Cryptographically Secure Hash Algorithm with Very Specific Property First of all, I apologize if this is a stupid question, but I don't have any background in cryptography.

I'm looking for a secure hash algorithm $H$ with the property $H(K_1+n)\oplus H(K_2+n)=H(K_1\oplus K_2 + n)$.  Would it be feasible to implement a hash algorithm like this, and could a hash algorithm even be secure if it had this property?  If not, would a hash algorithm with the property $H(K_1\oplus n)\oplus H(K_2 \oplus n)\oplus H(K_3\oplus n)=H(K_1\oplus K_2\oplus K_3\oplus n)$ be any better?

Thanks in advance for any comments.
No such function with either property would meet the requirements of a secure hash function; either of those properties would make it easy to find preimages, that is, given a value $H(x)$, you can find a value $y$ with $H(y) = H(x)$.

First off, I assume that $n$ is a constant for the hash function; if we were to assume that the first property holds for any $n$, then it is easy to see that the only function that has that property is $H(x)=0$ for all $x$.

To find a preimage of a function with the first property, we would compute the hashes:

$$H(1 + n)$$
$$H(2 + n)$$
$$H(4 + n)$$
$$H(8 + n)$$
$$...$$
$$H(2^k + n)$$

Then, when given a value $H(x)$, we use Gaussian Elimination to find a subset of the above list ${a, b, c, ..., z}$ with $H(a + n) \oplus H(b + n) \oplus H(c + n) \oplus ... \oplus H(z+n) = H(x)$
Then, because of the property, we know that $H(a \oplus b \oplus c \oplus ... \oplus z + n) = H(x)$.

With a function with the second property, it is only slightly trickier.  In that case, we would compute the list:

$$H(1 + n) \oplus H(0 + n)$$
$$H(2 + n) \oplus H(0 + n)$$
$$H(4 + n) \oplus H(0 + n)$$
$$H(8 + n) \oplus H(0 + n)$$
$$...$$
$$H(2^k + n) \oplus H(0 + n)$$

and proceed as previously.