#
XOR Chain and Perfect Secrecy at the Dawn of the Quantum Era^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. Research Motivation

- 1.
- Scientific: There has been little research on key reusing under OTP. According to [7], a key that is derived from quantum key distribution (QKD) can be reused without risk if an attacker’s presence is not discovered during quantum transmission. On the other hand, it has not been demonstrated that OTP perfect secrecy cannot be efficiently achieved. Indeed, chaos systems have been used as pseudo-random number generators (PRNGs), and chaotic cryptography-based systems have been investigated [8,9]. Yet, keys and ciphertext can exhibit short periods, and there is no systematic method for detecting weak keys [10]. It has been demonstrated that OTP is equivalent to finding the initial condition on a pair of binary maps [11]. In this work, we will introduce a cryptosystem that we call 3-encryption, which uses the rule of triple cancellation to achieve perfect secrecy.
- 2.
- Technological: Because it is well-known that current public key data protection mechanisms do not resist quantum cryptanalysis, the development of new cryptographic security schemes must be prioritized. Despite the fact that the National Institute of Standards and Technology (NIST) has published a set of post-quantum algorithms, the security evaluations and discussions of such algorithms (regarding potential vulnerabilities) are ongoing. These algorithms are used in encryption, signing, and key establishment. Nevertheless, our basic scheme can be simultaneously used for block chaining, data encryption, and digital signatures. As a result, we envision a multifunctional cryptographic platform that is capable of providing integrated security services.
- 3.
- Security: We will demonstrate that the encryption scheme is capable of achieving perfect secrecy. Surprisingly, our algorithm’s security properties are evaluated using XOR, hash, and integer addition, which are simple to analyze, and require no complex mathematical formalism. In the appendix of this document, we will show how the encryption method achieves perfect secrecy.

#### 1.2. State of the Art

- -
- -
- -
- Error-correcting codes are the foundation of code-based cryptography, functioning well for public key encryption. To ensure security, nearly all implemented algorithms in this class employ large keys. The McEliece public key encryption system and the Niederreiter cryptosystem are the most representative examples of code-based cryptography [16,17].
- -
- -
- Isogeny-based cryptography relies on the difficulty of finding a certain mapping (called isogeny) between two given supersingular elliptic curves. The Diffie–Hellman and elliptic curve Diffie–Hellman key-exchange methods can be replaced with the supersingular isogeny Diffie–Hellman key exchange (SIKE) as a quantum-resistant alternative [20].
- -

#### 1.3. Perfect Secrecy

- -
- The number of possible keys is greater than or equal to the number of possible plaintexts.
- -
- The key is selected uniformly at random from the key space.
- -
- A key should only be used once.

#### 1.4. Triple XOR Cancellation Rule

#### 1.5. Proposal of Our Approach

- -
- Digital signature: It is impossible for the adversary to send a message pretending to be from her because each user has a public key in their name. By using the XOR signature algorithm, users can make sure that the current message is connected to every previous message in the chain, starting with the user identification message. This topic will be discussed in Section 2.
- -
- XOR chain: The XOR signature model has allowed us to define a new approach to the blockchain system that we call the XOR chain. We will first present a hash function-based game called Crypto Bingo in order to conceptualize the XOR chain. Section 3 covers this subject.
- -
- Data encryption: The messages are encrypted so that it is impossible for the attacker to see their content; however, authorized recipients can recover the messages in original plain text. Section 4 of the document will address XOR encryption. In Appendix A, we show the perfect secrecy demonstration.

## 2. Digital Signature

#### 2.1. The Hash Chain Protocol

- 1.
- Alice and Bob generate their hash chain, which allows them to define their public and private keys.
- 2.
- They share their public keys over the public channel.
- 3.
- Alice computes ${f}^{{l}_{n}-h}\left({s}_{a}\right)$ and publishes it along m.
- 4.
- Bob—or any user who wants to verify the signature—just computes h, then ${f}^{h}\left({f}^{{l}_{n}-h}\left({s}_{a}\right)\right)=={f}^{{l}_{n}}\left({s}_{a}\right)$.
- 5.
- Alice shares a new public key.

- -
- An attacker can only exploit the hash values at the right-hand side of the signature hash in the current hash chain, which implies that ${h}_{i}^{\prime}<{h}_{i}$.
- -
- After a message is signed, some hash points to the left of the signature hash always remain unused in the previous hash chain.

- 1.
- Given that Alice has previously published ${f}^{{l}_{n}}\left({s}_{1}\right)$, to sign the message ${m}_{1}$, Alice computes ${\delta}_{1}$ to obtain ${f}^{{\delta}_{1}}\left({s}_{0}\right)$, then she publishes it along ${m}_{1}$.
- 2.
- Using ${m}_{0}$ and ${m}_{1}$, Bob computes ${\delta}_{1}$ and verifies two conditions:
- (i)
- ${f}^{{l}_{n}-{h}_{0}-{\delta}_{1}}\left({f}^{{\delta}_{1}}\left({s}_{0}\right)\right)={f}^{{l}_{n}-{h}_{0}}\left({s}_{0}\right)$.
- (ii)
- ${f}^{{h}_{1}}\left({f}^{{l}_{n}-{h}_{1}}\left({s}_{1}\right)\right)={f}^{{l}_{n}}\left({s}_{1}\right)$.

#### 2.2. Digital Signatures Based on HMAC

- 1.
- $A\to B:\phantom{\rule{4pt}{0ex}}<m{>}_{{f}^{{l}_{n}-i}\left({s}_{a}\right)}$. Alice signs the message (m) by applying the HMAC function and using her private key ${f}^{{l}_{n}-i}\left({s}_{a}\right)$.
- 2.
- $A\leftarrow B:\phantom{\rule{4pt}{0ex}}{f}^{{l}_{n}-j}\left({s}_{b}\right)$. Bob sends his private key ${f}^{{l}_{n}-j}\left({s}_{b}\right)$ to Alice.
- 3.
- $A\to B:\phantom{\rule{4pt}{0ex}}m,{f}^{{l}_{n}-j}$. Alice verifies Bob’s authenticator because she computes ${f}^{j}\left({f}^{{l}_{n}-j}\left({s}_{b}\right)\right)$, which returns Bob’s public key ${f}^{{l}_{n}}\left({s}_{b}\right)$. Then she sends ${f}^{{l}_{n}-i}\left({s}_{a}\right)$ and the message (m) to Bob.
- 4.
- $A\to B:\phantom{\rule{4pt}{0ex}}m,{f}^{{l}_{n}-i}$. Bob verifies Alice’s authenticator ${f}^{i}\left({f}^{{l}_{n}-i}\left({s}_{a}\right)\right)=={f}^{{l}_{n}}\left({s}_{a}\right)$

