1 Montgomery Modular Multiplication in Hard- ware

FEI KEMT
The Golić Design Golić’s goal is to provide digital TRNG built from logic gates
only. Such design is cost effective and suitable for implementation on any digital
chip. In article from Golić [70] the author proposes two new elements applied in
design of TRNG showed in Figures 5 – 4(a) and 5 – 4(b): the Galois ring oscillator
(GARO) and Fibonacci ring oscillator (FIRO).
(a) Galois ring oscillator (b) Fibonacci ring oscillator
Figure 5 – 4 Ring oscillator structures proposed by Golić.
Adding more complex feedback loop in the ring oscillator (RO) makes also its
behaviour more complex and therefore more suitable for TRNG where the random-
ness coming from jitter spreads faster. In comparison to classical RO, the usage of
GARO and FIRO yields a higher level of entropy and robustness of the generator.
Additional entropy of the generator comes from frequent metastability effects in the
sampling gate.
In [44] Golić and Dichtl show results of practical implementation of TRNG using
the oscillators presented above. The authors prove the randomness of the solution by
analysis of the generator output after repeated restarts of the circuit. The standard
deviation of the output signal voltage raises quickly after the restart and stabilises
on significantly large level which assure randomness of the sample taken in this time
period.
The Kohlbrenner and Gaj Design The principle similar to PLL-based genera-
tor [60] was proposed by Kohlbrenner and Gaj in [79]. Instead of PLL circuitry that
is not present in all FPGAs, the authors use a pair of oscillator rings implemented
in programmable logic area of FPGA. Since the principle expects a tight pair of
frequencies generated by rings, the oscillators must be matched precisely. That re-
quires also proper positioning of the rings inside the FPGA and manual corrections
in placements and routing.
The authors investigated also the influence of temperature on RO. The frequency
of a RO tends to wander as the chip’s temperature varies. It is important to place
the ROs in a pair close to each other so the difference between the frequencies is
reduced due to minimal difference in temperature.
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