Entropically secure encryption with faster key expansion (Poster)

An encryption scheme is called perfect if the cipher reveals no information whatsoever about the message that was encrypted. The quantum Vernam cipher, also known as Quantum One-Time Pad (QOTP) or private quantum channel, is a perfect quantum encryption scheme. In order to perfectly encrypt any $n$-qubit state, the necessary and sufficient key length is $2n$ bits. For someone who does not know this key, the state after encryption equals the fully mixed state regardless of the original state.

It has been shown that entropic security setting allows us to have information-theoretically secure QOTP with much shorter keys than $2n$, if the adversary has high uncertainty about the message states. Adversary's uncertainty is quantified using conditional quantum min-entropy. Informally, an encryption scheme is entropically-secure if having the ciphertext does not provide a non-negligible advantage to an attacker in learning any information about the message given that he has high uncertainty about the message. Desrosiers and Dupuis constructed an entropically secure quantum encryption scheme with a key length of $n-t+2\log\fr1\qe$ where $n$ is the number of the message states, $t$ is the conditional quantum min entropy of the message state $\sH_{\infty}(X|E)$ and $\qe$ is the leakage of the cipher. They expand the short key to $2n$ using a public string with a length of $2n$ then apply QOTP. Their key expansion method utilizes universal XOR hash function which consists of Galois Field multiplication in GF$(2^{2n})$.

We have also introduced an entropically secure quantum scheme with a new key expansion method and proved the same shortest key length $n-t+2\log\fr1\qe$. Our key expansion method is faster than the previous ones. This speedup depends on the entropy of the message and can be multi folds in case of high min-entropy of the message. We encrypt a bipartite state $\qf^{AE}\in\cD(\cH_A\otimes\cH_E)$ using a key $k\in\bits^\ell$ where $\ell > n$, and $\ell$ is an even integer. In short, we expand a key $k$ using two random public strings $u\in\bits^\ell$, $v\in\bits^{2n-\ell}$ by applying GF operations, and then appending the outcome to the key itself: $k \| (uk+v)_{\rm lsb}$. GF operations are in GF($2^\ell$) and lsb means taking the last $2n-\ell$ bits. Then the expanded key is used in QOTP encryption. The cipherstate consists of $n$ qubits and $2n$ classical bits.