There are no good constant-time data structures

You do not need constant-time comparisons of password hashes; to see why, reason through how you'd attack them.

An attack against an HMAC authenticator provides full, incremental control over the hash. No such control exists for a password hash; you can't step from "AAAAAA" to "AAAAAB" without a devastating break in the hash function itself. The capability to predict the input that generates "AAAAAB" implies a total failure of preimage resistance.

You can end up in a situation where timing leaks are relevant to password authentication. To do that, you need to design your own password storage system, and it needs to be badly flawed; for instance, you could literally use a general-purpose fast lookup hash and a chaining hash table. This is yet another reason to simply use bcrypt or scrypt to store passwords; doing so takes timing off the table.

In reality, though, you could just use a salted SHA1 hash (don't, though) and still be safe from this attack.

Counterexample: Cuckoo hashmap, at least for lookup.

Calculate both hashes. Index the table via each hash. Check both table indexes to see if either match.

The only timing information this leaks is the length of the input string, and that an adversary already knows.

Now, there will be some cache information potentially leaked, although there are ways around that too. But this is constant time disregarding cache, and is sublinear.

As for the question of "a nice constant-time comparison algorithm for (say) 160-bit values", it's relatively easy. Do a pairwise comparison of constant-size chunks of the value, mapping <, =, and > to 01, 10, and 11, respectively. Then recurse as necessary. Of course, this assumes you have a constant-time comparison function for a single chunk.

The following should be constant-time relatively to the contents of its inputs:

    // return -1 if x < y, else 0
    static unsigned lt(unsigned x, unsigned y) {
      return -((x - y) >> (sizeof(x)*8-1));
    }

    // return -1 if x != y, else 0
    static unsigned ne(unsigned x, unsigned y) {
      const unsigned d = (x - y) | (y - x);
      return -(d >> (sizeof(d)*8-1));
    }

    // return 1 if x[0..n-1] is lexicographically less than y[0..n-1], else 0
    int less_than(const unsigned char * x, const unsigned char * y, size_t n) {
      unsigned flag = 0;
      unsigned done = 0;
      for(size_t i = 0; i < n; ++i) {
        flag |= lt(x[i], y[i]) & ~done;
        done = ne(x[i], y[i]);
      }
      return flag & 1;
    }

    //quantize run duration to mitigate timing attacks
    checkPasswordWrapper(...) {
        result = checkPassword(...)
        until (getMilliseconds() % 250) {
            sleep(1) //Needs optimization
        }
        return result
    }

Before the hash lookup, calculate a set time into the future, say 300 ms. Then do the hash lookup. Then wait until the predetermined time and return the result.

Waking up a thread at a specific time is just real-time programming 101.

Another idea: if you have some context information that lets you identify that queries are coming from the same entity (same IP address, same tty, same operating system user, whatever), you can impose a shadow ban after 10 unsuccessful tries: fail even if there is a hash "hit", and randomize the times to feed the attacker junk timing data.

There's still the problem of cache noise.

Edit: Actually, even if by coincidence you did get FOOBAZ on the same bucket as FOOBAR, you wouldn't be able to easily distinguish from a single comparison with 5 equal characters from multiple comparisons with smaller number of equal characters. Also, if you're getting that many collisions to the point that this is a problem, you've sized your hash table incorrectly.

[0]: http://en.wikipedia.org/wiki/Trie