Is it possible to combine digital signature to provide message addition? Let's assume we are using the textbook RSA where $Sig(x)=x^d$. Alice has public key $(e,N)$, and private key $(d,p,q)$.

Now, if Alice sends $Sig(5)$ and $Sig(10)$ to Bob, where $5$ and $10$ is just numbers, is there a way to combine them on Bob's side to get $Sig(15)$? 

If this is not possible in RSA, is it possible in other schemes like, for example, ElGamal) ?

Please note that I'm looking for a solution to handle this via "addition", not "multiplication".
No, in textbook RSA signature with $\operatorname{Sig}(x)=x^d\bmod N$, there is no method to deduce $\operatorname{Sig}(15)$ from $\operatorname{Sig}(5)$ and $\operatorname{Sig}(10)$.



It is possible to deduce $\operatorname{Sig}(50)$, by using the general fact that in textbook RSA signature, if $x$ and $y$ are positive integers (with $xy$ below the limit for messages that can be signed, if any), then $\operatorname{Sig}(xy)\;=\;\operatorname{Sig}(x)\operatorname{Sig}(y)\bmod N$. But notice that being able to obtain an admissible signature for a message from the signature of other messages is considered a break of a signature scheme. That's why textbook RSA signature with $\operatorname{Sig}(x)=x^d\bmod N$ is not a secure signature scheme (rather, it is a building block towards one).

Further, if we have a signature scheme such that $\operatorname{Sig}(x+y)$ can be deduced from $\operatorname{Sig}(x)$ and $\operatorname{Sig}(y)$, then the signature of any message $x&gt;0$ can be deduced from $\operatorname{Sig}(1)$ with $O(\log(x))$ deductions. That's why the literature tends to consider such additively homomorphic signature schemes only to repel them.



If we really want an additively homomorphic signature scheme, we can define one related to RSA, as $\operatorname{Sig'}(x)=\operatorname{Sig}(g^x\bmod N)=(g^x\bmod N)^d\bmod N=(g^d\bmod N)^x\bmod N$ for some fixed public $g$ generator of the groups $\mathbb Z_p^*$ and $\mathbb Z_q^*$ (it is easier and best that $(p-1)/2$ and $(q-1)/2$ are primes). In order to verify the alleged signature $s$ of $x$ per this system, it is checked that $s$ is the textbook RSA signature for $g^x\bmod N$, that is $0\le s&lt;N$ and $s^e\bmod N\,=\,g^x\bmod N$. For all non-zero $a$ and $b$ and messages $x$ and $y$ it holds that $\operatorname{Sig'}(ax+by)\;=\;\operatorname{Sig'}(x)^a\operatorname{Sig'}(y)^b\bmod N$. This is comes at the price of extreme insecurity: revealing the signature of $x$ and $y$ allows the efficient computation of the signature of any multiple of $\gcd(x,y)$.