A circuit that does multiplication of two signals when some scaling down is OK
A diode can be used as a multiplier in its most basic form. Consider the 1N4148 from Fairchild. I'm naming fairchild because their data sheet contains really useful data: -



Consider what the forward resistance is at each applied voltage level. For instance, when 250mV is applied the foward current is 1 uA making the resistance 250 kohms. At ~380 mV the forward current is 10 uA - a resistance of 38 kohm. At ~500 mV the current is 100 uA and the resistance is 5 kohm. At 620 mV the forward current is 1 mA (R = 620 ohms).

Now if a small AC signal is applied via a resistor to the diode and a bias voltage generator (also fed from another resistor) controls the DC on the forward conducting diode, the AC level can be modulated over a considerable range.

Not very good for DC though because the terminal voltage on the anode contains the modulated AC signal and some amount of DC content.

So, the next step is to use two forward biased diodes with anti-phase AC signals applied to each. Both diodes are fed from the same bias voltage generator via there own individual resistors. The output needs to be differentially measured across both anodes and things look much better - the AC modulated signal is significantly less distorted and the DC offset (due to the bias generators) is virtually zero.

This can be further improved (quite a lot actually) by using  4 diodes that allow full four-quadrant multiplication but the complexity of the "circuit" is such that just trying to avoid transistors is impossible. You need good differential amplifiers, precision resistors and a lot of care to ensure diodes are temperature stable with each other.

It can be done but it's probably easier to buy a chip that does it.