#### The HMAC Signature Method

#### 2.3. Digital XOR Signature

#### 2.3.1. Security Analysis

#### 2.3.2. Key Renewal

- -
- $\{{x}_{0}\phantom{\rule{4pt}{0ex}}\oplus \phantom{\rule{4pt}{0ex}}{k}_{0}\phantom{\rule{4pt}{0ex}}\oplus \phantom{\rule{4pt}{0ex}}{k}_{1}\}$, $\{{y}_{0}\phantom{\rule{4pt}{0ex}}\oplus \phantom{\rule{4pt}{0ex}}{k}_{0}\phantom{\rule{4pt}{0ex}}\oplus \phantom{\rule{4pt}{0ex}}{k}_{2}\}$, $\{{k}_{1}\phantom{\rule{4pt}{0ex}}\oplus \phantom{\rule{4pt}{0ex}}{k}_{2}\}$
- -
- $\{{x}_{0}\phantom{\rule{4pt}{0ex}}\oplus \phantom{\rule{4pt}{0ex}}{k}_{0}\phantom{\rule{4pt}{0ex}}\oplus \phantom{\rule{4pt}{0ex}}{k}_{1}\}$, $\{{y}_{0}\phantom{\rule{4pt}{0ex}}\oplus \phantom{\rule{4pt}{0ex}}{k}_{0}\phantom{\rule{4pt}{0ex}}\oplus \phantom{\rule{4pt}{0ex}}{{k}_{2}}^{\prime}\}$, $\{{k}_{1}\phantom{\rule{4pt}{0ex}}\oplus \phantom{\rule{4pt}{0ex}}{{k}_{2}}^{\prime}\}$
- -
- $\{{x}_{0}\phantom{\rule{4pt}{0ex}}\oplus \phantom{\rule{4pt}{0ex}}{k}_{0}\phantom{\rule{4pt}{0ex}}\oplus \phantom{\rule{4pt}{0ex}}{k}_{1}\}$, $\{{y}_{0}\phantom{\rule{4pt}{0ex}}\oplus \phantom{\rule{4pt}{0ex}}{k}_{0}\phantom{\rule{4pt}{0ex}}\oplus \phantom{\rule{4pt}{0ex}}{{k}_{2}}^{\u2033}\}$, $\{{k}_{1}\phantom{\rule{4pt}{0ex}}\oplus \phantom{\rule{4pt}{0ex}}{{k}_{2}}^{\u2033}\}$
- -
- ⋮

## 3. Blockchain

#### 3.1. Crypto Bingo

- -
- In the first round, the root player computes and registers ${h}_{01}=f\left({g}_{01}\right)$ into the game, where ${g}_{01}={f}^{{x}_{01}}\left({w}_{01}\right)\left|\right|{t}_{01}$ and ${w}_{01}={z}_{0}$ (instead of ${z}_{0}\left|\right|{{h}_{01}}^{\prime}$ because there is no a previous winner). In addition, players compute and register ${h}_{i1}$ into the game. Then the root player announces ${g}_{01}$ and the player whose ${g}_{i1}$ failed the fewest number of bits wins. This value corresponds to the minimum Hamming distance, denoted as $\delta $. It is clear that the probability of obtaining the correct bits (e.g., 256 bits) is quite low. The winner is allowed to place his block into the chain. Let ${z}_{1}$ be the winning player with ${h}_{11}$. To be verified, the root player publishes $({x}_{01},{t}_{01})$, while player 1 announces $({x}_{11},{t}_{11})$. Then, all players verify that they correspond to ${h}_{01}$ and ${h}_{11}$, respectively.

