CHATTANOOGA, Tenn. (AP) — CHATTANOOGA, Tenn. (AP) — Astec Industries Inc. (ASTE) on Wednesday reported a third-quarter loss of $6.6 million, after reporting a profit in the same period a year earlier. The Chattanooga, Tennessee-based company said it had a loss of 29 cents per share. Losses, adjusted for non-recurring costs, were 1 cent per share.

Astec Industries: Q3 Earnings Snapshot

The milling machine is used to grind the top layer of asphalt pavement before resurfacing. Astec Industries reported a $6.6 million loss in the third quarter on Wednesday. Astec Industries Inc. reported Wednesday a third-quarter loss of $6.6 million after ...

Astec Industries loses $6.6 million in the third quarter

Astec Industries (ASTE) delivered earnings and revenue surprises of -101.56% and 8.01%, respectively, for the quarter ended September 2023. Do the numbers hold clues to what lies ahead for the stock?

Astec Industries (ASTE) Reports Q3 Loss, Lags Revenue Estimates

Below are the earnings highlights for Astec Industries (ASTE): Earnings: -$6.6 million in Q3 vs. $0.7 million in the same period

Astec Industries Q3 Earnings Summary

Nov. 01, 2023 (GLOBE NEWSWIRE) -- Astec Industries, Inc. (Nasdaq: ASTE) announced today its financial results for the third quarter of 2023. "During the third quarter, order activity normalized amidst ongoing positive sentiment from customers" said Jaco ...

Astec Reports Third Quarter 2023 Results

Investors will have to decide whether strong performance is enough to overcome their fears about the global economy. Machinery giant Caterpillar is slated to report third-quarter earnings on Tuesday morning. Wall Street is looking for earnings per share of $4.80 from sales of $16.6 billion, according to FactSet.

Astec Industries, Inc. (NASDAQ:ASTE) Stock's Been Sliding But Fundamentals Look Decent: Will The Market Correct The Share Price In The Future?

Astec (ASTE) declares $0.13/share quarterly dividend, in line with previous. Forward yield 1.29% Payable Dec. 1; for shareholders of record Nov. 10; ex-div Nov.

Astec declares $0.13 dividend

CHATTANOOGA, Tenn., Oct. 30, 2023 (GLOBE NEWSWIRE) --  (NASDAQ:ASTE) announced that its Board of Directors declared a quarterly dividend of $0.13 per share. The dividend is to be paid on or about December 1,

Astec Announces Quarterly Dividend

Astec Industries Inc. is a company that has been at the forefront of innovation in infrastructure solutions for over 50 years. Founded in 1972 by J. Don Brock, the company has grown from a small asphalt plant manufacturer to a global leader in the industry.

Astec Industries Inc.: Pioneering Innovation in Infrastructure Solutions

Astec Industries Inc. has long been at the forefront of innovation in the industrial equipment manufacturing industry. With a rich history spanning several decades, the company has consistently pushed the boundaries of what is possible, introducing ...

