Sistema de Gestión de Calidad

Hoeffding’s curve decays exponentially, but it is not sharp enough to capture the tail of the distribution of 𝐹̅_𝐴. Also, compared to the first three inequalities, tightness is achieved for larger sample sizes n.

5.6.4. Bounding the mean time – Two-sided bound

In this subsection, the behavior of event C in Equation (5.63) is examined. The probability of the absolute difference between the sample mean 𝜇̂ and the theoretical mean 𝜇 not deviating more than 𝜀, where 𝜀 ≥ 0, will be bounded using the two-sided versions of Chebyshev’s (5.9), Hoeffding’s (5.30) and Bennett’s (5.59) inequalities. By the strong law of large numbers, it is obtained that

𝑃 [lim

∑

(|𝑇

− 𝐸[𝑇

]|)

𝑖=1

= 0] → 1

The probability in Equation (5.67) indicates that with enough samples, the empirical mean is a good approximation to its true mean. According to the strong law of large numbers, the sample mean converges almost surely to the expected mean. A quantitative version of the law of large numbers for bounded variables is the investigation of that rate of convergence [85], by bounding the tail of 𝐹̅_𝐶. The decay rate, given by a bound, will provide useful information about the impact of finite values n on the convergence |𝜇̂ − 𝜇| → 𝜀, 𝜀 ≥ 0.

Figure 5.20 depicts the right tail 𝐹̅𝐶 bounded by inequalities given by (5.9), (5.30) and

(5.59). For all variations of the sample size n, the three inequalities admit an exponential decay, with Bernstein’s and Chebyshev’s being able to mimic the tail of 𝐹̅_𝐶. Bernstein’s appears sharper to the tail, while Hoeffding’s performance is somewhat

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Figure 5.20: Right tail of 𝐹̅_𝐶 vs bounds – Case 1.

poor compared to the rest of the inequalities. Therefore, under the condition that the right tail of 𝐹̅_𝐶 is examined, i.e. 𝜀 is large enough, the following relation holds

𝐹̅

= 𝑃(|𝜇̂ − 𝜇| ≥ 𝜀) ≤ 𝑃

≤ 𝑃

≪ 𝑃

Table 5.5 highlights Bernstein’s ability to provide a tight bound even for small n

values. For 1,000 encryptions, 𝑃(|𝜇̂ − 𝜇| ≥ 3,000) = 0.005 as resulted from the generated distribution 𝐹̅𝐶. Under the same parameterization on n and 𝜀, Bernstein’s

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Table 5.5: 𝐹̅_𝐶, Bernstein’s, Chebyshev’s and Hoeffding’s values for various n and 𝜀.

𝒏 𝜺 (𝒏𝒔) = 𝜶𝝁 𝑭̅𝑪 𝑷𝑩𝒆𝒓𝒏𝒔𝒕𝒆𝒊𝒏 𝑷𝑪𝒉𝒆𝒃𝒚𝒔𝒉𝒆𝒗 𝑷𝑯𝒐𝒆𝒇𝒇𝒅𝒊𝒏𝒈 1,000 3,000 0.0005 0.06 0.1 1 1,000 4,000 0.0001 0.006 0.07 0.9 10,000 1,000 0.002 0.03 0.1 1 10,000 1,200 0.0003 0.005 0.08 1 100,000 300 0.005 0.04 0.1 1 100,000 400 0.0002 0.002 0.07 0.9 1,000,000 100 0.003 0.02 0.1 1 1,000,000 150 0 0.0001 0.05 0.6

Chebyshev’s values are relatively close to Bernstein’s, in terms of tightness, for small or moderate n, with the tightness relaxed when n gets larger, failing to follow Bernstein’s sharpness.

5.7. Discussion

In this chapter an adaptive security scheme that suggests a new approach in the area of low energy encryption has been presented. The method relies on the application of Chernoff – type bounds on the tail probability that the mean time of n encryptions will exceed the expected time by a desired threshold so that the probability on an extreme value can be bounded. This method has also been used to provide a bound for the probability that the overall execution time of n encryptions will not exceed n times the mean encryption time of a single execution. Two-sided inequalities have been

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calculated and presented for those exponential bounds, for both the mean and overall encryption time. This provides knowledge about the rate of convergence of the sample mean to the true mean or the sum of n encryptions times to n times the mean, as concentration inequalities quantify what is known from the law of large numbers. Another advantage of the presented methodology is that relaxes the CLT’s assumption in Chapter 4, as number of executions can be considered finite. CLT presented in Chapter 4, is an asymptotic result and assures that the time distribution of the n

encryptions is approximately Gausian as 𝑛 → ∞. In this chapter, the scope was to develop a framework for investigating the trade-off between the number of encryptions

n, the level of accuracy 𝜀 and the tightness of each inequality to the right tail of the

distribution.

Performance evaluation of the probabilistic concentration inequalities was presented for both upper and two-sided bounds of mean and sum. This framework is flexible, Chebyshev’s, Hoeffding’s, Bennett’s and Bernstein’s bounds are distribution free and no assumption needs to be made for the distribution.

Bennett’s and Bernstein’s inequalities are approximately equal as presented. The investigation highlighted their superior performance compared to the other two inequalities, as they bound tightly the right tail of the distribution even for small or moderate sample size n. This makes it possible to set up and optimise energy efficient encryption schemes for policy makers with relatively small number of expected encryptions. A typical example of an encryption scheme with sample size equal to 1,000 executions might be the daily contactless payments of a central coffee shop or a supermarket. On the other hand, an expected number of transactions of order 106 _or

more, might be the daily usage of contactless passes in London’s tube. In both situations, Bernstein’s bound will result in tight predictions.

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Although Hoeffding’s inequality did not perform well compared to Bernstein’s, it could contribute at the early stages of an encryption scheme, where no data may be available for a performance analysis. This inequality requires the time to be independent random variable and bounded, i.e. the best and worst case scenario of the algorithm’s encryption time. Having that knowledge, rough approximations of the scheme’s performance could be made for n encryptions that the algorithm is expected to be executed during a specified time window, and by incorporating knowledge from Chapter 4 regarding the limiting behaviour of scheme’s encryption time, an optimal set up will be enabled. During the operation of the scheme, the recorded data will provide the decision maker with statistical confidence that the sample variance is a valid approximation of the true variance. Finally, having knowledge of the variance, tighter predictions can be made by applying Bernstein’s bound, while any modifications at the scheme will be made at that stage.

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