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1

No, assuming your code is running in an environment completely controlled by an adversary (no TPM etc.). Note that if the instructions were encrypted in some way we assume an adversary could simply intercept them after they are decrypted. Your best shot is to obfuscate the code making it cumbersome for someone else to grasp the logic behind it. This can ...

0

Assume we are working in a cyclic group $G$ with generator $g$ and let $A$ denote the public key in use. From the definition of ElGamal encryption, we have $R_i = g^{r_i}$ and $c_i=A^{r_i}\cdot m_i$, where $r_i$ is some random number, for $i\in\{1,2\}$. Hence, with $R:=R_1\cdot R_2$ and $c:=c_1\cdot c_2$ (where $\cdot$ denotes $G$'s operation), we have ...

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Regardless whether input is small, $n$ must be large enough to avoid factorization. Next, $r$ must be sampled from a large space to avoid decryption by trial-and-error. Some crypto and big-numbers library (bouncycastle, openssl, crypto..) might be handy to implement such an algorithm. It would be safe to choose an implementation rather than write it from ...

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I'd say the people dealing in homomorphic encryption are at the forefront of applied mathematical research (Shai, Smart, Gentry, Boneh...). So world-class certainly wouldn't hurt! Pick up a paper cited by HELib (for one example starting place), read it. Fail to understand it, find a cited paper, read it... repeat until eventually you find a text book ...

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