716
7 Countermeasures Against Hardware Trojans
in case of interest. As can be seen from the above, as the k increases, the probability
Prob [n,k,l] of non-detection of unauthorized components after executing the cycle
l of random tests (without replacements) will reduce significantly. The probability
Prob[n,k,l] also reduces significantly as the number of the used tests I increases. In
fact, the value 1−Prob[n,k,l] measures the so-called production fairness and increases
together with l.
For a responsible developer of the microcircuit, in order to obtain the desired
level of confidence at least equal to a, it is necessary to find the smallest numerical
value of the parameter l such that (1−Prob [n, k, l]) > a. Since, according to the
mathematicians, obtaining an exact formula in the closed form for a type 1 equation is
an extremely difficult task, the acceptable solution is often found by numeric methods
(or using classical approximations in case of high values of the n parameter).
(2) Assuming that the parameter k is evenly distributed, it is possible to immediately
calculate the possibility that the first unauthorized copy of the microcircuit will
be detected during the l + 1st test. Mathematically, it will look as follows [116]:
Prob [n, k, l + 1] = Pr ob [n, k, l] ·
l (l − 1)(k − 1)
N − 1
.
The authors of [128–130] in a consolidated manner concluded that the expected
number of such tests for detection of the first unauthorized copy is
∞
k=1
n(k−1)+1
l=1
l · Prob [n, k, l],
if the first unsuccessful result occurred at the l level, then the mathematical
expectation for k is qual to
E[k] =
∞
k=1
k · Prob [n, k, l].
In conclusion, it should be noted that all of the above methods today are used
in semiconductor business to one extent or another. It’s a different matter that in
practice the results of this study are not usually published in scientific and technical
press; they are left for closed trials, where technical specialists are usually present
only as experts or witnesses.
7 Countermeasures Against Hardware Trojans
in case of interest. As can be seen from the above, as the k increases, the probability
Prob [n,k,l] of non-detection of unauthorized components after executing the cycle
l of random tests (without replacements) will reduce significantly. The probability
Prob[n,k,l] also reduces significantly as the number of the used tests I increases. In
fact, the value 1−Prob[n,k,l] measures the so-called production fairness and increases
together with l.
For a responsible developer of the microcircuit, in order to obtain the desired
level of confidence at least equal to a, it is necessary to find the smallest numerical
value of the parameter l such that (1−Prob [n, k, l]) > a. Since, according to the
mathematicians, obtaining an exact formula in the closed form for a type 1 equation is
an extremely difficult task, the acceptable solution is often found by numeric methods
(or using classical approximations in case of high values of the n parameter).
(2) Assuming that the parameter k is evenly distributed, it is possible to immediately
calculate the possibility that the first unauthorized copy of the microcircuit will
be detected during the l + 1st test. Mathematically, it will look as follows [116]:
Prob [n, k, l + 1] = Pr ob [n, k, l] ·
l (l − 1)(k − 1)
N − 1
.
The authors of [128–130] in a consolidated manner concluded that the expected
number of such tests for detection of the first unauthorized copy is
∞
k=1
n(k−1)+1
l=1
l · Prob [n, k, l],
if the first unsuccessful result occurred at the l level, then the mathematical
expectation for k is qual to
E[k] =
∞
k=1
k · Prob [n, k, l].
In conclusion, it should be noted that all of the above methods today are used
in semiconductor business to one extent or another. It’s a different matter that in
practice the results of this study are not usually published in scientific and technical
press; they are left for closed trials, where technical specialists are usually present
only as experts or witnesses.