A Closer Look at Astec Industries Inc.’s Groundbreaking Technologies

Astec Industries

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Table of Links Abstract and 1 Introduction 2 Related Work 2.1 Neural Codec Language Models and 2.2 Non-autoregressive Models 2.3 Diffusion Models and 2.4 Zero-shot Voice Cloning 3 Hierspeech++ and 3.1 Speech Representations 3.2 Hierarchical Speech Synthesizer 3.3 Text-to-Vec 3.4 Speech Super-resolution 3.5 Model Architecture 4 Speech Synthesis Tasks 4.1 Voice Conversion and 4.2 Text-to-Speech 4.3 Style Prompt Replication 5 Experiment and Result, and Dataset 5.2 Preprocessing and 5.3 Training 5.4 Evaluation Metrics 5.5 Ablation Study 5.6 Zero-shot Voice Conversion 5.7 High-diversity but High-fidelity Speech Synthesis 5.8 Zero-shot Text-to-Speech 5.9 Zero-shot Text-to-Speech with 1s Prompt 5.10 Speech Super-resolution 5.11 Additional Experiments with Other Baselines 6 Limitation and Quick Fix 7 Conclusion, Acknowledgement and References 5.4 Evaluation Metrics For the reconstruction and resynthesis tasks, we conducted seven objective metrics: a log-scale Mel error distance (Mel), perceptual evaluation of speech quality (PESQ)[5] , Pitch, periodicity (Period.), voice/unvoice (V/UV) F1 score, and logscale F0 consistency F0c. We used the official implementation of CARGAN [60] for pitch, periodicity, and U/UV F1[6] . For F0c, we calculated the L1 distance between the log-scale ground-truth and the predicted F0 in the HAG. For VC, we used two subjective metrics: naturalness mean opinion score (nMOS) and voice similarity MOS (sMOS) with a CI of 95%; and three objective metrics for naturalness: UTMOS [69], character error rate (CER) and word error rate (WER); two objective metrics for similarity: automatic speaker verification equal error rate (EER), and speaker encoder cosine similarity (SECS). We utilized the open-source UTMOS[7] which is an MOS prediction model for a naturalness metric. Although this can not be considered an absolute evaluation metric, we believe that it is a simple way to estimate the audio quality of synthetic speech. Additionally, this method does not require ground-truth audio or labels to estimate the score. Therefore, we highly recommend using this simple metric during validation by adding a single line. For CER and WER, we utilized the Whisper’s official implementation. We used a large model with 1,550 M parameters and calculated the CER and WER after text normalization, as presented in the official implementation. We utilized a pre-trained automatic speaker verification models [42][8] which was trained with a large-scale speech dataset, VoxCeleb2 [14]. In [13], the effectiveness of metric learning in automatic speaker verification was demonstrated. Furthermore, [42] introduced online data augmentation, which decreased the EER from 2.17% to 1.17%. In addition, we utilized the pre-trained speaker encoder, Resemblyzer [9] to extract a speaker representation, and we calculated the cosine similarity between the speaker representation of the target speech and synthetic speech. For TTS, we additionally utilize a prosody MOS (pMOS). Sixty samples were randomly selected for each model. The nMOS was rated by 10 listeners on a scale of 1-5, and the sMOS and pMOS were rated by 10 listeners on a scale of 1-4. A confidence interval of 95% was reported for MOS. This paper is available on arxiv under CC BY-NC-SA 4.0 DEED license. [5] https://github.com/ludlows/PESQ [6] https://github.com/descriptinc/cargan [7] https://github.com/tarepan/SpeechMOS [8] https://github.com/clovaai/voxceleb_trainer [9] https://github.com/resemble-ai/Resemblyzer Authors:
(1) Sang-Hoon Lee, Fellow, IEEE with the Department of Artificial Intelligence, Korea University, Seoul 02841, South Korea;
(2) Ha-Yeong Choi, Fellow, IEEE with the Department of Artificial Intelligence, Korea University, Seoul 02841, South Korea;
(3) Seung-Bin Kim, Fellow, IEEE with the Department of Artificial Intelligence, Korea University, Seoul 02841, South Korea;
(4) Seong-Whan Lee, Fellow, IEEE with the Department of Artificial Intelligence, Korea University, Seoul 02841, South Korea and a Corresponding author. