1 Montgomery Modular Multiplication in Hard- ware

FEI KEMT
we can mention a generator kept in metastable state whose stable output value will
be influenced by noise conditions inside the generator. In such case, the primary
source of randomness is the metastability and the secondary source is the noise.
Metastability A fundamental building block of digital circuits, the flip-flop (FF)
has two well-defined stable states - high and low level usually denoted as 1 and 0
(see Figure 5 – 2). Under certain conditions the device may get into a state which
cannot be described by any of the above defined states. This condition is called
metastability.
stable state 0
Metastable state
stable state 1
Figure 5 – 2 Illustration of stable states (0 and 1) and undefined metastable state
The most common way to get a device into the metastability is to violate the
setup 7 and hold 8 times of the device. That can be achieved by choosing the frequen-
cies of the clock and input signals of the FF in a ratio that results into changes of
the input signal level that are too close to edges of the clock signal. Other option
is that the frequencies of the signals are the same, but the phases are aligned in a
way that causes FF’s setup and hold time violation.
Keeping the FF close to metastability and then allowing it to resolve produces a
binary sequence that depends on noise conditions inside the FF in the time of release.
If the origin of the noise is a thermal motion, then its random nature suggests that
repeatedly clocking a FF forced into metastability will produce a succession of binary
bits with little correlation between any pair in the sequence [75].
7Setup time is defined as the minimum time before sampling edge by which the sampled signal
must be stable
8Hold time is defined as the minimum time after sampling edge during which the sampled signal
must be stable
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