1 Montgomery Modular Multiplication in Hard- ware

FEI KEMT
5 True Random Number Generator - preliminar-
ies
Random values play a crucial role in several areas of science. In dependency on field
of application the requirements for parameters of random sequence and generator
of sequence itself may vary. Focusing on the sequence origin we distinguish between
truly- and pseudo-random sequences. The construction of generators decides on
their suitability for commercial or research applications.
In the following chapter we provide an introduction to the topic of randomness
and random values (Section 5.1) while focusing on generators applicable in cryptog-
raphy. In Section 5.2 we mention typical sources for generation of random sequences
in digital circuits. In Section 5.3 we summarise design ideas of the PLL-based gen-
erator we will analyse in the following chapter. In Section 5.4 we explain testing
techniques applied in order to evaluate generators and in Section 5.5 we discuss is-
sues related to attacks on RNGs. Finally, in Section 5.6 we summarise the chapter.
5.1 Randomness
We start with topic called randomness, and the most natural questions that come
in our minds may look like: How to define the randomness? Where comes it from?
Or how can we prove that a sequence is random?
The randomness of the world we live in has been a scientific and philosophical
topic for long time. Famous remark of Albert Einstein says that “God does not
play dice with the universe” what might convince us about determinism of our
environment. However, several physical phenomena present in physical world are
proved to have a random nature e.g. probabilistic nature of quantum mechanics,
thermal and shot noise in electronic components, or nuclear decay.
The fundamental problem of randomness is in fact that even with exact definition
it is very difficult to prove whether any finite numeric sequence is random or not. The
randomness of a source is evaluated through the parameters of sequence generated
using that source. The way how the values of sequence are extracted from the source
depends on applied harvesting mechanism. The optimal harvesting does not disturb
the random physical process and extracts as much entropy as possible.
The entropy H of a random variable X with n outcomes �
xi : i = 1, . . . , n �
is
defined as negative logarithm of the probability of the process’s most likely output
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