the entropy of a source producing n symbols will be maximum when
Answers
Explanation:
4.13.1.1 Case Study: Ring Oscillator-based Randomizer
Ring oscillators have been thoroughly studied and modeled. The entropy source of ring oscillators is the variation in delay, or jitter, across the circuit, causing the state of the ring to be unpredictable. Readers unfamiliar with the concept of a ring oscillator should take a moment to read any of the commonly found descriptions, such as Wikipedia, available on the Internet.
Figure 4.26 presents a ring oscillator with three delay elements: D0, D1, and D2. Each delay element consists of a simple logic inverter, except for the first, D0, which uses a NAND gate to multiplex an external enable line that holds the ring oscillator in a steady state prior to oscillation.
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Figure 4.26. Three-element ring oscillator.
When enabled, the ring oscillator has 2N states, where N is the number of delay elements in the ring. A timing diagram of this three-stage ring oscillator is shown in Figure 4.27.
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Figure 4.27. Timing diagram for three-element ring oscillator.
In the diagram in Figure 4.27, Dn represent the delay elements, P is the period, and d is the delay of one element. Only a single delay element changes state at a time; thus, the ring oscillator produces a square wave output that is regular when viewed over a short interval. However, over a longer period, drift and jitter cause the ring oscillator to be less predictable, especially if the jitter is greater than the period of the oscillator. Although not proven here, it can be shown that the next state of a ring oscillator S1 after time T, discounting any jitter, is related to the initial state S0 by the following equation:
S1 = (S0 + T/d) mod (2N)
For example, if d = 5 ns, N = 5, S0 = 0, and T = 500,000 ns (a long sample), then S1 is (0 + 500,000 / 5) mod 10 = 0. With jitter, the next state is
S1 = (S0 + (T + J)/d) mod (2N)
Jitter, J, is defined as the peak variation in time required for the oscillator to complete N cycles and varies randomly with noise due to temperature and voltage changes within transistors (see Figure 4.28). The preceding equation shows that increased jitter causes the next state to be unpredictable.