In may application, I have a series of real number $$\{v_1, v_2, \ldots, v_N\}, v_i\in\mathbb{R}$$. To protect them, I add a random noise $$r\in\mathbb{R}$$ to each themselves like one-time-pad encryption scheme. But due to some more requirements of optimization we only want to generate and use a random noise $$r$$ instead of generating one time pad. Is this model is secure and which level of security can I obtain?

• In my key random noise is similar to random key. Apr 12 '20 at 12:20
• Ask yourself if your could recognize ciphertext for $\{0,1\}$ from ciphertext for $\{0,2\}$. If the answer is yes, that's a valid Choosen Plaintext attack. Various issues: $\{v_1, v_2, \ldots, v_N\}$ is the notation for a set (where order does not matter), when most of the time plaintext is a vector. In crypto, it is seldom appropriate to use reals (elements of $\Bbb R$) for plaintext and ciphertext, for they have no fixed-size representation. A random elements of $\Bbb R$ is not well defined; and it it was, there's no clear way to share one between two actors;
– fgrieu
Apr 12 '20 at 19:44
• Either it is a one-time pad or it isn't, either distributed over a number of bits or some other domain. As long as the domain is long enough and the random number is fully random for each $v_i$ it should count as a one-time pad. As "fully random" is tricky, you should be fine with "random enough" practically speaking. But it is unclear if "random noise" counts as RNG. Apr 13 '20 at 2:17

Conceptually, your scheme is 110% information theoretic secure, as it's a one time pad($$\downarrow$$). But since we tend to use octet based computer architecture these days, $$\mathbb{Z}$$ is more appropriate.
Note 1. Ensure $$|r| \ge |v|$$.
Note 2. Don't want to patronise, but the one time pad is thee most contentious and misunderstood aspect of cryptography ever invented. Especially on this forum. Is $$r$$ truly random or just pseudo-random? It's worrisome that I read 'generate' in a kinda algebraic context...