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Related Work 2.1 Neural Codec Language Models and 2.2 Non-autoregressive Models 2.1 Neural Codec Language Models and 2.2 Non-autoregressive Models 2.3 Diffusion Models and 2.4 Zero-shot Voice Cloning 2.3 Diffusion Models and 2.4 Zero-shot Voice Cloning 3 Hierspeech++ and 3.1 Speech Representations 3 Hierspeech++ and 3.1 Speech Representations 3.2 Hierarchical Speech Synthesizer 3.2 Hierarchical Speech Synthesizer 3.3 Text-to-Vec 3.3 Text-to-Vec 3.4 Speech Super-resolution 3.4 Speech Super-resolution 3.5 Model Architecture 3.5 Model Architecture 4 Speech Synthesis Tasks 4.1 Voice Conversion and 4.2 Text-to-Speech 4.1 Voice Conversion and 4.2 Text-to-Speech 4.3 Style Prompt Replication 4.3 Style Prompt Replication 5 Experiment and Result, and Dataset 5 Experiment and Result, and Dataset 5.2 Preprocessing and 5.3 Training 5.2 Preprocessing and 5.3 Training 5.4 Evaluation Metrics 5.4 Evaluation Metrics 5.5 Ablation Study 5.5 Ablation Study 5.6 Zero-shot Voice Conversion 5.6 Zero-shot Voice Conversion 5.7 High-diversity but High-fidelity Speech Synthesis 5.7 High-diversity but High-fidelity Speech Synthesis 5.8 Zero-shot Text-to-Speech 5.8 Zero-shot Text-to-Speech 5.9 Zero-shot Text-to-Speech with 1s Prompt 5.9 Zero-shot Text-to-Speech with 1s Prompt 5.10 Speech Super-resolution 5.10 Speech Super-resolution 5.11 Additional Experiments with Other Baselines 5.11 Additional Experiments with Other Baselines 6 Limitation and Quick Fix 6 Limitation and Quick Fix 7 Conclusion, Acknowledgement and References 7 Conclusion, Acknowledgement and References 5.4 Evaluation Metrics For the reconstruction and resynthesis tasks, we conducted seven objective metrics: a log-scale Mel error distance (Mel), perceptual evaluation of speech quality (PESQ)[5] , Pitch, periodicity (Period.), voice/unvoice (V/UV) F1 score, and logscale F0 consistency F0c. We used the official implementation of CARGAN [60] for pitch, periodicity, and U/UV F1[6] . For F0c, we calculated the L1 distance between the log-scale ground-truth and the predicted F0 in the HAG. For VC, we used two subjective metrics: naturalness mean opinion score (nMOS) and voice similarity MOS (sMOS) with a CI of 95%; and three objective metrics for naturalness: UTMOS [69], character error rate (CER) and word error rate (WER); two objective metrics for similarity: automatic speaker verification equal error rate (EER), and speaker encoder cosine similarity (SECS). We utilized the open-source UTMOS[7] which is an MOS prediction model for a naturalness metric. Although this can not be considered an absolute evaluation metric, we believe that it is a simple way to estimate the audio quality of synthetic speech. Additionally, this method does not require ground-truth audio or labels to estimate the score. Therefore, we highly recommend using this simple metric during validation by adding a single line. For CER and WER, we utilized the Whisper’s official implementation. We used a large model with 1,550 M parameters and calculated the CER and WER after text normalization, as presented in the official implementation. We utilized a pre-trained automatic speaker verification models [42][8] which was trained with a large-scale speech dataset, VoxCeleb2 [14]. In [13], the effectiveness of metric learning in automatic speaker verification was demonstrated. Furthermore, [42] introduced online data augmentation, which decreased the EER from 2.17% to 1.17%. In addition, we utilized the pre-trained speaker encoder, Resemblyzer [9] to extract a speaker representation, and we calculated the cosine similarity between the speaker representation of the target speech and synthetic speech. For TTS, we additionally utilize a prosody MOS (pMOS). Sixty samples were randomly selected for each model. The nMOS was rated by 10 listeners on a scale of 1-5, and the sMOS and pMOS were rated by 10 listeners on a scale of 1-4. A confidence interval of 95% was reported for MOS. This paper is available on arxiv under CC BY-NC-SA 4.0 DEED license. This paper is available on arxiv under CC BY-NC-SA 4.0 DEED license. available on arxiv [5] https://github.com/ludlows/PESQ https://github.com/ludlows/PESQ [6] https://github.com/descriptinc/cargan https://github.com/descriptinc/cargan [7] https://github.com/tarepan/SpeechMOS https://github.com/tarepan/SpeechMOS [8] https://github.com/clovaai/voxceleb_trainer https://github.com/clovaai/voxceleb_trainer [9] https://github.com/resemble-ai/Resemblyzer https://github.com/resemble-ai/Resemblyzer Authors: (1) Sang-Hoon Lee, Fellow, IEEE with the Department of Artificial Intelligence, Korea University, Seoul 02841, South Korea; (2) Ha-Yeong Choi, Fellow, IEEE with the Department of Artificial Intelligence, Korea University, Seoul 02841, South Korea; (3) Seung-Bin Kim, Fellow, IEEE with the Department of Artificial Intelligence, Korea University, Seoul 02841, South Korea; (4) Seong-Whan Lee, Fellow, IEEE with the Department of Artificial Intelligence, Korea University, Seoul 02841, South Korea and a Corresponding author. Authors: Authors: (1) Sang-Hoon Lee, Fellow, IEEE with the Department of Artificial Intelligence, Korea University, Seoul 02841, South Korea; (2) Ha-Yeong Choi, Fellow, IEEE with the Department of Artificial Intelligence, Korea University, Seoul 02841, South Korea; (3) Seung-Bin Kim, Fellow, IEEE with the Department of Artificial Intelligence, Korea University, Seoul 02841, South Korea; (4) Seong-Whan Lee, Fellow, IEEE with the Department of Artificial Intelligence, Korea University, Seoul 02841, South Korea and a Corresponding author.

