Design of a Quantum One Way Trapdoor Function

Published: 2021-09-13 10:30:11
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Of late security has become a key concern of data transmission mechanism over a communications channel. In an asymmetric cryptographic system, a public key is shared across an insecure medium. This makes the data exchange vulnerable to potential threat from various attackers. This paper proposes the design of a one-way trapdoor function based upon the principles of quantum computing. A quantum public key is used for encryption and a classical private key is used during decryption of the secret message. The mapping between numbers used in the classical paradigm and their corresponding quantum states is established through the proposed one way trapdoor function.
One of the foremost challenges of data transmission in today’s world is security. An electronic system that is to be reliable needs adequate data security in order to operate in full working order. The security of data depends on a good encryption algorithm. In case of symmetric key encryption, the same key is used for both encryption as well as decryption. In case of asymmetric key encryption a pair of keys namely a private and a public key is used. Data is encrypted with the help of public key and decrypted by the private key. Therefore generation of the pair of keys (public and private) is critically important in any cryptosystem.Numerical trapdoor one-way functions i.e., functions that are “easy” to compute, but “hard” to invert without some additional information (the so-called trapdoor information) are commonly used in asymmetric cryptography. The main feature of these mathematical functions is that they provide the genuine users with a tractable problem, while any illicit user or adversary is faced with a problem that is computationally hard. This barrier of complexity between the valid users and the unauthorized users is the main concept behind a lot of modern public-key cryptosystems.
This paper introduces a one-way trapdoor function based on the axioms of quantum theory. The unit of quantum information is a quantum bit or qubit, vis-à-vis a bit (which takes a value of either 0 or 1) in a classical system. A qubit can exist as the superposition of the quantum |0> and |1> states, which depict the ground and excited states of a single electron. The proposed algorithm presents a quantum one way trapdoor function for the generation of the pair of keys by rotationof qubits.
Rest of this paper is organized as follows: Section 2 discusses various related works in this domain. Section 3 provides an overview of quantum cryptography. Section 4 gives a mathematical background of one way trapdoor functions. Section 5 focuses on the rotation of qubits based on quantum trapdoor functions. Section 6 introduces the proposed algorithm. Section 7discusses various security aspects of the given procedure and Section 8 concludes the discussions.
Related Works
In the year 1984 Charles H. Bennett and Gilles Brassard [1] first proposed the protocol on quantum cryptography named as BB84 where BB stands for Bennett and Brassard respectively. The algorithm was based on Heisenberg’s Uncertainty principle. In the year 1992 Charles H. Bennett [2] again proposed a much simplified version of BB84 which required only two polarization states and named it as B92. In the year 2002 Ching-Nung Yang and Chen-Chin Kuo [3] suggested an improved quantum key exchange protocol using BB84 and B92. One of them was to increase the idealized maximum efficiency to 28.6% with the average complexity order n2, the other was to increase to an efficiency of 42.9% and thereby has the average complexity order of n2.86. In the year 2011 M. Houshmand and S. Hosseini-Khayat [4] proposed an entanglement-based quantum key distribution in whichan updated version of Cabello’s definition of efficiency of quantum key distribution protocols was used to compare between their protocol and BB84. An arrangement of qubit pairs were obtained by separating the stream of qubits thereby giving less information about the key bit than BB84. Initially the participants publically agree on two 2-qubit unitary transformations, say U1 and U2. Here all transmitted qubits are used to differentiate from BB84 and half of the qubits are discarded on average. In the protocol one classical bit is used to acknowledge the receipt of each qubit and one classical bit is used for determining the basis of each of the group of qubits, thus giving an advantage over an eavesdropper in case of an intercept resent attack. In the year 2014, Abdurrahman Aldhaheri, Khaled Elleithy, Majid Alshammari Hussam [5] proposed a novel secure quantum key distribution algorithm, which overcame the weaknesses in BB84 and B92 protocols. It was obtained by removing the need for two communicating parties to confirm their used basis over a public channel. In their work they exchanged the session key over the quantum channel. Moreover the users’ authentication and confidentiality was maintained by exchanging random basis and nonce. In the year 2013 Ammar Odeh, Khaled Elleithy, Muneer Alshowkan, Ema Abdelfattah [6] presented a new model for Quantum Key Distribution (QKD) [7] between three parties in which there was a trusted centre that provided clients with the requisite secret information to securely communicate between each other. In this algorithm there was no need for any physical channel for checking the qubits sequence. The algorithm had two parts – The first part deals with the user authentication and quantum bases distribution and the second part the data transfer over the quantum channel. The algorithm improved the efficiency by removing the unnecessary rounds to check the quantum bases and provide authentication.
