Random Generators

(Last update May 13 2017)

Random numbers are often needed for test signals, for dithering undesired carriers and to create seeds for the generation of cryptographic keys.
It may sound surprising but random data, for analog thinking people let's call it noise, is an important and very hard to achieve signal!
Analog Noise Generators
In theory every physical process generates noise. This is a basic property of quantum physics. So the easiest way to achieve a noise signal is to simply amplify thermal noise of a resistor or shot noise of a semiconductor junction. Zener diodes too can be very noisy. So a first draft circuit could look like this:
Fig. 1 analog noise generator

Zener noise of Q1 gets amplified by OP1. At pin AOUT an analog random signal is available. Comparator COMP1 converts the analog signal into a random bit stream available at output DOUT. But is it really pure noise? No!
In a true random signal it should be impossible to guess what will be a future state knowing past states. The circuit shown suffers from various imperfections:

1. Zener noise is not pure noise. Depending on the electrical field inside the junction certain amplitudes are preferred.
2. Amplifier OP may have an offset that leads to different likelyhoods for a logic 1 or a logic 0 at output DOUT.
3. To reduce the effect of offsets the DC gain of the amplifier is 1 while the AC gain is 10. The chance of a 0101.. pattern is surely higher than the chance of a 00000011111111 pattern below the cut off frequency of R2 and C.
4. Neither OP1 nor COMP1 are unlimited fast. So every signal will have successors with identical data (at DOUT) or similar voltage (at AOUT)
5. Since OP1 and COMP1 have a limited supply rejection any modulation of the supply will change the statistics of this analog generator.
6. Slightest electromagnetic interference will also change the statistics.

To achieve reasonably good random signals using physical effects is tricky because it requires:
- Amplifiers with excellent supply rejection
- Fast Amplifiers.
- Low Offset.
- Good shielding against unwanted periodic (or otherwise predictable) disturbance.
- High bandwidth circuits.

Conclusion: Creating noise or random numbers from physical effects is expensive and error prone.

Digital Noise Generators
Digital noise generators basically produce very long data streams with a quasi random sequence of ones and zeros. With an 8 bit shift register and some XOR gates in the feedback path the data stream is already 256 bit long. With a 16 bit shift register the stream starts to repeat after 65536 bit. Making the register longer and longer we get closer to real random (although we never reach real random).
Fig. 2: Digital Random Generator

The example of figure 2 produces a stream of 256 bit length. After reset the content of the shift register is 00000000. To start the random sequence a 1 is needed as a seed. This 1 is produced by the or gates in the upper feedback path.
Once the shift register is running the lower feed back path with the XNOR gates produces the quasi random sequence.
If you want to play with some behavioral verilog code, here it is:

    input CLK, RESET;
    output OUT;
    reg[7:0] SH;
    reg xnor1, xnor2, xnor3;
    wire OUT;
    always @ (RESET or posedge CLK) begin
        if (RESET==1) #1 SH=0;
        else begin
            if (SH==0) #1 SH[0]<=1;
            else begin
    assign OUT = SH[6];

However the statistics of this digital approach is not quite clean either. Since the signal is periodic the output signal can be regarded as a spectrum with discrete lines with a spacing of the spectral lines of 1/256 of the clock frequency.
The following waveforms shos the signals and the internal states of the shift register. The bus SH[7:0] is displayd as an analog trace to graphically represent the states.

Fig. 3: Signals of an 8 bit quasi random shift register

The two cursors mark begin and end of one period (clock period is 10 time quantas).