The 7 Objective Metrics We Conducted for the Reconstruction and Resynthesis Tasks

ZUG, Switzerland, December 5th, 2024, Chainwire/--Holyheld, a leading Swiss crypto payments startup, announced today that it has secured funding to accelerate BRRR protocol with investment from Toyota Ventures, TempleDAO, Tomahawk VC, Prismatic Capital, Zee Prime Capital, Polygon, Kosmos Ventures, Moonlanding Ventures, as well as from angel investors from leading projects including Lido, Olympus, Dinero, Paragon, Superfluid, vfat, Inverse Finance, Daedalus, DCV, Generative Ventures, and Vamient. In an effort to fill the critical infrastructure necessary for users to transact globally with merchants instantly, Holyheld’s BRRR protocol aims to connect public blockchain networks, traditional payment, and card networks to create a global layer of real-time payments. “This opportunity consolidates our lead in the market, and fuels our drive to redefine how real-time clearing of payments should work,” said Anton Mozgovoy, Holyheld CEO. “With our technology of global stablecoin orchestration, we’re well-positioned to transform how users and businesses conduct business today with payments of tomorrow.” BRRR’s protocol is used to power Holyheld’s app, a leader in the European market in the segment of Web3 payments apps, which offers a cash account with a debit card available to residents of 30 European countries. Users of the Holyheld app can connect any self-custody wallet and use their cryptocurrency for daily purchases. This enables traditional consumer behaviors, like swiping or tapping a credit card or mobile device for payment, while users maintain self-custody of their funds. “This funding is an essential step as we work towards our vision of enabling crypto customers and fintechs to build new payment settlement and reconciliation solutions using crypto.” Dr. Anton Zagorodnikov, Holyheld co-founder and CTO, added. The new funding will allow Holyheld to continue expanding the protocol to more clients and payment networks globally. Holyheld has integrated over one hundred leading crypto protocols across 15 blockchain networks. The company’s BRRR has also orchestrated over $100M in volume since its launch in April of 2023. “Holyheld turns any self-custody wallet into a personal IBAN and a debit card, which enables users to have a frictionless digital wallet experience,” said Chris Abshire, principal, Toyota Ventures. “We are excited to work with Holyheld’s team as they build the leading on-chain clearing house.” To learn more about Holyheld, users can visit https://holyheld.com/ or https://holyheld.com/BRRR About Holyheld Founded in 2022, Holyheld is committed to enabling crypto customers and fintech to build new payment settlement and reconciliation solutions using stablecoins. With offices in Zug, Switzerland and Vilnius, Lithuania, the company’s first product is a Web3 debit card that allows users to make traditional payments with digital assets. Holyheld’s Blockchain Reconciliation and Remittance Record (BRRR) protocol enables consumers who are invested in blockchain-related assets to have significant purchasing power in the real world. To learn more about Holyheld, users can visit https://holyheld.com/ or https://holyheld.com/BRRR. Contact Holyheld Labs hello@holyheld.com This story was distributed as a release by Chainwire under HackerNoon’s Business Blogging Program. Learn more about the program here ZUG, Switzerland, December 5th, 2024, Chainwire/--Holyheld, a leading Swiss crypto payments startup, announced today that it has secured funding to accelerate BRRR protocol with investment from Toyota Ventures, TempleDAO, Tomahawk VC, Prismatic Capital, Zee Prime Capital, Polygon, Kosmos Ventures, Moonlanding Ventures, as well as from angel investors from leading projects including Lido, Olympus, Dinero, Paragon, Superfluid, vfat, Inverse Finance, Daedalus, DCV, Generative Ventures, and Vamient. In an effort to fill the critical infrastructure necessary for users to transact globally with merchants instantly, Holyheld’s BRRR protocol aims to connect public blockchain networks, traditional payment, and card networks to create a global layer of real-time payments. “This opportunity consolidates our lead in the market, and fuels our drive to redefine how real-time clearing of payments should work,” said Anton Mozgovoy, Holyheld CEO. “With our technology of global stablecoin orchestration, we’re well-positioned to transform how users and businesses conduct business today with payments of tomorrow.” “This opportunity consolidates our lead in the market, and fuels our drive to redefine how real-time clearing of payments should work,” said Anton Mozgovoy, Holyheld CEO. “With our technology of global stablecoin orchestration, we’re well-positioned to transform how users and businesses conduct business today with payments of tomorrow.” BRRR’s protocol is used to power Holyheld’s app, a leader in the European market in the segment of Web3 payments apps, which offers a cash account with a debit card available to residents of 30 European countries. Users of the Holyheld app can connect any self-custody wallet and use their cryptocurrency for daily purchases. This enables traditional consumer behaviors, like swiping or tapping a credit card or mobile device for payment, while users maintain self-custody of their funds. “This funding is an essential step as we work towards our vision of enabling crypto customers and fintechs to build new payment settlement and reconciliation solutions using crypto.” Dr. Anton Zagorodnikov, Holyheld co-founder and CTO, added. “This funding is an essential step as we work towards our vision of enabling crypto customers and fintechs to build new payment settlement and reconciliation solutions using crypto.” Dr. Anton Zagorodnikov, Holyheld co-founder and CTO, added. The new funding will allow Holyheld to continue expanding the protocol to more clients and payment networks globally. Holyheld has integrated over one hundred leading crypto protocols across 15 blockchain networks. The company’s BRRR has also orchestrated over $100M in volume since its launch in April of 2023. “Holyheld turns any self-custody wallet into a personal IBAN and a debit card, which enables users to have a frictionless digital wallet experience,” said Chris Abshire, principal, Toyota Ventures. “Holyheld turns any self-custody wallet into a personal IBAN and a debit card, which enables users to have a frictionless digital wallet experience,” said Chris Abshire, principal, Toyota Ventures. “We are excited to work with Holyheld’s team as they build the leading on-chain clearing house.” “We are excited to work with Holyheld’s team as they build the leading on-chain clearing house.” To learn more about Holyheld, users can visit https://holyheld.com/ or https://holyheld.com/BRRR https://holyheld.com/ https://holyheld.com/ https://holyheld.com/BRRR https://holyheld.com/BRRR About Holyheld Founded in 2022, Holyheld is committed to enabling crypto customers and fintech to build new payment settlement and reconciliation solutions using stablecoins. With offices in Zug, Switzerland and Vilnius, Lithuania, the company’s first product is a Web3 debit card that allows users to make traditional payments with digital assets. Holyheld’s Blockchain Reconciliation and Remittance Record (BRRR) protocol enables consumers who are invested in blockchain-related assets to have significant purchasing power in the real world. To learn more about Holyheld, users can visit https://holyheld.com/ or https://holyheld.com/BRRR . Blockchain Reconciliation and Remittance Record (BRRR) protocol Blockchain Reconciliation and Remittance Record (BRRR) protocol https://holyheld.com/ https://holyheld.com/ https://holyheld.com/BRRR https://holyheld.com/BRRR Contact Holyheld Labs hello@holyheld.com This story was distributed as a release by Chainwire under HackerNoon’s Business Blogging Program. Learn more about the program here This story was distributed as a release by Chainwire under HackerNoon’s Business Blogging Program. Learn more about the program here here here

Holyheld Raises Funding To Launch Blockchain Reconciliation And Remittance Record

Holyheld تجمع التمويل لإطلاق سجل التسوية والتحويلات المالية عبر تقنية Blockchain

Holyheld, Blockchain uzlaşması və pul köçürmə rekordunu işə salmaq üçün vəsait toplayır

Holyheld recauda fondos para lanzar un sistema de reconciliación y registro de remesas basado en blockchain

Holyheld lève des fonds pour lancer un service de réconciliation et de transfert de fonds basé sur la blockchain