#### 3.2. XOR Chain

- -
- Round 0 (registration phase): Using his numbers, $\{{{k}_{0}}^{{z}_{0}},{{k}_{1}}^{{z}_{0}},{{k}_{2}}^{{z}_{0}},{{x}_{0}}^{{z}_{0}},{{y}_{0}}^{{z}_{0}}\}$, the root player, denoted as ${z}_{0}$, registers $\{{{c}_{0}}^{{z}_{0}},{{l}_{0}}^{{z}_{0}},{{r}_{0}}^{{z}_{0}}\}$ in the public DB, where ${{c}_{0}}^{{z}_{0}}={{k}_{0}}^{{z}_{0}}\oplus {{k}_{1}}^{{z}_{0}}$, ${{l}_{0}}^{{z}_{0}}={{x}_{0}}^{{z}_{0}}\oplus {{k}_{0}}^{{z}_{0}}\oplus {{k}_{2}}^{{z}_{0}}$ and ${{r}_{0}}^{{z}_{0}}={{y}_{0}}^{{z}_{0}}\oplus {{k}_{1}}^{{z}_{0}}\oplus {{k}_{2}}^{{z}_{0}}$. In addition, ${z}_{0}$ stores ${{h}_{0}}^{{z}_{0}}$ in DB, where ${{h}_{0}}^{{z}_{0}}={{x}_{0}}^{{z}_{0}}\oplus {{y}_{0}}^{{z}_{0}}$. But ${z}_{0}$ computes ${{h}_{0}}^{{z}_{0}}$, so that is the root of his Merkle tree. Then, all players select their own set of numbers, for example, ${z}_{1}$ chooses $\{{{k}_{0}}^{{z}_{1}}$, ${{k}_{1}}^{{z}_{1}}$, ${{k}_{2}}^{{z}_{1}}$, ${{x}_{0}}^{{z}_{1}}$, and ${{y}_{0}}^{{z}_{1}}\}$, so that they match the public keys of ${z}_{0}$.
- -
- Round 1: All nodes, for example, ${z}_{1}$ using $\{{{x}_{0}}^{{z}_{1}},{{y}_{0}}^{{z}_{1}},{{x}_{1}}^{{z}_{1}},{{y}_{1}}^{{z}_{1}}\}$, compute ${{h}_{0}}^{{z}_{1}}={{x}_{0}}^{{z}_{1}}\oplus {{y}_{0}}^{{z}_{1}}$ and ${{h}_{1}}^{{z}_{1}}={{x}_{1}}^{{z}_{1}}\oplus {{y}_{1}}^{{z}_{1}}$. But ${z}_{1}$ chooses his numbers, so that ${{h}_{1}}^{{z}_{1}}$ yields the root of his Merkle tree. Then, ${z}_{1}$ publishes $\{{{h}_{0}}^{{z}_{1}},{{h}_{1}}^{{z}_{1}}\}$ in DB to the rest of the nodes. Now, node ${z}_{0}$ publishes $\{{{l}_{1}}^{{z}_{0}},{{r}_{1}}^{{z}_{0}}\}$ in DB, where ${{l}_{1}}^{{z}_{0}}={{k}_{1}}^{{z}_{0}}\oplus {{k}_{3}}^{{z}_{0}}\oplus {{x}_{0}}^{{z}_{0}}\oplus {{x}_{1}}^{{z}_{0}}$ and ${{r}_{1}}^{{z}_{0}}={{y}_{1}}^{{z}_{0}}\oplus {{k}_{2}}^{{z}_{0}}\oplus {{k}_{3}}^{{z}_{0}}$. All nodes, for example, ${z}_{1}$ using ${{k}_{3}}^{{z}_{1}}$, obtain ${{l}_{1}}^{{z}_{1}}={{k}_{1}}^{{z}_{1}}\oplus {{k}_{3}}^{{z}_{1}}\oplus {{x}_{0}}^{{z}_{1}}\oplus {{x}_{1}}^{{z}_{1}}$ and ${{r}_{1}}^{{z}_{1}}={{y}_{1}}^{{z}_{1}}\oplus {{k}_{2}}^{{z}_{1}}\oplus {{k}_{3}}^{{z}_{1}}$. But ${z}_{1}$ chooses ${{k}_{3}}^{{z}_{1}}$, so that ${{l}_{1}}^{{z}_{1}}={{l}_{1}}^{{z}_{0}}$ (as illustrated in Figure 9). Then, ${z}_{1}$ publishes $\{{{l}_{1}}^{{z}_{1}},{{r}_{1}}^{{z}_{1}}\}$ in DB to the rest of the nodes. If the nodes (or the majority of them) agree that ${{r}_{1}}^{{z}_{1}}$ has the minimum distance to ${{r}_{1}}^{{z}_{0}}$, then ${z}_{1}$ wins the first round. Now, DB removes the auxiliary data from the other nodes but permanently stores $\{{{l}_{1}}^{{z}_{1}},{{r}_{1}}^{{z}_{1}}\}$, and $\{{{h}_{0}}^{{z}_{1}},{{h}_{1}}^{{z}_{1}}\}$ to enable all nodes to verify ${z}_{1}$ because ${{l}_{1}}^{{z}_{1}}\oplus {{r}_{1}}^{{z}_{1}}={{h}_{0}}^{{z}_{1}}\oplus {{h}_{1}}^{{z}_{1}}$. Before the next round takes place, each node, say ${z}_{2}$, selects its numbers again, ensuring that ${{h}_{1}}^{{z}_{2}}={{h}_{1}}^{{z}_{1}}$, where ${{h}_{1}}^{{z}_{1}}$ is the root of the Merkle tree that belongs to ${z}_{1}$. However, ${{h}_{2}}^{{z}_{2}}$ is the root of the Merkle tree of the ${z}_{2}$ node.

## 4. Data Encryption

#### 4.1. CBC Mode Encryption

#### 4.2. 3-Encryption

#### 4.3. Perfect Secrecy

- 1.
- Every encryption key must be distinct from any previous key.
- 2.
- The number of available keys is greater than or equal to the number of messages in the system.
- 3.
- Every key is chosen randomly and the probability must be the same for all keys.

- —
- ${m}_{j}={l}_{r}-{r}_{j}-{r}_{j-1}+{m}_{j-1}+{x}_{j}$ where ${x}_{j}=({x}_{j}-{x}_{j-1}-{m}_{j-1})+{x}_{j-1}+{m}_{j-1}$ requires 6 additions and 4 terms to be stored: $({x}_{j}-{x}_{j-1}-{m}_{j-1}),{x}_{j-1},{m}_{j-1},{r}_{j-1}$.
- —
- ${m}_{j}=({m}_{j}-{m}_{j-1}-{x}_{j})+{m}_{j-1}+{x}_{j}$ where ${x}_{j}=({x}_{j}-{x}_{j-1}-{m}_{j-1})+{x}_{j-1}+{m}_{j-1}$ requires 4 additions and 3 terms to be stored: $({x}_{j}-{x}_{j-1}-{m}_{j-1}),{x}_{j-1},{m}_{j-1}$.

## 5. Discussion and Future Work

- 1.
- Digital signature: To avoid analysis by Grover’s quantum search technique, we shall assume that the hash code must be at least 256 bits long. Thus, the size of the private key, $\{{k}_{0},{k}_{1},{k}_{2}\}$, achieves $3\times 256=768$ bits. In addition, the size of the public key, $\{{x}_{0}\oplus {k}_{0}\oplus {k}_{2}\},\{{y}_{0}\oplus {k}_{1}\oplus {k}_{2}\},\{{k}_{0}\oplus {k}_{1}\}$ reaches $3\xb7256=768$ bits. These key sizes are really small compared to current public key cryptosystems. Figure 5 in Section 2.3 shows the storage requirements: 1024 bits for Alice, Bob occupies 512 bits, and DB requires 256 bits. The signature process uses three random numbers, five XOR operations, and one hash computation. Finally, signature verification needs four XOR operations, one hash, and one comparison. The main drawback of our method involves the need for a central node, which can introduce delays in the signature process.
- 2.
- Blockchain: In the blockchain system [40], miners calculate and maintain a unified chain of all transactions on the network. In the distributed database, a table entry is 768 bits in size. In contrast, the table entry size is 8192 bits in the HMAC chain and hash chain algorithms. In this scenario, every user calculates and maintains his own independent chain of transactions. The table entry size in the XOR chain introduced here is 768 bits. Similar to blockchain, network nodes compute and store copies of a unified chain of transactions. However, some of the major advantages of our scheme over blockchain are the following: (1) it does not require proof of work (PoW), and (2) it is immune to quantum cryptanalysis. The comparison of such parameters is given in Table 7.
- 3.
- The 3-encryption: Figure 12 in Section 4.2 shows the storage prerequisites for 3-encryption. If we assume that both the message segment and key size are 256 bits, Alice requires 768 bits and Bob needs the same. The encryption process demands the generation of one random number, five (adding) operations, and one multiplication. Decryption takes four (adding) operations. The encryption method holds promise for exceptional execution performance since it does not necessitate numerous rounds of substitution and permutation, unlike the symmetric CBC chained mode. However, as previously stated, our system lacks a data permutation or dispersion mechanism; thus, if a message is repeated in the next round, the previous one can be inferred.

