# Development of Quantum Protocol Modification CSLOE–2022, Increasing the Cryptographic Strength of Classical Quantum Protocol BB84

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Protocol BB84

_{K}matrix of order m × n, is also agreed upon. Both matrices can be known in advance or can be determined during the execution of the protocol, and then can be sent via the classical channel. In turn, the matrix (r + m) × n, the rows of which are the rows P

_{C}and P

_{K}, if taken together, must have the rank r + m. Alice randomly selects the sequences of bits: strings i from 2n – bit, b ∈ F

_{2}

^{2n}, where F

_{2}defines the field of 2 elements {0; 1}, that is, field of integer numbers modulo 2. Then, the state $|{i}^{b}\rangle =|{i}_{1}^{{b}_{1}}\rangle \dots |{i}_{2n}^{{b}_{2n}}\rangle $ is encoded. For each bit, the coordinate basis is randomly selected, rectilinear or diagonal, with which the bit will be encoded. When transmitting the photon from Alice, Bob will inform her about the photon receiving, but will not measure it.

^{B}, and thus output the bit that Alice sent correctly, in the case of absence of noise and signs of eavesdropping on the communication line from the side of an attacker.

_{s}and b

_{s}

_{.}For every $j\in \left[1\dots 2n\right]$, such, that the value s

_{j}= 0, the participants of the data transfer process Alice and Bob publish the value of the bit with jth number.

_{a}, then they interrupt the protocol execution. The preliminarily fixed parameter p

_{a}of the protocol, in fact, is the relationship of permitted bit flips, intended for the testing process.

^{B}.

^{B}as necessary. String $\xi =x{P}_{C}^{T}$ is named the syndrome of string x (relative to P

_{C}). If bits correspond, then they are rejected, and other bits create the common secret m–bit key.

_{j}to the composite system. This is true because the theorem on prohibition of cloning (anti–cloning theorem) [7] guarantees that it is impracticable to reproduce a particle of an unknown state.

## 3. Protocol BB84–Info–Z

- –
- Generalized numbers of bits n, n
_{z}and n_{x}(n–the numbers of informational bits, where Z and X are the test bits, accordingly); - –
- Section P = (s, z, b) to divide n–bit string i into three non–overlapping sets (I, T
_{Z}и T_{X}); - –
- Two special thresholds, which are separate (p
_{A,}_{z}and p_{A,}_{x}) in place of one threshold (p_{A}).

- –
- numbers denoted as n, n
_{z}and n_{X}(specified as the relationship N = n + n_{z}+ n_{x}); - –
- thresholds of errors denoted as P
_{A}_{,Z}and P_{A,X}, r × n (which correspond to the linear code of error correction C); - –
- matrices of privacy enhancement m × n (representing the linear key of generation function).

_{C}and P

_{K}matrices are assembled as linearly independent.

_{z}; | B | = n

_{x}; and | s + z + b | = N.

- –
- I (information bits, where s
_{j}= 1) size n; - –
- T
_{Z}(test bits Z, where z_{j}= 1) with size n_{z}; - –
- T
_{X}(test bits X, where b_{j}= 1) with size n_{x}.

_{1}…b

_{N}via the classical channel to Bob. Bob gauges and measures every qubit that he has received and saved. When measuring ith qubit, Bob measures it in the Z–coordinate basis if b

_{i}= 0, and measures it in the X–coordinate basis if b

_{i}= 1. This string of bits that Bob measured is designated as i

^{B}. If noise and eavesdropping are absent, then the bit string is equal to i

^{B}= i.

_{j}= 1, while test Z and X bits (which will be applied for testing) are n

_{Z}+ n

_{X}with s

_{j}= 0. Substrings are denoted by i and b, and correspond to information bits i

_{S}and b

_{S}, accordingly.

_{Z}· p

_{a},

_{Z}test Z–bits are, or more than n

_{X}· p

_{a},

_{X}test X–bits between them are different, then they interrupt this protocol, where p

_{a},

_{Z}and p

_{a},

_{X}are preliminarily coordinated thresholds of errors. Alice and Bob keep values of residual n bits (information bits where s

_{j}= 1) a strict secret. The bit chain of Alice is designated as x = i

_{s}, and the bit string of Bob is denoted as x

^{B}.

_{C}matrix), which includes r bits and is determined as $\xi =x{P}_{C}^{T}$. Using the value ξ, Bob rectifies the errors in his string of bits x

^{B}(it is similarly x). The final key is m–bit sequence and is determined as $k=x{P}_{K}^{T}$. Alice and Bob, together, calculate it. It is obvious that the protocols are very similar.

## 4. Description of Cyberattack of Eve and the Properties of It

_{j}to the composite system. Then, she stores her quantum probes in the quantum memory for the subsequent measurements, and dispatches to Bob his part of the system [11].

_{j}; they are determined by Eve in advance and, thus, are corrected and fixed for all feasible variants and options of i, b and s.