Holyheld raccoglie fondi per lanciare la riconciliazione blockchain e il registro delle rimesse

Holyheld が資金を調達し、ブロックチェーンによる決済と送金記録を開始

Holyheld Azwi Misolo Pona Ko Lancer Dossier Ya Reconciliation Na Remise Ya Blockchain

Holyheld piesaista finansējumu, lai uzsāktu blokķēdes saskaņošanas un pārskaitījumu ierakstu

Holyheld Qullqita Huñun Blockchain Reconciliación hinaspa Remesas Record nisqa qallarinanpaq

Holyheld барои оғози сабти мусолиҳаи Blockchain ва интиқоли маблағ маблағ ҷамъ мекунад

ሆሊሄልድ መዝገብ ዕርቂን ሓዋላን ብሎክቸይን ንምጅማር ምወላ ኣዋህሊሉ።

Holyheld збирає фінансування для запуску блокчейн-звірки та запису грошових переказів

This paper is available on arxiv under CC 4.0 license. Authors: (1) Xiaoyu Ai, School of Electrical Engineering & Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia; (2) Robert Malaney, School of Electrical Engineering & Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia. Table of Links Abstract & Introduction System Overview Overcoming the limitation of key reconciliation Final Key Rate Experimental Results Discussion Conclusion & References Appendix VII. CONCLUSION In this work, we have carried out a full-blown analysis and experimental implementation of a Slice Reconciliation scheme applied to a specific CV QKD protocol (with post-selection) under simulated channel conditions anticipated for satelliteto-Earth channels. We have provided the optimal solution for the classical reconciliation process for this CV-QKD protocol in the context of massive parallelisation under the finite key regime. More specifically, we have identified the optimal block length when a large-code block is to be subdivided so as to improve the final secure key rate in bits per second. Although our results were based on a specific CV-QKD protocol and a specific GPU architecture, the type of analysis we have introduced here will apply in general terms a large suite of CV-QKD protocols run over any form of architecture that offers massive parallelisation. Our results, therefore, pave the way to optimal reconciliation system design for a wide range of practical CV-QKD systems that operate in the finite key regime. As the demand on the finite key size grows (better security thresholds), and technology advances lead to larger quantum signalling rates, the importance of optimised multithreaded CV-QKD reconciliation will grow. REFERENCES [1] N. Hosseinidehaj, Z. Babar, R. Malaney, S. X. Ng, and L. Hanzo, “Satellite-Based Continuous-Variable Quantum Communications: Stateof-the-Art and a Predictive Outlook,” IEEE Communications Surveys & Tutorials, vol. 21, no. 1, 881–919, 2018. [2] C. H. Bennett and G. Brassard, “Quantum Cryptography: Public Key Distribution and Coin Tossing,” in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing,, vol. 175, 8, 1984. [3] H. Weinfurter, “Quantum Communication Experiments with Discrete Variables,” Quantum Information: From Foundations to Quantum Technology Applications, 369–381, 2016. [4] L. Gy ¨ongy ¨osi, L. Bacsardi, and S. Imre, “A Survey on Quantum Key Distribution,” Infocommunications Journal, vol. 11, no. 2, 14–21, 2019. [5] S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, et al., “Advances in Quantum Cryptography,” Advances in Optics and Photonics, vol. 12, no. 4, 1012–1236, 2020. [6] B. Korzh, C. C. W. Lim, R. Houlmann, N. Gisin, M. J. Li, D. Nolan, B. Sanguinetti, R. Thew, and H. Zbinden, “Provably Secure and Practical Quantum Key Distribution over 307 km of Optical Fibre,” Nature Photonics, vol. 9, no. 3, 163–168, 2015. [7] T. A. Eriksson, T. Hirano, B. J. Puttnam, G. Rademacher, R. S. Lu´ıs, M. Fujiwara, R. Namiki, Y. Awaji, M. Takeoka, N. Wada, et al., “Wavelength Division Multiplexing of Continuous Variable Quantum Key Distribution and 18.3 Tbit/s Data Channels,” Communications Physics, vol. 2, no. 1, 1–8, 2019. [8] S.