**Table 7.**Block chaining schemes are compared below. The sizes are written in bits. PQ stands for post-quantum while PoW stands for proof of work.

Blockchain | Hash Chain | HMAC Chain | This Work | |
---|---|---|---|---|

Table Size Entry | 768 | 8192 | 8192 | 768 |

PQ | no | yes | yes | yes |

PoW | yes | no | no | no |

- 1.
- Keep a distributed copy of the database among the users of the system, which is the approach used by the blockchain system.
- 2.
- Offline operation. The protocol operates without requiring intervention from the central node, which would guarantee the continuity of the system in the event of failures.

## 6. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Perfect Secrecy Proof

## References

- Nielsen, M.A.; Chuang, I.L. Quantum Computation and Quantum Information; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Dattani, N.S.; Bryans, N. Quantum factorization of 56153 with only 4 qubits. arXiv
**2014**, arXiv:1411.6758. [Google Scholar] - Dridi, R.; Alghassi, H. Prime factorization using quantum annealing and computational algebraic geometry. arXiv
**2016**, arXiv:1604.05796. [Google Scholar] - Shor, P.W. Algorithms for quantum computation: Discrete logarithms and factoring. In Proceedings of the 35th Annual Symposium on Foundations of Computer Science, Santa Fe, NM, USA, 20–22 November 1994; pp. 124–134. [Google Scholar]
- Grover, L.K. A fast quantum mechanical algorithm for database search. In Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, Philadelphia, PA, USA, 22–24 May 1996; pp. 212–219. [Google Scholar]
- Nagaraj, N.; Vaidya, V.; Vaidya, P.G. Re-visiting the One-Time Pad. arXiv
**2005**, arXiv:cs/0508079. [Google Scholar] - Damgård, I.; Pedersen, T.B.; Salvail, L. A quantum cipher with near optimal key-recycling. In Proceedings of the Advances in Cryptology–CRYPTO 2005: 25th Annual International Cryptology Conference, Santa Barbara, CA, USA, 14–18 August 2005. Proceedings 25; Springer: Cham, Switzerland, 2005; pp. 494–510. [Google Scholar]
- Baptista, M. Cryptography with chaos. Phys. Lett. A
**1998**, 240, 50–54. [Google Scholar] [CrossRef] - Jakimoski, G.; Kocarev, L. Chaos and cryptography: Block encryption ciphers based on chaotic maps. IEEE Trans. Circuits Syst. I Fundam. Theory Appl.
**2001**, 48, 163–169. [Google Scholar] [CrossRef] - Dachselt, F.; Schwarz, W. Chaos and cryptography. IEEE Trans. Circuits Syst. I Fundam. Theory Appl.
**2001**, 48, 1498–1509. [Google Scholar] [CrossRef] - Nagaraj, N. One-Time Pad as a nonlinear dynamical system. Commun. Nonlinear Sci. Numer. Simul.
**2012**, 17, 4029–4036. [Google Scholar] [CrossRef] - Ajtai, M. Generating hard instances of lattice problems. In Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, Philadelphia, PA, USA, 22–24 May 1996; pp. 99–108. [Google Scholar]
- Hoffstein, J.; Pipher, J.; Silverman, J.H. NTRU: A ring-based public key cryptosystem. In Proceedings of the International Algorithmic NUMBER Theory Symposium; Springer: Berlin/Heidelberg, Germany, 1998; pp. 267–288. [Google Scholar]
- Buchmann, J.; Dahmen, E.; Hülsing, A. XMSS-a practical forward secure signature scheme based on minimal security assumptions. In Proceedings of the Post-Quantum Cryptography: 4th International Workshop, PQCrypto 2011, Taipei, Taiwan, 29 November–2 December 2011. Proceedings 4; Springer: Berlin/Heidelberg, Germany, 2011; pp. 117–129. [Google Scholar]
- Bernstein, D.J.; Hopwood, D.; Hülsing, A.; Lange, T.; Niederhagen, R.; Papachristodoulou, L.; Schneider, M.; Schwabe, P.; Wilcox-O’Hearn, Z. SPHINCS: Practical stateless hash-based signatures. In Proceedings of the Annual International Conference on the Theory and Applications of Cryptographic Techniques; Springer: Berlin/Heidelberg, Germany, 2015; pp. 368–397. [Google Scholar]
- McEliece, R.J. A public-key cryptosystem based on algebraic. Coding Thv
**1978**, 4244, 114–116. [Google Scholar] - Niederreiter, H. Knapsack-type cryptosystems and algebraic coding theory. Prob. Contr. Inform. Theory
**1986**, 15, 157–166. [Google Scholar] - Matsumoto, T.; Imai, H. Public quadratic polynomial-tuples for efficient signature-verification and message-encryption. In Proceedings of the Advances in Cryptology—EUROCRYPT’88: Workshop on the Theory and Application of Cryptographic Techniques Davos, Switzerland, 25–27 May 1988 Proceedings 7; Springer: Berlin/Heidelberg, Germany, 1988; pp. 419–453. [Google Scholar]
- Ding, J.; Schmidt, D. Rainbow, a new multivariable polynomial signature scheme. In Proceedings of the International Conference on Applied Cryptography and Network Security; Springer: Berlin/Heidelberg, Germany, 2005; pp. 164–175. [Google Scholar]
- Jao, D.; De Feo, L. Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies. In Proceedings of the Post-Quantum Cryptography: 4th International Workshop, PQCrypto 2011, Taipei, Taiwan, 29 November–2 December 2011. Proceedings 4; Springer: Berlin/Heidelberg, Germany, 2011; pp. 19–34. [Google Scholar]
- Standard, A.E. Federal Information Processing Standards Publication 197. FIPS PUB. 2001; pp. 3–46. Available online: https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.197.pdf (accessed on 12 October 2023).
- Campagna, M.; Hardjono, T.; Pintsov, L.; Romansky, B.; Yu, T. Kerberos revisited quantum-safe authentication. In Proceedings of the ETSI Quantum-Safe-Crypto Workshop, Nice, France, 26–27 September 2013; pp. 26–27. [Google Scholar]
- Bernstein, D.J.; Lange, T. Post-quantum cryptography. Nature
**2017**, 549, 188–194. [Google Scholar] [CrossRef] - Alagic, G.; Apon, D.; Cooper, D.; Dang, Q.; Dang, T.; Kelsey, J.; Lichtinger, J.; Miller, C.; Moody, D.; Peralta, R.; et al. Status Report on the Third Round of the Nist Post-Quantum Cryptography Standardization Process; US Department of Commerce, NIST: Gaithersburg, MD, USA, 2022.
- Laboratory, I.T. PQC Standardization Process: Third Round Candidate Announcement. 2020. Available online: https://csrc.nist.gov/news/2020/pqc-third-round-candidate-announcement (accessed on 12 October 2023).
- Chen, L.; Chen, L.; Jordan, S.; Liu, Y.K.; Moody, D.; Peralta, R.; Perlner, R.; Smith-Tone, D. Report on Post-Quantum Cryptography; US Department of Commerce, National Institute of Standards and Technology: Gaithersburg, MD, USA, 2016; Volume 12.
- Persichetti, E. NIST Round 3 Finalists. 2020. Available online: https://pqc-wiki.fau.edu/w/Special:DatabaseHome (accessed on 12 October 2023).
