A new class of Fibonacci sequence based error correcting codes

Esmaeili, Morteza; Moosavi, M.; Gulliver, T. Aaron

doi:10.1007/s12095-015-0178-x

Cited by 23 publications

(18 citation statements)

References 5 publications

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“…Further improvements of the scheme might include the reduction of the the communication cost, either by reducing the cheating probability from 2/3 to 1/2 or by finding a way of reducing the response size in the case of the challenge b = 1. Finally, it would be interesting to compare our protocol with other instantiations of it using generalizations of the p-Fibonacci sequences, such as those based on Fibonacci polynomials sequences [19] or Fibonacci M p -sequences [20].…”

Section: Discussionmentioning

confidence: 99%

“…In 2006, Stakhov [17] introduced a new technique, based on the so-called Fibonacci p-numbers or p-sequence, further analyzed in [18], to obtain error-correcting codes. A few generalizations of his theory were explored in [17] (Fibonacci (p, m)-sequence) [19] (Fibonacci polynomial sequence), and [20] (Fibonacci M p -sequence), but apart from that, the theory behind these codes has not received much attention, probably due to the fact that their error model is very unlikely to be applicable in real life for error correction. The aim of this paper is to show that this type of code deserves to be investigated more deeply, as it could be very useful in cryptographic applications.…”

Section: Our Contributionmentioning

confidence: 99%

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An Application of p-Fibonacci Error-Correcting Codes to Cryptography

2021

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In addition to their usefulness in proving one’s identity electronically, identification protocols based on zero-knowledge proofs allow designing secure cryptographic signature schemes by means of the Fiat–Shamir transform or other similar constructs. This approach has been followed by many cryptographers during the NIST (National Institute of Standards and Technology) standardization process for quantum-resistant signature schemes. NIST candidates include solutions in different settings, such as lattices and multivariate and multiparty computation. While error-correcting codes may also be used, they do not provide very practical parameters, with a few exceptions. In this manuscript, we explored the possibility of using the error-correcting codes proposed by Stakhov in 2006 to design an identification protocol based on zero-knowledge proofs. We showed that this type of code offers a valid alternative in the error-correcting code setting to build such protocols and, consequently, quantum-resistant signature schemes.

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Section: Discussionmentioning

confidence: 99%

Section: Our Contributionmentioning

confidence: 99%

An Application of p-Fibonacci Error-Correcting Codes to Cryptography

2021

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“…If we denote v n := t n−2 − t n−5 + 2(t n−4 − t n−3 ) we get by (28) and (30) the following very computationally effective form of the decoding matrix…”

Section: Example Of Coding/decoding By Matrix Based On (2 5)-distancmentioning

confidence: 99%

“…, which is again the matrix in the form (6), when we imagine F p (n) replaced by F m,t,p (n). Recently, many other papers that modify Stakhov's method in some way have been published (see, e.g., [28][29][30][31][32][33]). We showed that much effort was devoted to a different generalization of (4) by introducing some new coefficients in the Fibonacci p-numbers recurrence for the purpose to design a coding/decoding system in the last ten years.…”

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confidence: 99%

On Coding by (2,q)-Distance Fibonacci Numbers

Matoušová

Trojovský

2020

Mathematics

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In 2006, A. Stakhov introduced a new coding/decoding process based on generating matrices of the Fibonacci p-numbers, which he called the Fibonacci coding/decoding method. Stakhov’s papers have motivated many other scientists to seek certain generalizations by introducing new additional coefficients into recurrence of Fibonacci p-numbers. In 2013, I. Włoch et al. studied (2,q)-distance Fibonacci numbers F2(q,n) and found some of their combinatorial properties. In this paper, we state a new coding theory based on the sequence (Tq(n))n=−∞∞, which is an extension of Włoch’s sequence (F2(q,n))n=0∞.

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“…According to work by Esmaeili et al [26], one can write a Fibonacci Q-matrix of dimension 2 as follows: The following relations can be used to create consecutive Fibonacci matrices:…”

Section: Block-diagonal Fibonacci Matrix D 2kmentioning

confidence: 99%

High-rate and high-capacity measurement-device-independent quantum key distribution with Fibonacci matrix coding in free space

Lai

Luo

Pieprzyk

et al. 2018

Sci. China Inf. Sci.

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This paper proposes a high-rate and high-capacity measurement-device-independent quantum key distribution (MDI-QKD) protocol with Fibonacci-valued and Lucas-valued orbital angular momentum (OAM) entangled states in free space. In the existing MDI-OAM-QKD protocols, the main encoding algorithm handles encoded numbers in a bit-by-bit manner. To design a fast encoding algorithm, we introduce a Fibonacci matrix coding algorithm, by which, encoded numbers are separated into segments longer than one bit. By doing so, when compared to the existing MDI-OAM-QKD protocols, the new protocol can effectively increase the key rate and the coding capacity. This is because Fibonacci sequences are used in preparing OAM entangled states, reducing the misattribution errors (which slow down the execution cycle of the entire QKD) in QKD protocols. Moreover, our protocol keeps the data blocks as small as possible, so as to have more blocks in a given time interval. Most importantly, our proposed protocol can distill multiple Fibonacci key matrices from the same block of data, thus reducing the statistical fluctuations in the sample and increasing the final QKD rate. Last but not the least, the sender and the receiver can omit classical information exchange and bit flipping in the secure key distillation stage.

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A new class of Fibonacci sequence based error correcting codes

Cited by 23 publications

References 5 publications

An Application of p-Fibonacci Error-Correcting Codes to Cryptography

An Application of p-Fibonacci Error-Correcting Codes to Cryptography

On Coding by (2,q)-Distance Fibonacci Numbers

High-rate and high-capacity measurement-device-independent quantum key distribution with Fibonacci matrix coding in free space

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