-Y. Shen, M.-W. Dai, X.-T. Zheng, Q.-Y. Sun, G.-C. Guo, and Z.- F. Han, “Free-Space Continuous-Variable Quantum Key Distribution of Unidimensional Gaussian Modulation Using Polarized Coherent States in an Urban Environment,” Physical Review A, vol. 100, 012325, 2019. [9] L. Gyongyosi and S. Imre, “Secret Key Rates of Free-Space Optical Continuous-Variable Quantum Key Distribution,” International Journal of Communication Systems, vol. 32, no. 18, e4152, 2019. [10] P. Jouguet, S. Kunz-Jacques, and A. Leverrier, “Long-Distance Continuous-Variable Quantum Key Distribution with a Gaussian Modulation,” Physical Review A, vol. 84, no. 6, 062317, 2011. [11] P. Jouguet, S. Kunz-Jacques, T. Debuisschert, S. Fossier, E. Diamanti, R. All´eaume, R. Tualle-Brouri, P. Grangier, A. Leverrier, P. Pache, et al., “Field Test of Classical Symmetric Encryption with Continuous Variables Quantum Key Distribution,” Optics Express, vol. 20, no. 13, 14030–14041, 2012. [12] Q. Liao, Y. Guo, D. Huang, P. Huang, and G. Zeng, “Long-Distance Continuous-Variable Quantum Key Distribution Using Non-Gaussian State-Discrimination Detection,” New Journal of Physics, vol. 20, no. 2, 023015, 2018. [13] D. Guo, C. He, T. Guo, Z. Xue, Q. Feng, and J. Mu, “Comprehensive High-Speed Reconciliation for Continuous-Variable Quantum Key Distribution,” Quantum Information Processing, vol. 19, no. 9, 1–19, 2020. [14] Y. Zhang, Z. Chen, S. Pirandola, X. Wang, C. Zhou, B. Chu, Y. Zhao, B. Xu, S. Yu, and H. Guo, “Long-Distance Continuous-Variable Quantum Key Distribution over 202.81 km of Fiber,” Physical Review Letters, vol. 125, no. 1, 010502, 2020. [15] A. Leverrier, F. Grosshans, and P. Grangier, “Finite-Size Analysis of a Continuous-Variable Quantum Key Distribution,” Physical Review A, vol. 81, no. 6, 062343, 2010. [16] F. Furrer, T. Franz, M. Berta, A. Leverrier, V. B. Scholz, M. Tomamichel, and R. F. Werner, “Continuous Variable Quantum Key Distribution: Finite-Key Analysis of Composable Security against Coherent Attacks,” Physical Review Letters, vol. 109, no. 10, 100502, 2012. [17] S. Kish, E. Villase˜nor, R. Malaney, K. Mudge, and K. Grant, “Feasibility Assessment for Practical Continuous Variable Quantum Key Distribution over the Satellite-to-Earth Channel,” Quantum Engineering, vol. 2, no. 3, e50, 2020. [18] M. Tomamichel, C. C. W. Lim, N. Gisin, and R. Renner, “Tight FiniteKey Analysis for Quantum Cryptography,” Nature Communications, vol. 3, no. 1, 1–6, 2012. [19] Z. 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Guo, “ContinuousVariable Quantum Key Distribution with Rateless Reconciliation Protocol,” Physical Review Applied, vol. 12, no. 5, 054013, 2019. [24] M. Milicevic, C. Feng, L. M. Zhang, and P. G. Gulak, “Quasi-Cyclic Multi-Edge LDPC Codes for Long-Distance Quantum Cryptography,” NPJ Quantum Information, vol. 4, no. 1, 1–9, 2018. [25] S.-S. Yang, Z.-G. Lu, and Y.-M. Li, “High-Speed Post-Processing in Continuous-Variable Quantum Key Distribution Based on FPGA Implementation,” Journal of Lightwave Technology, vol. 38, no. 15, 3935–3941, 2020. [26] H. Li and Y. Pang, “FPGA-Accelerated Quantum Computing Emulation and Quantum Key Distillation,” IEEE Micro, vol. 41, no. 4, 49–57, 2021. [27] C. Weedbrook, A. M. Lance, W. P. Bowen, T. Symul, T. C. Ralph, and P. K. Lam, “Quantum Cryptography without Switching,” Physical Review Letters, vol. 93, no. 17, 170504, 2004. [28] N. Hosseinidehaj, A. M. Lance, T. Symul, N. Walk, and T. C. 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Optimised Multithreaded CV-QKD Reconciliation for Global Quantum Networks: Conclusion & References

