What asymetric scheme provides the shortest signature, while being secure?
You cannot have a secure signature scheme in less than 50 bits. Demonstration: the attacker can just enumerate all sequences of 50 bits until a match is found. Indeed, one point of digital signatures is that the verification algorithm can be computed by just everybody, since it uses only the public key (which, by definition, is public).
Best you can hope, theoretically, is a signature of n bits for a security of 2n. Traditional threshold of infeasibility was n = 80 bits, but the relentless advance of technology tends to raise that. Modern cryptographers tend to jump to 128 bits, because that's a power of two, hence beautiful (Kant notwithstanding, most aesthetic judgements are relative). 100 bits ought to be safe for quite some time, though.
Among known signature algorithms which are believed to be secure, BLS is right now the best in class; it will produce signatures of size 2n bits for a security level of 2n, so you are contemplating signatures of size 160 to 200 bits at least. There are algorithms which can go below (down to about 1.4*n) but they have a rather long history of breakage and fixing and breakage again (I am talking about SFLASH and its ilk), so their use is not really recommended, and there's no directly usable standard.
Another possibility is to use RSA with ISO/IEC 9796-2 padding. As the linked article shows, there are some subtleties with regards to its usage. This signature scheme is a scheme "with recovery" meaning that while the signature is rather bulky (1024 bits if you use a 1024-bit RSA key), it can also embed some of the data which is signed; so, if your problem is about appending a signature to a message, then this scheme will induce relatively low overhead. As the article explains, you get 261 protection for a signature overhead of 22 bytes, aka 176 bits; to get good security, you would have to use a hash function with an output size of, say, 224 bits, leading to a signature overhead of 240 bits -- not as good as the 200 bits of an equivalent-strength BLS, but better than the 400 bits of DSA or ECDSA (for still the same strength level).
The problem with a short signature is that it makes it easier to get a collision. You also can't quite as easily make it prohibitively long to run (by using a slow hash or increasing iterations) as the file size can differ by orders of magnitude. If you make it slow enough to safely sign "hi there", then it is going to have a really hard time signing a 50 gigabyte bluray release. Thus, the easiest option left to us is to increase the number of bits and thus increase the number of tries required to get a collision if the algorithm is not otherwise broken.
It is also worth noting that the size of the key has nothing to do with the size of the signature. The size of the hash is what determines the signature size because a signature is a hash of the file which is then encrypted with the private key such that the public key can decrypt it and compare it to it's own calculation of the hash. As long as you don't use an encryption scheme that requires padding, the length of the signature is completely independent of key size.
Edit: Ok, for the updated question it sounds like you are not actually looking for a signature. A signature takes a hash of a file and then encrypts it with the private key, thus a) others can verify the file has not changed and b) others can verify you attest to the file. This works because someone can't make another file that fits the same hash value.
What you are looking for is a challenge/response that can verify that the key is held by the person trying to get in. This is generally done with a symmetric key rather than an asymmetric key and a time stamp is encrypted and then hashed to reduce the length. The problem is the hash to reduce the length will throw off a public/private system because both sides have to be able to perform the action. I suppose a short output could be derived from a private key signature, but I have a feeling it would be significantly weaker (no provable reason, but it hasn't been done to my knowledge) than using a symmetric key.