- Castryck, W.; Decru, T. An Efficient Key recovery Attack on SIDH (Preliminary Version). Cryptology ePrint Archive 2022. Available online: https://eprint.iacr.org/2022/975 (accessed on 12 October 2023).
- Beullens, W. Breaking Rainbow Takes a Weekend on a Laptop. Cryptology ePrint Archive, Paper 2022/214. 2022. Available online: https://eprint.iacr.org/2022/214 (accessed on 12 October 2023).
- Beullens, W. Improved cryptanalysis of UOV and rainbow. In Proceedings of the Annual International Conference on the Theory and Applications of Cryptographic Techniques; Springer International Publishing: Cham, Switzerland, 2021; pp. 348–373. [Google Scholar]
- Rivest, R.L.; Shamir, A.; Adleman, L. A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM
**1978**, 21, 120–126. [Google Scholar] [CrossRef] - PUB, F. Digital Signature Standard (DSS). FIPS PUB. 2000; pp. 186–192. Available online: https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf (accessed on 12 October 2023).
- Moody, D.; Alagic, G.; Apon, D.C.; Cooper, D.A.; Dang, Q.H.; Kelsey, J.M.; Liu, Y.K.; Miller, C.A.; Peralta, R.C.; Perlner, R.A.; et al. Status Report on the Second Round of the NIST Post-Quantum Cryptography Standardization Process; US Department of Commerce, NIST: Gaithersburg, MD, USA, 2020.
- Merkle, R.C. Secrecy, Authentication, and Public Key Systems; Stanford University: Stanford, CA, USA, 1979. [Google Scholar]
- Lizama-Pérez, L.A.; Montiel-Arrieta, L.J.; Hernández-Mendoza, F.S.; Lizama-Servín, L.A.; Eric, S.A. Public hash signature for mobile network devices. Ing. Investig. Tecnol.
**2019**, 20, 1–10. [Google Scholar] [CrossRef] - Lizama-Perez, L.A. Digital signatures over hash-entangled chains. SN Appl. Sci.
**2019**, 1, 1568. [Google Scholar] [CrossRef] - Schneier, B. Description of a new variable-length key, 64-bit block cipher (Blowfish). In Proceedings of the International Workshop on Fast Software Encryption; Springer: Berlin/Heidelberg, Germany, 1993; pp. 191–204. [Google Scholar]
- Rogaway, P. Evaluation of Some Blockcipher Modes of Operation. Cryptography Research and Evaluation Committees (CRYPTREC) for the Government of Japan. 2011. Available online: https://www.cs.ucdavis.edu/~rogaway/papers/modes-cryptrec.pdf (accessed on 12 October 2023).
- Bujari, D.; Aribas, E. Comparative analysis of block cipher modes of operation. In Proceedings of the International Advanced Researches & Engineering Congress, Osmaniye, Turkey, 16–18 November 2017; pp. 1–4. [Google Scholar]
- Nakamoto, S. Bitcoin: A Peer-to-Peer Electronic Cash System. Decentralized Business Review. 2008, p. 21260. Available online: https://assets.pubpub.org/d8wct41f/31611263538139.pdf (accessed on 12 October 2023).
- Johar, S.; Ahmad, N.; Asher, W.; Cruickshank, H.; Durrani, A. Research and applied perspective to blockchain technology: A comprehensive survey. Appl. Sci.
**2021**, 11, 6252. [Google Scholar] [CrossRef] - Kearney, J.J.; Perez-Delgado, C.A. Vulnerability of blockchain technologies to quantum attacks. Array
**2021**, 10, 100065. [Google Scholar] [CrossRef] - Vujičić, D.; Jagodić, D.; Ranđić, S. Blockchain technology, bitcoin, and Ethereum: A brief overview. In Proceedings of the 2018 17th International Symposium Infoteh-Jahorina (Infoteh), East Sarajevo, Bosnia and Herzegovina, 21–23 March 2018; pp. 1–6. [Google Scholar]
- Paulavičius, R.; Grigaitis, S.; Igumenov, A.; Filatovas, E. A decade of blockchain: Review of the current status, challenges, and future directions. Informatica
**2019**, 30, 729–748. [Google Scholar] [CrossRef] - Papageorgiou, O.; Sedlmeir, J.; Fridgen, G.; Vlachos, I.; Kostopoulos, N.; Damvakeraki, T.; Noszek, Z.; Papoutsoglou, I.; Anania, A.; Belotti, M.; et al. Energy Efficiency of Blockchain Technologies; European Union Blockchain Observatory & Forum. 2021. Available online: https://www.eublockchainforum.eu/sites/default/files/reports/Energy%20Efficiency%20of%20Blockchain%20Technologies_1.pdf (accessed on 12 October 2023).
- Fernandez-Carames, T.M.; Fraga-Lamas, P. Towards post-quantum blockchain: A review on blockchain cryptography resistant to quantum computing attacks. IEEE Access
**2020**, 8, 21091–21116. [Google Scholar] [CrossRef] - Edwards, M.; Mashatan, A.; Ghose, S. A review of quantum and hybrid quantum/classical blockchain protocols. Quantum Inf. Process.
**2020**, 19, 184. [Google Scholar] [CrossRef] - Ruggeri, C. Quantum Key Distribution in Softwarised Networks. Ph.D. Thesis, Politecnico di Torino, Turin, Italy, 2020. [Google Scholar]
- Ikeda, K. qBitcoin: A peer-to-peer quantum cash system. In Proceedings of the Intelligent Computing: Proceedings of the 2018 Computing Conference, Volume 1; Springer International Publishing: Cham, Switzerland, 2019; pp. 763–771. [Google Scholar]
- Gottesman, D.; Chuang, I. Quantum digital signatures. arXiv
**2001**, arXiv:quant-ph/0105032. [Google Scholar] - Shannon, C.E. Communication theory of secrecy systems. Bell Syst. Tech. J.
**1949**, 28, 656–715. [Google Scholar] [CrossRef] - Feutrill, A.; Roughan, M. A Review of Shannon and Differential Entropy Rate Estimation. Entropy
**2021**, 23, 1046. [Google Scholar] [CrossRef] [PubMed] - Martin, K.M. Everyday cryptography. The Australian Mathematical Society; Oxford University Press: Oxford, UK, 2012; pp. 231–234. [Google Scholar]
- Shimeall, T.; Spring, J. Introduction to Information Security: A Strategic-Based Approach; Newnes: Oxford, UK, 2013. [Google Scholar]
- Lizama-Pérez, L.A. Digital signatures over HMAC entangled chains. Eng. Sci. Technol. Int. J.
**2021**, 32, 101076. [Google Scholar] [CrossRef] - Krawczyk, H.; Canetti, R.; Bellare, M. HMAC: Keyed-Hashing for Message Authentication. 1997. Available online: https://www.rfc-editor.org/rfc/rfc2104 (accessed on 12 October 2023).
- Yan, B.; Tan, Z.; Wei, S.; Jiang, H.; Wang, W.; Wang, H.; Luo, L.; Duan, Q.; Liu, Y.; Shi, W.; et al. Factoring integers with sublinear resources on a superconducting quantum processor. arXiv
**2022**, arXiv:2212.12372. [Google Scholar] - Ehrsam, W.F.; Meyer, C.H.; Smith, J.L.; Tuchman, W.L. Message Verification and Transmission Error Detection by Block Chaining. U.S. Patent 4,074,066, 14 February 1978. [Google Scholar]
- Trappe, W. Introduction to Cryptography with Coding Theory; Pearson Education: London, UK, 2020. [Google Scholar]