This paper is available on arxiv under CC 4.0 license. Authors: (1) Xiaoyu Ai, School of Electrical Engineering & Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia; (2) Robert Malaney, School of Electrical Engineering & Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia. Table of Links Abstract & Introduction System Overview Overcoming the limitation of key reconciliation Final Key Rate Experimental Results Discussion Conclusion & References Appendix VI. DISCUSSION CV-QKD. High-speed implementations realised in [61], [62] have used multidimensional reconciliation [37]. This multidimensional scheme is preferred for low SNR - but not so for the higher SNRs available via the Gaussian post-selection technique - a technique likely to be more useful in the satellite context [1]. 1) Reconciliation schemes for satellite-based In CV-QKD, Alice and Bob have to discard a block of reconciled bits if they detect a reconciliation failure (coding error) for that block. To compensate for the discarded bits, additional quantum signals need to be transmitted and reconciled, causing unwanted delays. Such delays can be problematic for satellitebased systems since the satellite is not always visible to the ground station. The FPGA-based reconciliation of [25] may suffer from this problem due to limited precision of arithmetic operations leading to higher reconciliation failures. As shown in many GPU-based works (including this work), GPU-based reconciliation offers less probability of reconciliation failure. 2) Low probability of reconciliation failures.

Optimised Multithreaded CV-QKD Reconciliation for Global Quantum Networks: Discussion

This paper is available on arxiv under CC 4.0 license. Authors: (1) Xiaoyu Ai, School of Electrical Engineering & Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia; (2) Robert Malaney, School of Electrical Engineering & Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia. Table of Links Abstract & Introduction System Overview Overcoming the limitation of key reconciliation Final Key Rate Experimental Results Discussion Conclusion & References Appendix APPENDIX

Optimised Multithreaded CV-QKD Reconciliation for Global Quantum Networks: Appendix

This paper is available on arxiv under CC 4.0 license. Authors: (1) Xiaoyu Ai, School of Electrical Engineering & Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia; (2) Robert Malaney, School of Electrical Engineering & Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia. Table of Links Abstract & Introduction System Overview Overcoming the limitation of key reconciliation Final Key Rate Experimental Results Discussion Conclusion & References Appendix V. EXPERIMENTAL RESULTS A. Decoding Error and Time B. The Optimal NR For this experiment, we pre-built a database to store LDPC codes with their code rates ranging from 0.01 to 0.8 and their block lengths ranging from 106 to 109 . For code rates less than 0.1, Multi-Edge-Type LDPC codes (degree distributions outlined in [22]) were used. These achieve a lower pDecode compared to irregular LDPC codes with the same code rate. For code rates greater than or equal to 0.1, we adopted the irregular LDPC codes (degree distributions outlined in [57]). At such code rates, these latter codes have the same pDecode performance as the Multi-Edge-Type counterparts, but allow for faster code construction. C. Final-key Effect for a Given No [15] Recall, we are particularly interested in the satellite-to-Earth channel. As in other works, we assume losses for this channel are dominated by diffraction effects, and therefore the transmissivity can be held constant. We further assume post-selections, using a bright classical beam sent along with the quantum signals (but different polarisation), remove any significant transmissivity deviations. As discussed elsewhere [17], some receiver/transmitter apertures, coupled to detailed phase-screen simulations of satellite downlink channels, render the constant-transmissivity assumption reasonable [56]. If the transmissivity is highly variable the optimal block length, NR, can be calculated by an expectation over the transmissivity density function. [16] We adopted the following method to determine ch. For m LDPC codes with NR = 106 and T = 0.9, we obtained the total number of arithmetic operations for those codes. Next, we measured the elapsed time to reconcile a block of 106 quadrature values. We then obtained ch by dividing the number of arithmetic operations to this measured elapsed time. [17] These trapping sets are the primary reason that additional decoding iterations are consumed for only a marginal decrease of the decoding error, i.e. the error floor effect [58].

Optimised Multithreaded CV-QKD Reconciliation for Global Quantum Networks: Experimental Results

This paper is available on arxiv under CC 4.0 license. Authors: (1) Xiaoyu Ai, School of Electrical Engineering & Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia; (2) Robert Malaney, School of Electrical Engineering & Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia. Table of Links Abstract & Introduction System Overview Overcoming the limitation of key reconciliation Final Key Rate Experimental Results Discussion Conclusion & References Appendix IV. FINAL KEY RATE We now present the penalty incurred for the division of N in the finite-key regime, and then propose an optimisation procedure to find the optimal NR which maximises the final key rate in bits per second. A. Analysis of the Final Key Rate For the protocol considered, in the finite-key regime the final key rate in bits per pulse, K, under the assumption of Gaussian collective attacks is adopted as [28], [49], [50][11] Thus far, we have been analysing the final key rate in bits per pulse. However, the final key rate in bits per second is more interesting in our context - the system will not be viable if the reconciliation takes too long to complete. From this point forward, we use a dashed symbol to distinguish a final key rate that is given in bits per second. Taking the decoding complexity into account, and ignoring the time taken to acquire the quantum signals, we can write the final key rate, K′ Finite, as B. Optimised LDPC Blocklength for CV-QKD Reconciliation where Eq. 30 can be solved via a numerical root-finding algorithm [54]. If we consider the standard CV-QKD settings, we obtain a stationary point at NR = 3.6×107 bits (the value of K′ Finite at this NR is discussed later). [11] This key-rate formulation was developed in [49] with a typographical error corrected in [50]. Eq. 19 is from [28] which acknowledged the correction and simplified the final key rate formulation (see footnote 2 of [51] for more details). A general discussion on the use of other key-rate formulations (e.g. [52], [53]) within our framework is given later. [12] Recalling Eq. 18, we note that for a given LDPC code, Ej is only dependent on the degree distribution pairs and Dj is only a function of the degree distribution pairs and γ. [13] In using this technique it is important to check that the inequality holds for the chosen parameter range of interest. This is done for all calculations we show here, but also for a much wider range not shown. For example, we find for ǫ = 10−9 , N = 109 and m = 5, the inequality 28 holds for any combination of the remaining parameters selected from the ranges VA ∈ [1, 34], T ∈ [0, 1] and ξ = [0, 0.05]. [14] The values of ξch and ξd in the standard CV-QKD settings are predicted values after accounting for all noise terms [17].