Application of Quantum Computing on Cryptographic Systems
Quantum cryptography [8] applies the simple idea of physics to develop a cryptosystem that is entirely secure besides being compromised without the information of the sender or the receiver of the messages. The word quantum refers to the fundamental comportment of the smallest particles of matter and energy: quantum theory [9] explains everything that exists and nothing can be in violation of it. Quantum key distribution [10] is built on the principles of quantum physics and that of classical information theory. The key to be distributed must be common and secret. Quantum key distribution guarantees long-term secrecy of confidential data transmission. A fundamental and fairly important feature of quantum mechanics is quantum entanglement [11], a physical phenomenon that occurs when quantum particles behave in such a way that the quantum state of each particle cannot be defined separately.
One Way Trapdoor Function
Let A be a set of numbers and Q be the set of quantum states [12] of a system. Then a one way function can be depicted as a mapping M: A→Q that is simple to evaluate but hard to invert. A quantum one way trapdoor function [13]is one, which is reversible by means of some a-priori (trapdoor) information.
M : A → Q
x M(x)
easy given t
Assuming A ∈Zn :={0,1,2,…,n– 1|n∈ N} as the input set of integers for the one way trapdoor function and the quantum state |φa˃ be the corresponding output, with the initial state being|0˃ in the Hilbert space H; then for any randomly chosen A ∈ Zn, a rotation operator O ̂:H→H may be defined as follows: |0˃ □(→┴O ̂ )|φs˃.
Hence all possible sets of output states of the one way trapdoor function will be Q ≡ {|φa˃ | A ∈Zn}, Q ∈ H. So, if M:Zn Q is a one-to-one mapping, then there exists a unique A ∈Zn such that |0˃ O |φa˃ and the cardinality of Zn is same as that of Q.
Rotations of Qubit Using Quantum Trapdoor Function
An arbitrary single qubit state lying on the x − z plane can be written as: |ɸ(θ)> = cos (θ/2) |0z> + e^iφsin (θ/2) |1z>,where 0 ≤ θ ≤ π and 0 ≤ φ≤2π define a point on a unit sphere, known as the Bloch sphere [14], as illustrated below.
Therefore a qubit can represent a range of states on the x−z Bloch plane as against the conventional way of storing a discrete variable by means of two real values, given by “0” and “1”. Now a rotation about the y-axis R ̂(θ)= e^(-iy ̂ θ/2) is the Pauli Y operator, that yields |ɸ(θ)>= R ̂(θ) |0z>, where y ̂= i(|1z>
Since the Pauli operators [15] are both unitary and Hermitian, the rotation vector can be expanded as: R ̂(θ) = e^(-iy ̂ θ/2) = cos⁡〖θ/2 I-i sin⁡〖θ/2 y ̂ 〗 〗=(■(cos⁡〖θ/2〗&〖-sin〗⁡〖θ/2〗@sin⁡〖θ/2〗&cos⁡〖θ/2〗 )) Where I is the identity matrix.
Since the input to the quantum trapdoor function is a random integer A, which is evenly spread over Z2n for n∈N, and a qubit is initially prepared as |0z>, an n-bit string is adequate to find the input A for a fixed n. For n ∈N and A ∈Z2n, the qubit state is rotated by Aθn around the y-axis with θn =π/2n−1.
Therefore for any constant n∈N, the one way trapdoor function Qn={| ɸA(θn)>|A ∈Z2n, θn= π/2n−1}, can be written as: |ɸA(θn)> ≡R ̂(Aθn)|0z>= cos(Aθn/2) |0z> + sin(Aθn/2) |1z> (1)
Hence, both Z2n and Qn remain unknown if n is not known.
Hence for any given value of n and A, the mapping A→|ɸA(θn)> is easy to obtain as it rotates only a single-qubit. Inverse of the mapping A→|ɸA(θn)> is to recover A from the given qubit |ɸA(θn)> chosen randomly from the known set Qn. The inverse of the function is to identify the different non-orthogonal states [17] chosen randomly from Qn for a given value of n. As n increases the number of non-orthogonal states also increases and for n>>1, the nearest overlapping so obtained as the following: = cos(θn/2) → 1.
Therefore, a projective Von Neumann cannot discriminate all of the states for n >>1, as the number of possible results in such a calculation is limited by the dimensions of the qubit.
