What sort of tasks can an ASIC be developed to do? I know requiring a lot of memory is one thing that makes an algorithm ASIC resist. Are there any others?

Could an ASIC be developed to do Elliptic Curve Crypto? Like adding two points, or signing a piece of data? Could an ASIC be developed to solve polynomial equations?
Taken from ASICs and Decentralization FAQ by Andrew Poelstra:

Are ASIC’s evil?
No, dedicated hardware brings us closer to the thermodynamic limit,
and is therefore eventu- ally a good thing for mining
decentralization. Also, because ASIC’s produce more hashes for the
same amount of energy, they produce stronger proofs-of-work with
proportionally less environmental impact.
However, ASIC’s bring with them a risk of manufacturer centralization,
such as what we saw with Bitcoin in the early days of ASIC mining.
Market forces eventually broke this monopoly and one thing which sped
up the process is that Bitcoin uses the SHA2 hasing algorithm, which
was designed for easy development of dedicated hardware.
Therefore, relatively little startup capital is needed to develop
Bitcoin ASIC’s. Further, regardless of one’s personal feeling toward
ASIC’s, they are inevitable. Dedicated hardware will always be more
efficient than general-purpose hardware (exactly because it is closer
to the thermodynamic limit) and Bitcoin’s incentives are aligned for
ever-increasing efficiency.
Is ASIC resistance desirable?
No. ASIC resistance typically involves increasing algorithmic
complexity to discourage ASIC developers. However, ASIC’s are still
inevitable; all ASIC resistance does is increase the startup capital
required and therefore increase centralization of manufacturing.
Further, increasing the complexity of proof generation often means
also increasing the com- plexity of proof validation, often
disproportionately slow. This discourages (unpaid) non- mining
validators, which also increases centralization.
Is ASIC resistance possible?
ASIC resistance, in the sense of making life difficult for ASIC
manufacturers (and therefore reducing the number of distinct
manufacturers) is possible. But it is impossible to create an
algorithm which runs at the same speed on general-purpose and
dedicated hardware (since general-purpose hardware contains many
extraneous features, e.g. communication buses for peripherals), and so
ultimately ASIC resistance is futile.
(Schemes such as “the developers will just change the proof-of-work
algorithm if ASIC’s appear” do not even make sense — in a
decentralized currency the developers have no such power, while in a
centralized currency proof-of-work is a completely unnecessary waste
of power.)
Is memory hardness desirable?
No. Memory hardness has the effect of increasing ASIC board footprint,
weakening the heat- dissipation decentralization provided by the
thermodynamic limit. Further, it increases the capital costs of mining
equipment relative to the energy costs, which also encourages cen-
tralization (since established miners have amortized their equipment
more than new miners). These effects are amplified by the fact that
SRAM is both several times faster and several times more expensive
than DRAM.
Also, memory hard proofs-of-work often require lots of memory on the
part of the verifiers which is bad for decentralization as already
discussed.
As an aside, since memory is far away and expensive to access on
general purpose computers, memory hardness actually increases the
benefit provided by ASIC’s! This is contrary to the goals of most
memory-hard advocates, and as we have seen above, memory-hardness
worsens the centralizing effects of ASIC’s while weakening the
decentralizing effects.
One more thing worth mentioning is time-memory tradeoff (TMTO). This
is a property of an algorithm which allows higher memory usage to be
traded for a heavier computational load. An algorithm which is highly
susceptible to TMTO has poorly defined memory hardness, which at the
very least complicates analysis. It may also cause an algorithm to
fail to be optimization free.