1 Montgomery Modular Multiplication in Hard- ware

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and effort needed for reproduction the output is lower. Mathematical analysis of
TRNGs tries to find deterministic dependencies inside the extraction method causing
pseudo-randomness.
As the parameters of TRNGs are highly dependent on the implementation, at-
tacking directly the hardware realisation can be more powerful.
Implementation attacks The second group, the implementation attacks, expects
a direct physical access to an implementation and is based on weaknesses caused by
implementation of the RNG. Implementation attacks are further divided to passive
and active attacks.
Passive attacks usually called side-channel attacks, benefit from a side channel in-
formation gained from the physical implementation. The power consumption,
execution time or electromagnetic emanations can provide additional useful
information about RNG internal status or processed data.
Active attacks require an involvement of the attacker into changes of the standard
working conditions, operation flow or design of the original implementation of
the RNG. The non-invasive active attacks apply non-permanent changes of ex-
ternal parameters for RNG e.g. supply voltage, temperature, with motivation
to achieve non-standard - biased RNG output. With more resources one can
execute an invasive attack and change the physical structure of the implemen-
tation. The attacker tries to destroy the source of randomness and make the
output of the RNG constant or to get directly the output of generator.
5.6 Conclusions
In this chapter we have introduced the topic of random numbers. The extraction
of random bits in digital environment is a crucial topic in the area of system imple-
mentations with public-key cryptography. The randomness itself and typical three
sources of randomness: noise, metastability and jitter were described. In order to
provide an overview on the actual status in the research we have collected descrip-
tions of the recently published design proposals and implementations of TRNG.
A typical design of TRNG implemented in a digital device includes a source
of randomness from which a digitised noise signal can be harvested by a proper
mechanism. We have explained the importance of research in the areas of the
harvesting mechanisms and postprocessing. The positive results of statistical tests
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