# what is the probability for an adversary to find the new key after adding new entropy in a group where computational diffie hellman is hard?

Let's say I have an Elliptic curve group $$E(\mathbb{F}_q)$$ with base Point $$G$$ and large prime order $$n$$. Computational Diffie-Hellman is assumed to be hard in that group.
$$H: \{0,1\}^*\rightarrow \{0,1\}^\lambda$$, where $$\lambda$$ is fixed, is a collision resistant hash function.

• I have a symmetric encryption key $$k\leftarrow H(X+Y)$$, with $$X,Y \in E(\mathbb{F}_q)$$
• Let's suppose after a certain time, the value $$\alpha = X+Y$$ is leaked, and I decide to update $$k$$ as follows: $$k\leftarrow H(X+Y+Z)$$ where $$Z\in \mathbb{F}_q$$ is a point sampled at random
• Then (ElGamal Encryption) I sample at random $$r\in \mathbb{Z}^*_n$$ and compute $$c_1 = r.G$$ and $$c_2 = X+Y+Z+r.pk_R$$, where $$pk_R$$ is the public key of the receiver. Then, I send $$c_1, c_2$$ to the receiver over an authenticated line.

My question is the following: if a polynomial-time bounded adversary $$\mathcal{A}$$ obtains $$\alpha, c_1, c_2$$, what will be its probability to find/compute the new symmetric key $$k\leftarrow H(X+Y+Z)$$?

• Well, you have asked an easy question in a long way. Isn't is simply $H(a +b)$ is secure if $a$ is known? That depends on the size of $b$, if it is less than the output size of the hash function than, the pre-image search is faster theoretically but not necessarily practically broken if $len(b)>112$. – kelalaka Feb 7 at 21:07
• Thank you for the comment. – vxek Feb 7 at 22:10