1 Montgomery Modular Multiplication in Hard- ware

FEI KEMT
Equation 6.11 can be viewed as a stochastic model of the generator, since it permits
to estimate a probability of the generator output value as a function of the mean
values of critical samples (which depend on the jitter characteristics). However, the
model is valid if and only if critical samples are independent.
The proposed model shows that (as it could be expected) the bias of the generator
output decreases with the increasing number of critical samples (note that this
number is related to the jitter variation). It can be seen that if the mean value of
any of these samples is equal to 0.5, the bias on the generator output is equal to
zero and does not depend on the remaining samples. Finally, the sign of the bias
depends on the number of samples having a mean value equal to one (K 1 D).
The advantage of the proposed model lies in the fact that the model can also be
used as a proof of mutual statistical independence of the critical samples. To evaluate
the statistical independence, the output mean value and the mean value of critical
samples are measured and the validity of the model expressed in Equation 6.11
is verified. If the test fails, the random variables (critical samples) are mutually
dependent.
Model Verification The validity of the model has been tested on real data in
order to confirm the model empirically. We have tested outputs of seven TRNG
configurations implemented in Altera Stratix devices. The Table 6 – 8 presents the
chosen parameters of the tested configurations (KM, KD, FCLK and FCLJ) and the
corresponding results – mean value of critical samples (E[pi]), mean value of the
generator output (m = E [x(nTQ)]), number of samples equal to one (K 1 D) and
number of critical samples (K p
D).
The mean value of the output bitstream m = E [x(nTQ)] is computed as an
arithmetic mean of 512,000 successive bits at the output of the TRNG. The mean of
the model E[pi] is calculated using the Equation 6.11, while employing probabilities
of the critical samples pi accumulated after Q = 1000 periods TQ.
As it can be seen, the model is very precise for a small number of critical samples,
since both mean values are very similar. For a higher number of critical samples,
the mean value tends to the ideal value 0.5. Note that the model provides correct
information about the statistical deviation of the output bitstream in configurations
1, 2 and 5. The model gives acceptable results corresponding closely to the mean
value of the generated sequence in tests 6 and 7. It should be noted that in config-
urations 3 and 4, the model outputs do not agree with the generator outputs (most
probably) because of statistical dependence between critical samples.
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