In the paper we propose a method of quantum hashing which mostly combines the existing constructions of universal hash families with quantum one-way functions. Next, we define the concept of a quantum hash generator and present an approach for building a large number of various quantum hash functions. The construction is based on the integration of a classical e-universal hash family and a given family of functions quantum hash generator. The proposed construction combines the properties of robust representation of information by classical error-correcting codes together with the possibility of highly reliable representation of information by quantum systems. In particular, we present a quantum hash function based on the Reed-Solomon code and prove that this construction is optimal in the sense of the number of qubits needed.
In the paper, we define the concept of the quantum hash generator and offer design, which allows to build a large amount of different quantum hash functions. The construction is based on composition of classical ∈ -universal hash family and a given family of functions-quantum hash generator. In particular, using the relationship between ∈ -universal hash families and Freivalds fingerprinting schemas we present explicit quantum hash function and prove that this construction is optimal in the sense of number of qubits needed for construction.
In the letter we define the notion of a quantum resistant ( -resistant) hash function which consists of a combination of pre-image (one-way) resistance ( ε -resistance) and collision resistance ( δ -resistance) properties. We present examples and discussion that supports the idea of quantum hashing. We present an explicit quantum hash function which is ‘balanced’, one-way resistant and collision resistant and demonstrate how to build a large family of quantum hash functions. Balanced quantum hash functions need a high degree of entanglement between the qubits. We use a phase transformation technique to express quantum hashing constructions, which is an effective way of mapping hash states to coherent states in a superposition of time-bin modes. The phase transformation technique is ready to be implemented with current optical technology.
Cryptographic hash functions are fundamental primitives widely used in practice. For such a function
f:{0,1}n→{0,1}m, it is nearly impossible for an adversary to produce the hash
f(x) without knowing the secret message
x∈{0,1}n. Unfortunately, all hash functions are vulnerable under the side-channel attack, which is a grave concern for information security in practice. This is because typically
m≪n and an adversary needs only
m bits of information to pass the verification test. In sharp contrast, we show that when quantum states are used, the leakage allowed can be almost the entire secret. More precisely, we call a function that maps
n bits to
m qubits a quantum cryptographic function if the maximum fidelity between two distinct hashes is negligible in
n. We show that for any
k=n−ω(logn), all quantum cryptographic hash functions remain cryptographically secure when leaking
k bits of information. By the quantum fingerprinting constructions of Buhrman et al.~({\em Phys. Rev. Lett.} 87, 167902), for all
m=ω(logn), there exist such quantum cryptographic hash functions. We also show that one only needs
ω(log2n) qubits to verify a quantum cryptographic hash, rather than the whole classical information needed to generate one. Our result also shows that to approximately produce a small amount of quantum side information on a classical secret, it may require almost the full information of the secret. This large gap represents a significant barrier for proving quantum security of classical-proof extractors.
In the paper we define a notion of quantum resistant ((δ, є)-resistant) hash function which combine together a notion of pre-image (one-way) resistance (δ-resistance) property and the notion of collision resistance (є-resistance) properties. We present a discussion that supports the idea of quantum hashing oriented for cryptographical purposes. We propose a quantum setting of a classical digital signature scheme do demonstrate a theoretical possibilities and restrictions of (δ, є)-hashing. The assumption we use is that a set of qubits (quantum hash) we generate, send, and receive during the execution of a protocol can be stored for a certain (a large enough) amount of time; next, the scheme requires the high degree of entanglement between the qubits which makes such a quantum hash. These properties make quantum hash cryptographically efficient.