In earlier work on SKG the secrecy capacity S X of a fuzzy secret X is defined

# In earlier work on skg the secrecy capacity s x of a

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In earlier work on SKG, the secrecy capacity S ( X ) of a fuzzy secret X is defined as the theoretical maximum number of secure key bits that can be extracted from X [14], and it is shown that S ( X ) = I ( X ; X 0 ) with X and X 0 two noisy realisations of the same fuzzy secret. We calculate this mutual information bound of X N as I ( X N ; X 0 N )= H ( X N ) - H ( X N | X 0 N ) and consider both terms separately. We expand H ( X N ) as N i =1 H ( X i | X ( i - 1) ) , and H ( X i | X ( i - 1) ) x ( i - 1) Pr ( x ( i - 1) ) · H ( X i | x ( i - 1) ) = x ( i - 1) Pr ( x ( i - 1) ) · h ( p i ) x ( i - 1) Pr ( x ( i - 1) ) · h ( SR ( i - 1)) = h ( SR ( i - 1)) . From which it follows that: H ( X N ) N i =1 h ( SR ( i - 1)) . To evaluate H ( X N | X 0 N ) , we assume a simple but realistic noise model for the PUF response with bit errors occurring i.i.d. over the different response bits with probability p e Pr ( X i 6 = X 0 i ) . This is equivalent to a transmission over a binary symmetric channel and in that case H ( X N | X 0 N ) = N · h ( p e ) . Substituting both results in the secrecy capacity bound leads to S ( X N ) N i =1 h ( SR ( i - 1)) - N · h ( p e ) . An upper bound for the secrecy capacity can thus be calculated using the success rates SR ( i - 1) of a PUF response model and the bit
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0.00 0.25 0.50 0.75 1.00 N S(X N ) [bit] 0 500 1000 1500 2000 S(X N ) [bit] S(X N ) (Simple Arbiter PUF) S(X N ) (2-XOR Arbiter PUF) S(X N ) (Simple Arbiter PUF) S(X N ) (2-XOR Arbiter PUF) Fig. 3. Upper bounds on the secrecy capacity of Arbiter PUF responses. error probability p e estimated from the PUF’s statistics. Using our empirical results from Section III, we calculate this bound for increasing N . The result is shown in Fig. 3. Also shown is an upper bound on the incremental secrecy capacity S ( X N ) , which indicates a bound on how much S ( X N ) increases by considering a single additional response bit. It is clear that S ( X N ) decreases steadily as N grows and approaches 0 for N 5000 . The S ( X N ) upper bound of our simple Arbiter PUF implementation reaches 600 bits for N = 5000 and will not increase substantially for larger N . We also performed the S ( X N ) vs. N analysis for the 2- XOR Arbiter PUF results, considering the fact that from an information-theoretical viewpoint, S ( X N ) of a 2-XOR Ar- biter PUF response bit can never be larger than 2 × S ( X N ) of a simple Arbiter PUF as calculated earlier. Moreover, S ( X N ) of a single PUF response bit can never be larger than 1 . The results are also shown in Fig. 3. Again, the SKG results shown in Fig. 3 express rather loose upper bounds on the number of secure key bits which can be generated in practice. First of all, S ( X N ) expresses a theoret- ical maximum, but no efficient algorithms are known to reach this maximum. Secondly, we did not calculate S ( X N ) exactly but only an upper bound thereof. Finally, any improvement upon our ML attacks will further decrease these upper bounds.

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