier whose power consumption is data dependent and measurable.

5. CHES 2001C.D. Walter, UMIST4 History P. Kocher, J. Jaffe & B. Jun Introduction to Differential Power Analysis and Related Attacks Crypto 99 T. S. Messerges, E.A. Dabbish & R.H. Sloan Power Analysis Attacks of Modular Exponentiation in Smartcards CHES 99

6. CHES 2001C.D. Walter, UMIST5 Multipliers Switching a gate in the H/W requires more power than not doing so; On average, a Mult-Acc op n a×b+c has data dependent contributions roughly linear in the Hamming weights of a and b; Variation occurs because of the initial state set up by the previous mult-acc op n.

7. CHES 2001C.D. Walter, UMIST6 First Results This theory was checked by simulation and found to be broadly correct; Refinements were made to this model (which will be reported elsewhere); These give a more precise & detailed partial ordering.

8. CHES 2001C.D. Walter, UMIST7 Combining Traces I The long integer product A×B in an exponentiation contains a large number of small digit multiply-accumulates: a i ×b j +c k Identify the power subtraces of each a i ×b j +c k from the power trace of A×B; Average the power traces for fixed i as j varies: this gives a trace tr i which depends on a i but only the average of the digits of B.

9. CHES 2001C.D. Walter, UMIST8 Combining Traces a0b0a0b0 a0b1a0b1 a0b2a0b2 a0b3a0b3

10. CHES 2001C.D. Walter, UMIST9 Combining Traces a0b0a0b0

11. CHES 2001C.D. Walter, UMIST10 Combining Traces a0b0a0b0 a0b1a0b1

12. CHES 2001C.D. Walter, UMIST11 Combining Traces a0b0a0b0 a0b1a0b1 a0b2a0b2

13. CHES 2001C.D. Walter, UMIST12 Combining Traces a0b0a0b0 a0b1a0b1 a0b2a0b2 a0b3a0b3

14. CHES 2001C.D. Walter, UMIST13 Combining Traces

15. CHES 2001C.D. Walter, UMIST14 Combining Traces a 0  (b 0 +b 1 +b 2 +b 3 )/4 Average the traces:

16. CHES 2001C.D. Walter, UMIST15 b is effectively an average random digit; So trace is characteristic of a 0 only, not B. tr 0 Combining Traces a0ba0b _ _

17. CHES 2001C.D. Walter, UMIST16 Combining Traces II The dependence of tr i on B is minimal if B has enough digits; Concatenate the average traces tr i for each a i to obtain a trace tr A which reflects properties of A much more strongly than those of B; The smaller the multiplier or the larger the number of digits (or both) then the more characteristic tr A will be.

18. CHES 2001C.D. Walter, UMIST17 Combining Traces tr 0

19. CHES 2001C.D. Walter, UMIST18 Combining Traces tr 0 tr 1

20. CHES 2001C.D. Walter, UMIST19 Combining Traces tr 0 tr 1 tr 2

21. CHES 2001C.D. Walter, UMIST20 Combining Traces tr 0 tr 1 tr 2 tr 3

22. CHES 2001C.D. Walter, UMIST21 Question: Is the trace tr A sufficiently characteristic to determine repeated use of a multiplier A in an exponentiation routine? Combining Traces tr A

23. CHES 2001C.D. Walter, UMIST22 Distinguish Digits? Averaging over the digits of B has reduced the noise level; In m-ary exponentiation we only need to distinguish: –squares from multiplies –the multipliers A (1), A (2), A (3), …, A (m–1) For small enough m and large enough number of digits they can be distinguished in a simulation of clean data.

24. CHES 2001C.D. Walter, UMIST23 Distances between Traces tr 0 tr 1 d(0,1) = (  i=0 ( tr 0 (i)  tr 1 (i) ) 2 ) ½ n i n0 power

25. CHES 2001C.D. Walter, UMIST24 Simulation tr 0 tr 1 d(0,1) = (  i=0 ( tr 0 (i)  tr 1 (i) ) 2 ) ½ n i n0 gate switch count

26. CHES 2001C.D. Walter, UMIST25 Simulation Results 16-bit multiplier, 4-ary exp n, 512-bit modulus. d(i,j) = distance between traces for ith and jth multiplications of exp n. Av d for same multipliers 2428 gates SD for same multipliers 1183 Av d for different multipliers23475 gates SD for different multipliers 481

27. CHES 2001C.D. Walter, UMIST26 Simulation Results Equal exponent digits can be identified – their traces are close; Unequal exponent digit traces are not close; Squares can be distinguished from mult ns : their traces are not close to any other traces; There are very few errors for typical cases.

28. CHES 2001C.D. Walter, UMIST27 Exp nt Digit Values Pre-computations A (i+1)  A  A (i) mod M provide traces for known multipliers. So: We can determine which mult ive op ns are squares; We can determine the exp digit for each mult n ; Minor extra detail for i = 0, 1 and m–1; This can be done independently for each op n.

29. CHES 2001C.D. Walter, UMIST28 Some Conclusions The independence means attack time proportional to secret key length; Longer modulus means better discrimination between traces; No greater safety against this attack from longer keys.

30. CHES 2001C.D. Walter, UMIST29 Warning single exponentiationWith the usual DPA averaging already done, it may be possible to use a single exponentiation to obtain the secret key; So using exp nt d+rφ(M) with random r may be no defence.So using exp nt d+rφ(M) with random r may be no defence.

31. CHES 2001C.D. Walter, UMIST30 Final Conclusions Sliding Windows exp n method may be broken in this way; Like a Big Mac, you can nibble away at each secret exponent digit in turn and enjoy finding out its value.