**Figure 2.**Triple XOR cancellation rule. Here, ${k}_{0}$, ${k}_{1}$, and ${k}_{2}$ are the initial shared secret keys. The number ${x}_{0}$ is a random number chosen by Alice and ${y}_{0}$ is computed as ${y}_{0}={h}_{0}\oplus {x}_{0}$, where ${h}_{0}=f\left({m}_{0}\right)$, and f is the hash function applied to the message ${m}_{0}$.

**Figure 6.**The miners calculate the hash of the concatenation resulting from the previous transaction, a number used only once (nonce), and the root of the Merkle tree, which groups the miner’s transactions.

**Figure 7.**Players register ${h}_{ij}=f\left({g}_{ij}\right)$, then they announce ${g}_{ij}={f}^{{x}_{ij}}\left({w}_{ij}\right)\left|\right|{t}_{ij}$ and the pair $\{{x}_{ij},{t}_{ij}\}$.

**Figure 8.**Diagram of the process to compute ${h}_{ij}$. Input ${h}_{ij}^{\prime}$ represents the hash code of the winner in the previous round.

**Figure 9.**In round 1, ${z}_{1}$ publishes $\{{{h}_{0}}^{{z}_{1}},{{h}_{1}}^{{z}_{1}}\}$ in DB. He computes ${{l}_{1}}^{{z}_{1}}$ and ${{r}_{1}}^{{z}_{1}}$ but chooses ${{k}_{3}}^{{z}_{1}}$, so that ${{l}_{1}}^{{z}_{1}}={{l}_{1}}^{{z}_{0}}$. Then, ${z}_{1}$ publishes $\{{{l}_{1}}^{{z}_{1}},{{r}_{1}}^{{z}_{1}}\}$ in DB. The rest of the nodes agree that ${{r}_{1}}^{{z}_{1}}$ has the minimum distance to ${{r}_{1}}^{{z}_{0}}$; that is, ${{r}_{1}}^{{z}_{0}}\sim {{r}_{1}}^{{z}_{1}}$, and ${z}_{1}$ wins the first round.

**Figure 10.**CBC mode: Before the encryption process, the plaintext block and the previous ciphertext block are passed to the XOR function. As a result, each round generates a cipher block that is dependent on the previous plaintext blocks.

**Figure 11.**Boxes with bold outline imply that these numbers are private and will not be transmitted over the public channel.

**Figure 13.**In scenario (

**a**), with the help of central node R, any pair of nodes can exchange signed data. In (

**b**), Node 1 has an initial secret key with the other nodes in the network, so it can maintain encrypted communications with them.

**Table 1.**The public and private keys of the hash chain protocol are specified, where $i\left(j\right)\ge 1$.

User | Public Key | Private Keys |
---|---|---|

Alice | ${f}^{{l}_{n}}\left({s}_{a}\right)$ | ${f}^{{l}_{n}-i}\left({s}_{a}\right)$ |

Bob | ${f}^{{l}_{n}}\left({s}_{b}\right)$ | ${f}^{{l}_{n}-j}\left({s}_{b}\right)$ |

Private Key | Public Key |
---|---|

$\{{k}_{0},\phantom{\rule{4pt}{0ex}}{k}_{1},\phantom{\rule{4pt}{0ex}}{k}_{2}\}$ | $\{{x}_{0}\oplus {k}_{0}\oplus {k}_{2}\},\phantom{\rule{4pt}{0ex}}\{{y}_{0}\oplus {k}_{1}\oplus {k}_{2}\},\phantom{\rule{4pt}{0ex}}\{{k}_{0}\oplus {k}_{1}\}$ |

**Table 3.**Protocol execution for rounds $0\dots j$. The terms ${c}_{j}$, ${l}_{j}$, ${r}_{j}$, and ${d}_{j}$ are shown after each execution round. In the last column, we show the results of ${c}_{j}\oplus {l}_{j}\oplus {r}_{j}\oplus {d}_{j}$. The authentication rule is $f\left({m}_{j}\right)$ == ${h}_{j}$, where ${h}_{j}={h}_{j-1}\oplus {c}_{j}\oplus {l}_{j}\oplus {r}_{j}\oplus {d}_{j}$.

