correction: above circuit breaks when a < na

maybe a gilbert cell already does this and I was misunderstanding how it works...

The requirement that the output be signed while the inputs are relative to a reference would seem to suggest using differential amplifiers for each input signal and then using a standard multiplier circuit. Is there a particular reason you're trying to multiply without having access to negative voltages?

Maybe you could do it in three steps: logarithmic converter on each input, sum the two, and then exponential converter. Then you get multiplication by the properties of logarithms

See the AD538 datasheet for some inspiration. It requires dual supplies though

It's a hard problem in general and doing it properly for the four quadrant case would probably require a pretty complicated circuit. I think it can be done though.

Or another way: expand out your expression to i_out = v_a*v_b - v_a_middle*v_b - v_b_middle*v_a + v_a_middle*v_b_middle and then implement it by summing the 1-quadrant mixer output of v_a*v_b with an offset (corresponding to the product of the references) and then subtract scaled versions of v_a and v_b. Then you can do it with only a one-quadrant mixer and some summing/differencing

I would imagine that would be simpler than the 4-quadrant mixer version

Oh yeah also check out the AD633 datasheet

Or if you want to go down the gilbert cell route, here's an excellent video giving an intuitive explanation of how it works:

I don't have a particular circuit topology in mind for the implementation

But at that point I'd suggest using dual supplies and some kind of circuit that has a voltage-controlled gain.

Check out the Fairchild Application Note AN-6603

I've seen jfets used as voltage controlled resistors a lot. I don't know if you have access to those sorts of devices in your process, but maybe you can get access to a depletion mode mosfet which behaves very similarly. Although that sort of topology doesn't have the 4-quadrant multiplication you're looking for

Perhaps you could take inspiration from the gilbert cell in that you could: 1) create two signal paths - one which inverts the input and one which does not, 2) give each path a voltage-controlled gain, but in a way that makes each have high gain when the other has low gain, and 3) sum the output. I guess the difference here would be that you'd be using single-ended signals throughout instead of differential signals (which is what the gilbert cell does). And hopefully with the single-ended version it's a lot more linear than the gilbert cell