Encryption and Decryption of the Pain Text
Key Generation
Every user in the cryptosystem will generate a pair of keys, a Private Key (Kpv) and a Public Key (Kpb). The following algorithm illustrates the key generation process:
Algorithm KeyGen
Let n>>1 be a random positive integer.
Let A be a set of random integer strings of length N where A = {A1, A2, A3 …, AN} with Aj chosen independently from Z2n and A ∈Z2n, and let X be a random binary string of length N where X = {X1X2X3…XN} with Xi ∈ {0, 1}. Obtain N qubits in the state |0z> = 0⊗N by applying N parallel Hadamard operations.
For each qubit Aj in A
Rotate the qubit R ̂(j)(Ajθn) by angle θn= π/2n−1 if and only if Xj= 1,such that the jth qubit |ɸAj(θn)>j =R ̂(j)(Ajθn)|0z> takes the form |ɸA(θn)>≡R ̂(Aθn)|0z> = cos(Aθn/2) |0z> + sin(Aθn/2) |1z> where 0 ≤ θ
Private Key Kpv = {n, A} and Public Key Kpb = {N, |ɸ(pk)(θn)>} with the N qubits states |ɸ(pk) (θn)> ≡ 0⊗Nj|ɸAj(θn)> j.
Let Bob be the sender and Alice be the receiver. Now Bob wants to send Alice an r-bit message M = (M1,M2, …,Mr), with Mj∈ {0, 1} and r ≤ N. To encrypt the plain text without altering the order of the public-key qubits the following algorithmis employed:
Algorithm Encrypt
Obtain Alice’s authentic public key Kpb. If r > N, Bob requests Alice to increase the length of her public key. Encrypt the jthbit of his message Mj by the rotation R ̂j(mjπ) on the qubits of the public key, which becomes |ɸAj,Mj(θn)>j=R ̂(Mjπ)|ɸAj(θn)>j
Obtain the quantum encrypted message as: |ɸ(Kpy)Aj,M(θn)>=⊗Nj=1|ɸAj,Mj(θn)>j is sent to Alice.
The message is encoded in the first r qubits of the cipher text so that, in the decryption process Alice only focus onthis part of the cipher text, neglecting the remaining N − r qubits, which do not have any additional information.
The following algorithm is used by Alice to decrypt the cipher text |ɸ(Kpy)A,M(θn)> to obtain the message m.
Algorithm Decrypt
Apply R ̂(j)(Ajθn)-1 on the jth qubit of the cipher text. Measure each qubit of the cipher text in the basis {|0z>,|1z>}.
The foremost intention of an eavesdropper is to decrypt the plaintext from the cipher text meant for Alice. Whereas there is always an additional determined objective concerning to the retrieval of the private key from Alice’s public key [18]. A cryptosystem is said ɸAj,Mj(θn)>j to be broken if any of the two purposes is satisfied provided that the opponent has access to all of the texts intended for Alice. Without doing authentication [19] which is an integral measure of quantum cryptography any encryption is vulnerable to an impersonation attack. To focus on its importance it is described in the Encrypt algorithm that Bob should obtain an authentic copy of Alice’s public key [20]. The Private Key (Kpv) of each entity is Kpv ={n,A} where n>>1 is a random positive integer and A is a set of random integer strings of length N where A=(A1, A2, A3, …, AN) with Aj chosen independently from Z2n and A∈Z2n. The entropy for n is H(n) = log2(|N ̃|), where |N ̃|denotes the number of elements in N ̃.
Again the entropy for A is H(A|n) = Nn Hence we can infer that the entropy of the private key Kpv is, H(Kpv) = H(n) + H(A|n) = log2(|N ̃|) + Nn >> N To satisfy the above criteria it is sufficient to have either n≫1 or log2(|N ̃|) >>N.
In Section 6 both the conditions are satisfied simultaneously because is a randomly chosen from the integer set N with the constraint n>>1. Hence Eve’s information gain is very less as compared to H(Kpv), which therefore remains basically unknown her.
This paper proposes the design of a one way trapdoor function using rotation of quantum bits. It demonstrates the application of one way trapdoor functions in the context of a quantum public-key cryptosystem. In the encryption algorithm, the user creates a pair of asymmetric keys: a private key, that is purely classical, and a public key, that contains a set of independently generated qubits. The sender encrypts the message with the receiver’s public key through rotation the state of its qubits. An eavesdropper cannot decode the secret message without having knowledge of the recipient’s private key. The focus of the work is to illustrate how the axioms of quantum computing can establish a correct theoretical framework of designing a one way trapdoor function. The proposed framework establishes a security barrier between authentic users and eavesdroppers, using the foundations of quantum public-key encryption.

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