Round | ${\mathit{m}}_{\mathit{j}}$ | ${\mathit{c}}_{\mathit{j}}={\mathit{y}}_{\mathit{j}-1}\oplus $ | ${\mathit{l}}_{\mathit{j}}=$ | ${\mathit{r}}_{\mathit{j}}={\mathit{y}}_{\mathit{j}}\oplus $ | ${\mathit{d}}_{\mathit{j}}=$ | ${\mathit{c}}_{\mathit{j}}\oplus {\mathit{l}}_{\mathit{j}}\oplus $ |
---|---|---|---|---|---|---|

${\mathit{k}}_{\mathit{j}}\oplus {\mathit{k}}_{\mathit{j}+\mathbf{1}}$ | ${\mathit{k}}_{\mathit{j}}\oplus {\mathit{k}}_{\mathit{j}+\mathbf{2}}$ | ${\mathit{k}}_{\mathit{j}+\mathbf{1}}\oplus {\mathit{k}}_{\mathit{j}+\mathbf{2}}$ | ${\mathit{x}}_{\mathit{j}-\mathbf{1}}\oplus {\mathit{x}}_{\mathit{j}}$ | ${\mathit{r}}_{\mathit{j}}\oplus {\mathit{d}}_{\mathit{j}}$ | ||

0 | ${m}_{0}$ | ${k}_{0}\oplus {k}_{1}$ | ${x}_{0}\oplus {k}_{0}\oplus {k}_{2}$ | ${y}_{0}\oplus {k}_{1}\oplus {k}_{2}$ | — | ${h}_{0}$ |

1 | ${m}_{1}$ | ${y}_{0}\oplus {k}_{1}\oplus {k}_{2}$ | ${k}_{1}\oplus {k}_{3}$ | ${y}_{1}\oplus {k}_{2}\oplus {k}_{3}$ | ${x}_{0}\oplus {x}_{1}$ | ${h}_{0}\oplus {h}_{1}$ |

2 | ${m}_{2}$ | ${y}_{1}\oplus {k}_{2}\oplus {k}_{3}$ | ${k}_{2}\oplus {k}_{4}$ | ${y}_{2}\oplus {k}_{3}\oplus {k}_{4}$ | ${x}_{1}\oplus {x}_{2}$ | ${h}_{1}\oplus {h}_{2}$ |

3 | ${m}_{3}$ | ${y}_{2}\oplus {k}_{3}\oplus {k}_{4}$ | ${k}_{3}\oplus {k}_{5}$ | ${y}_{3}\oplus {k}_{4}\oplus {k}_{5}$ | ${x}_{2}\oplus {x}_{3}$ | ${h}_{2}\oplus {h}_{3}$ |

4 | ${m}_{4}$ | ${y}_{3}\oplus {k}_{4}\oplus {k}_{5}$ | ${k}_{4}\oplus {k}_{6}$ | ${y}_{4}\oplus {k}_{5}\oplus {k}_{6}$ | ${x}_{3}\oplus {x}_{4}$ | ${h}_{3}\oplus {h}_{4}$ |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

j | ${m}_{j}$ | ${y}_{j-1}\oplus {k}_{j}\oplus {k}_{j+1}$ | ${k}_{j}\oplus {k}_{j+2}$ | ${y}_{j}\oplus {k}_{j+1}\oplus {k}_{j+2}$ | ${x}_{j-1}\oplus {x}_{j}$ | ${h}_{j-1}\oplus {h}_{j}$ |

**Table 4.**We demonstrate the terms ${c}_{j}$, ${l}_{j}$, and ${r}_{j}$ after each execution round. The last column shows the result of ${c}_{j}\oplus {l}_{j}\oplus {r}_{j}$. The XOR chain rule is ${l}_{j}\oplus {r}_{j}$ == ${h}_{j}\oplus {h}_{j+1}$.

Round | ${\mathit{c}}_{\mathit{j}}={\mathit{y}}_{\mathit{j}-1}\oplus $ | ${\mathit{l}}_{\mathit{j}}={\mathit{k}}_{\mathit{j}}\oplus {\mathit{k}}_{\mathit{j}+2}\oplus $ | ${\mathit{r}}_{\mathit{j}}={\mathit{y}}_{\mathit{j}}\oplus $ | ${\mathit{c}}_{\mathit{j}}\oplus {\mathit{l}}_{\mathit{j}}\oplus {\mathit{r}}_{\mathit{j}}$ |
---|---|---|---|---|

${\mathit{k}}_{\mathit{j}}\oplus {\mathit{k}}_{\mathit{j}+\mathbf{1}}$ | ${\mathit{x}}_{\mathit{j}-\mathbf{1}}\oplus {\mathit{x}}_{\mathit{j}}$ | ${\mathit{k}}_{\mathit{j}+\mathbf{1}}\oplus {\mathit{k}}_{\mathit{j}+\mathbf{2}}$ | ||

0 | ${k}_{0}\oplus {k}_{1}$ | ${x}_{0}\oplus {k}_{0}\oplus {k}_{2}$ | ${y}_{0}\oplus {k}_{1}\oplus {k}_{2}$ | ${h}_{0}$ |

1 | ${y}_{0}\oplus {k}_{1}\oplus {k}_{2}$ | ${k}_{1}\oplus {k}_{3}\oplus {x}_{0}\oplus {x}_{1}$ | ${y}_{1}\oplus {k}_{2}\oplus {k}_{3}$ | ${h}_{0}\oplus {h}_{1}$ |

2 | ${y}_{1}\oplus {k}_{2}\oplus {k}_{3}$ | ${k}_{2}\oplus {k}_{4}\oplus {x}_{1}\oplus {x}_{2}$ | ${y}_{2}\oplus {k}_{3}\oplus {k}_{4}$ | ${h}_{1}\oplus {h}_{2}$ |

