# Generation and Distribution of Quantum Oblivious Keys for Secure Multiparty Computation

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## Abstract

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## Featured Application

**Private data mining.**

## Abstract

## 1. Introduction

#### 1.1. Secure Multiparty Computation and Oblivious Transfer

#### 1.2. State of the Art

## 2. Methods

#### 2.1. Generating the OTs

#### 2.2. Oblivious Key Distribution

## 3. Results and Discussion

#### 3.1. Security

**Theorem**

**1.**

**Proof.**

#### 3.2. Efficiency

**F**, $\tilde{n}$ bit comparisons (if using the technique proposed in [29] to find the required f), and one additional evaluation of f. Cryptographic hash functions are designed so that their time complexity is linear on the size of the input, which in this case is $\ell =4k+2\tilde{n}+4$. To compute the universal hashing, the construction in [29] requires $\tilde{n}k$ binary multiplications. Thus, the running time of ${\pi}_{COMH}$ is linear on the security parameter k. On the other hand, ${\pi}_{QOT}$ has two security parameters: n, associated to the size of the keys used to encrypt the transferred bits, and m, associated to the security of the measurement test done by Alice. The protocol requires $n+m$ qubit preparations and measurements, $n+m$ calls of the commitment scheme, and n bit comparisons. This leads to an overall time complexity of $O\left(k\right(n+m\left)\right)$ for the ${\pi}_{HOK}$ protocol, which is linear in all of its security parameters.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**In secure multiparty computation, N parties compute a function preserving the privacy of their own input. Each party only has access to their own input–output pair.

**Figure 2.**Quantum OT protocol based on secure commitments. The ⨁ denotes the bit XOR of all the elements in the family.

**Figure 3.**Oblivious keys. Alice has the string k and Bob the string $\tilde{k}$. For each party, the boxes in the left and right represent the bits of their string associated to the indices i for which ${x}_{i}$ equals 0 (

**left box**) or 1 (

**right box**). Alice knows the entire key, Bob only knows half of the key, but Alice does not know which half Bob knows.

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**MDPI and ACS Style**

Lemus, M.; Ramos, M.F.; Yadav, P.; Silva, N.A.; Muga, N.J.; Souto, A.; Paunković, N.; Mateus, P.; Pinto, A.N.
Generation and Distribution of Quantum Oblivious Keys for Secure Multiparty Computation. *Appl. Sci.* **2020**, *10*, 4080.
https://doi.org/10.3390/app10124080

**AMA Style**

Lemus M, Ramos MF, Yadav P, Silva NA, Muga NJ, Souto A, Paunković N, Mateus P, Pinto AN.
Generation and Distribution of Quantum Oblivious Keys for Secure Multiparty Computation. *Applied Sciences*. 2020; 10(12):4080.
https://doi.org/10.3390/app10124080

**Chicago/Turabian Style**

Lemus, Mariano, Mariana F. Ramos, Preeti Yadav, Nuno A. Silva, Nelson J. Muga, André Souto, Nikola Paunković, Paulo Mateus, and Armando N. Pinto.
2020. "Generation and Distribution of Quantum Oblivious Keys for Secure Multiparty Computation" *Applied Sciences* 10, no. 12: 4080.
https://doi.org/10.3390/app10124080