Constructing public key encryption schemes with desired ciphertext expansion
From your expanded comments, it appears that you hope to use encryption to do data compression; that is, you have an $N$ Megabyte input, and hope to store it in a 1 Megabyte storage, and later recover it in toto.

Sorry, but that's (probably) impossible.  If there are $2^n$ possible largeInstance's, then you cannot store them in less than $n$ bits, and be able to recover it later.

The only case where it might be achievable is if the largeInstance's have a great deal of redundancy (and so that even though they are $n \times N$ bits long, only $2^n$ values are actually possible); if so, you'd be better off looking at a data compression method, rather than an encryption method.