3 | ${y}_{2}\oplus {k}_{3}\oplus {k}_{4}$ | ${k}_{3}\oplus {k}_{5}\oplus {x}_{2}\oplus {x}_{3}$ | ${y}_{3}\oplus {k}_{4}\oplus {k}_{5}$ | ${h}_{2}\oplus {h}_{3}$ |

4 | ${y}_{3}\oplus {k}_{4}\oplus {k}_{5}$ | ${k}_{4}\oplus {k}_{6}\oplus {x}_{3}\oplus {x}_{4}$ | ${y}_{4}\oplus {k}_{5}\oplus {k}_{6}$ | ${h}_{3}\oplus {h}_{4}$ |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

j | ${y}_{j-1}\oplus {k}_{j}\oplus {k}_{j+1}$ | ${k}_{j}\oplus {k}_{j+2}\oplus {x}_{j-1}\oplus {x}_{j}$ | ${y}_{j}\oplus {k}_{j+1}\oplus {k}_{j+2}$ | ${h}_{j-1}\oplus {h}_{j}$ |

**Table 5.**The XOR chain is conformed by the succession $\{{{l}_{0}}^{{z}_{0}},{{r}_{0}}^{{z}_{0}},{{h}_{0}}^{{z}_{0}}\},\{{{l}_{1}}^{{z}_{1}},{{r}_{1}}^{{z}_{1}},{{h}_{1}}^{{z}_{1}}\},\dots \{{{l}_{j}}^{{z}_{j}},{{r}_{j}}^{{z}_{j}},{{h}_{j}}^{{z}_{j}}\}$, where ${{h}_{0}}^{{z}_{0}}={{h}_{0}}^{{z}_{1}}$, ${{h}_{1}}^{{z}_{1}}={{h}_{1}}^{{z}_{2}}\dots $.

j | ${\mathit{l}}_{\mathit{j}}$ | ${\mathit{r}}_{\mathit{j}}$ | ${\mathit{h}}_{\mathit{j}-1}$ | ${\mathit{h}}_{\mathit{j}}$ | |
---|---|---|---|---|---|

0 | ${{l}_{0}}^{{z}_{0}}$ | ${{r}_{0}}^{{z}_{0}}$ | — | ${{h}_{0}}^{{z}_{0}}$ | |

↙ | |||||

1 | ${{l}_{1}}^{{z}_{1}}$ | ${{r}_{1}}^{{z}_{1}}$ | ${{h}_{0}}^{{z}_{1}}$ | ${{h}_{1}}^{{z}_{1}}$ | |

↙ | |||||

2 | ${{l}_{2}}^{{z}_{2}}$ | ${{r}_{2}}^{{z}_{2}}$ | ${{h}_{1}}^{{z}_{2}}$ | ${{h}_{2}}^{{z}_{2}}$ | |

↙ | |||||

⋮ | ⋮ | ⋮ | ⋮ | ||

j | ${{l}_{j}}^{{z}_{j}}$ | ${{r}_{j}}^{{z}_{j}}$ | ${{h}_{j-1}}^{{z}_{j}}$ | ${{h}_{j}}^{{z}_{j}}$ | |

↙ |

**Table 6.**Protocol execution for rounds $0\dots j$. The terms ${m}_{j}$, ${c}_{j}$, ${l}_{j}$, and ${r}_{j}$ are shown after each execution round. The last column shows the result of ${l}_{j}-{r}_{j}-{r}_{j-1}$.

Round | ${\mathit{l}}_{\mathit{j}}=$ | ${\mathit{r}}_{\mathit{j}}=$ | ${\mathit{c}}_{\mathit{j}}=$ | ${\mathit{l}}_{\mathit{j}}-{\mathit{r}}_{\mathit{j}}-{\mathit{r}}_{\mathit{j}-1}$ |
---|---|---|---|---|

${\mathit{m}}_{\mathit{j}}+{\mathit{k}}_{\mathit{j}+2}-{\mathit{k}}_{\mathit{j}}$ | ${\mathit{m}}_{\mathit{j}-\mathbf{1}}+{\mathit{k}}_{\mathit{j}+\mathbf{2}}-{\mathit{k}}_{\mathit{j}+\mathbf{1}}$ | ${\mathit{m}}_{\mathit{j}-\mathbf{2}}+{\mathit{k}}_{\mathit{j}+\mathbf{1}}-{\mathit{k}}_{\mathit{j}}$ | ||

0 | ${m}_{0}+{k}_{2}-{k}_{0}$ | ${k}_{2}-{k}_{1}$ | ${k}_{1}-{k}_{0}$ | ${m}_{0}$ |

1 | ${m}_{1}+{k}_{3}-{k}_{1}$ | ${m}_{0}+{k}_{3}-{k}_{2}$ | ${k}_{2}-{k}_{1}$ | ${m}_{1}-{m}_{0}$ |

2 | ${m}_{2}+{k}_{4}-{k}_{2}$ | ${m}_{1}+{k}_{4}-{k}_{3}$ | ${m}_{0}+{k}_{3}-{k}_{2}$ | ${m}_{2}-{m}_{1}-{m}_{0}$ |

3 | ${m}_{3}+{k}_{5}-{k}_{3}$ | ${m}_{2}+{k}_{5}-{k}_{4}$ | ${m}_{1}+{k}_{4}-{k}_{3}$ | ${m}_{3}-{m}_{2}-{m}_{1}$ |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

j | ${m}_{j}+{k}_{j+2}-{k}_{j}$ | ${m}_{j-1}+{k}_{j+2}-{k}_{j+1}$ | ${m}_{j-2}+{k}_{j+1}-{k}_{j}$ | ${m}_{j}-{m}_{j-1}-{m}_{j-2}$ |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lizama-Pérez, L.A.
XOR Chain and Perfect Secrecy at the Dawn of the Quantum Era. *Cryptography* **2023**, *7*, 50.
https://doi.org/10.3390/cryptography7040050

**AMA Style**

Lizama-Pérez LA.
XOR Chain and Perfect Secrecy at the Dawn of the Quantum Era. *Cryptography*. 2023; 7(4):50.
https://doi.org/10.3390/cryptography7040050

**Chicago/Turabian Style**

Lizama-Pérez, Luis Adrián.
2023. "XOR Chain and Perfect Secrecy at the Dawn of the Quantum Era" *Cryptography* 7, no. 4: 50.
https://doi.org/10.3390/cryptography7040050