#### 4.1. Cyberattack of Eve on the Separate Qubit

#### 4.2. Spreading the Cyberattack to the Several Qubits–Collective Cyberattack

_{j}= 1. The set $\left\{j|{s}_{j}=1\right\}$ has n elements and is global. The global corresponding values s, b, x can now be saved as:

#### 4.3. Probability of Errors

#### 4.4. Probability of Errors Occurrence in the Conjugate Basis

## 5. Security Confirmation of Classical Protocol BB84 against the Collective Cyberattacks

#### Proof of Security

_{1}… c

_{2n}, appropriate to all qubit measurements in some valid coordinate basis b. Let $\overline{X}=\frac{\left|{C}_{I}\right|}{n}$ be the average value of selection respective to incorrect bits of information;

## 6. Security Proof for the Protocol BB84–Info–Z against the Collective Cyberattacks

#### 6.1. General Collective Cyberattack of Eve

_{j}to the composite system. Then, Eve holds in her quantum memory subsystem E

_{j}, which is the state of her quantum probe; next, she dispatches to Bob the subsystem T

_{j}. This subsystem is the qubit dispatched from Alice to Bob (it may be modified by Eve’s cyberattack U

_{j}).

_{j}of Eve is directed on the jth qubit, presented by orthonormal coordinate basis. The cyberattack described as:

_{j}, that was fastened and fixed to jth qubit.

#### 6.2. Proof of Security

_{TX}random variable matches the string of bits of the errors, on these bits. Hence, it is possible to consider the choice of n–bits indexes of information (Info) and n

_{x}–bits TEST–X as a random selection (after n, n

_{z}and n

_{x}numbers; and bit indexes TEST–Z, which was selected already) and to apply the theorem of Heffding [9].

_{1}… c

_{n + nx}, which consists of errors in the bits n + n

_{x}Info and TEST–X, if Info bits were coded in the coordinate basis X, then we can use Heffding’s theorem: let us take the sample with size n without changing from combination c

_{1}, …, c

_{n + nx}. In the above discussion [8], the following theorem is actually proven:

**Theorem.**

_{n}–bit keys, the distance between the states of Eve, appropriate to k and k’, meets the following requirements and boundaries:

#### 6.3. Reliability

_{Z}bits into two subsets, with sizes n and n

_{Z}(provided that the bits indexes of TEST–X was already selected). Thus, it matches Heffding’s sample.

## 7. Protocol CSLOE–2022 (BB84–CSLOE–2022)

^{b}, as Alice does, then he goes into the same state. Then, Bob must measure the same polarization in the string i

^{B}. Hence, he can output properly the bit that Alice was going to dispatch in that case, if there is no noise orsigns of eavesdropping in the communication channel.

#### 7.1. Cloning Methods

- (1)
- Optimal symmetric universal quantum copying machine (UQCM) proposed by Vladimir Bužek (Buzhek, in various sources is spelled differently)–Mark Hillery (BH) in 1996;
- (2)
- Symmetric universal quantum copying machine (UQCM) proposed by Nickolas Gisin–Serge Massar and their scientific group in 1997;
- (3)
- Asymmetric universal quantum copying machine (copier)–UQCM.

_{A}and F

_{B}, that confirms the inequality of non–cloning:

#### 7.2. Comparison of Protocols

## 8. Conclusions

- The rate of errors should be checked separately so that it remains below the threshold values ${p}_{a,z}$ and ${p}_{a,x}$ for bits TEST–Z and TEST–X; accordingly, when in the quantum protocol BB84, the threshold value of error rate p
_{a}is applied to all bits of TEST jointly [7]. - The indexes and information indicators of the interceptor Eve (in security terms) and probability of the error-correcting code failure (in reliability terms) differ from the indexes and indicators in the case of classical quantum protocol BB84 [8].

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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Bit of Alice | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | |

Basis of Alice | × | + | + | × | + | × | + | × | |

Polarization of Alice | ⬈ | ↑ | → | ⬉ | → | ⬉ | ↑ | ⬈ | |

Basis of Bob | × | × | × | + | + | × | + | + | |

Measurement of Bob | ⬈ | ⬉ | ⬈ | → | → | ⬉ | ↑ | → | |

Public discussion | × | + | + | * | + | × | + | * | |

Shared secret key | 0 | 1 | 0 | 0 |

Protocol | Error Threshold |
---|---|

BB84 | 11% |

BB84–Info–Z | 7.56% |

CSLOE–2022 | 11% |

Protocol | Working Distance |
---|---|

BB84 | 70 km |

BB84–Info–Z | 70 km |

CSLOE–2022 | 511 km |

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**MDPI and ACS Style**

Cherckesova, L.V.; Safaryan, O.A.; Beskopylny, A.N.; Revyakina, E. Development of Quantum Protocol Modification CSLOE–2022, Increasing the Cryptographic Strength of Classical Quantum Protocol BB84. *Electronics* **2022**, *11*, 3954.
https://doi.org/10.3390/electronics11233954

**AMA Style**

Cherckesova LV, Safaryan OA, Beskopylny AN, Revyakina E. Development of Quantum Protocol Modification CSLOE–2022, Increasing the Cryptographic Strength of Classical Quantum Protocol BB84. *Electronics*. 2022; 11(23):3954.
https://doi.org/10.3390/electronics11233954

**Chicago/Turabian Style**

Cherckesova, Larissa V., Olga A. Safaryan, Alexey N. Beskopylny, and Elena Revyakina. 2022. "Development of Quantum Protocol Modification CSLOE–2022, Increasing the Cryptographic Strength of Classical Quantum Protocol BB84" *Electronics* 11, no. 23: 3954.
https://doi.org/10.3390/electronics11233954