Optimised Multithreaded CV-QKD Reconciliation for Global Quantum Networks: Final Key Rate

This paper is available on arxiv under CC 4.0 license. Authors: (1) Xiaoyu Ai, School of Electrical Engineering & Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia; (2) Robert Malaney, School of Electrical Engineering & Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia. Table of Links Abstract & Introduction System Overview Overcoming the limitation of key reconciliation Final Key Rate Experimental Results Discussion Conclusion & References Appendix III. OVERCOMING THE LIMITATIONS OF KEY RECONCILIATION A. GPU-based SR The process of SR is to reconcile mN bits. One can naively use m LDPC matrices with NR = N for each matrix. However, due to practical hardware limitations, the process is better implemented by dividing N into Nd blocks of some smaller NR so that the same LDPC decoders can reconcile these blocks in parallel. This process resembles the idea of Single Program Multiple Data (see [39], [40] for more details). As illustrated in Fig. 2, we implement SR by creating Nd LDPC decoders loaded with the same LDPC matrix on Nd GPU threads and let these decoders reconcile Nd blocks in parallel. This helps to reduce the SR timescale and assists in meeting the time constraints, such as those posed in satellite-based scenarios. Section V-A will demonstrate in detail the advantage of using such parallelisation. B. The Penalty of Using Finite-Length LDPC Codes In SR, Bob needs to transmit syndrome bits to Alice based on the selected LDPC matrix with the code rate, Rj , for Sj via classical communications. For a given channel condition, selecting Rj closest to the capacity is the common approach to minimise the number of bits disclosed to the eavesdropper while Alice can still reconstruct Bob’s quantised bits without error [36]. Specifically, for a given T , we can obtain the SNR, γ, as [42] where Y is a vector of Bob’s quadrature values of length NR, M(Y ) is a mN-bit string obtained by applying the quantisation function M(·) to each quadrature value in Y , and Π(M(Y )) is the entropy function of M(Y ). Increasing m to values that render the quantisation error negligible is always possible, but this would require the individual LDPC codes for every j th slice to be near perfect (capacity-achieving) otherwise the efficiency β will be low; m = 5 is found to be a good pragmatic compromise, and is adopted here. Given five slices a constant quantisation size of the real line across 2 5 bins centered on zero is chosen. This size, which is dependent on the adopted γ, optimises β (see [31] for further discussion). The LDPC code rates, Rj , in Eq. 2 are the actual rates of the specific codes used for each slice (of length NR). Normally, in practice, NR is simply set to some value that allows target time-frames to be met, given that the decoding time is an increasing function of the block length [24]. We use Rj to obtain our experimental key rate in Eq. 32. A more nuanced value of that NR that optimises secure key rates is now analysed. Function A is termed the “channel dispersion” since it represents the reduction of the code rate from the channel capacity due to a tolerated decoding error probability. It is the “price to pay” for using a code with finite block length, for a given γ. C. Analysing the Computational Complexity of S With this all in place, it is now possible to solve for Dj as given by Eq. 1 Now we focus on the determination of Ej . When messages are propagated from the variable nodes to the check nodes, there are 2G multiplications and G additions [48]. When messages are propagating back to the variable nodes, there are 4G operations required (2G multiplications and 2G additions) [48]. Therefore, Ej is obtained by [46], [48] The decoding time of the whole reconciliation process, ∆t, is given by [7] We assume the least significant bit is l i 0 . [8] The rationale behind this is that the soft decoding output of Sj−1 provides a priori information on the reliability of each bit in Sj [30] [9] This approximation is accurate if the code achieves more than 80% of the capacity [43]. [10] In a BP decoder, a decoding iteration is one pass through the decoding algorithm

Overcoming the limitation of key reconciliation

This paper is available on arxiv under CC 4.0 license. Authors: (1) Xiaoyu Ai, School of Electrical Engineering & Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia; (2) Robert Malaney, School of Electrical Engineering & Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia. Table of Links Abstract & Introduction System Overview Overcoming the limitation of key reconciliation Final Key Rate Experimental Results Discussion Conclusion & References Appendix II. SYSTEM OVERVIEW Although, as just stated, our analysis will apply to most CV-QKD protocols, for detailed quantitative discussion we will consider only one specific CV-QKD protocol - the “noswitching” protocol [27]–[29] based on heterodyne detection. In this protocol, the quantum signal is encoded using Gaussian-modulated coherent states [27]. The main advantage of the no-switching protocol is that Alice and Bob can utlise all measurement results [28] (in most other protocols some results are discarded due to a random quadrature selection). We also adopt a Slice Reconciliation (SR) variant named Multi-Stage Hard Decoding[4] (MSHD) [30], [33], [34] for the classical reconciliation step, where the number of bits derived from each measurement outcome is m. We refer to this variant simply as SR in the following. It is worth noting that the optimisation analysis to follow is to some extent independent of the details of the reconciliation scheme. However, SR [30], [36] can be compared with the other well-known reconciliation scheme for CV-QKD - multidimensional reconciliation [37]. It is known that SR achieves higher reconciliation efficiency when the Signal-to-Noise Ratio (SNR) is greater than 1 [36]. At low SNR the opposite is true. For focus, here we adopt SR (as multistage hard decoding [36]) since in many satellite scenarios post-selection is used to filter out the low SNR quantum signals [1]. Our adopted scheme will be more useful in such scenarios. We now briefly describe the steps of the protocol, a diagram of which shown in Fig. 1. In the following, we assume the quantum signalling rate is much larger than the reconciliation rate. • Alice selects a fixed modulation variance VA. For each quantum signal to be transmitted to Bob, Alice randomly selects a number from a Gaussian distribution, N(0, VA), and then prepares a signal by displacing one of the quadrature components of a vacuum state by this random number. The process is repeated on the signal for the other quadrature. The signal is then transmitted to Bob. Step 1: Signal Preparation. • Bob performs heterodyne detection to obtain the two quadrature values (real numbers) for each received signal. Bob compares each measured quantum signal with a given cut-off threshold and informs Alice to discard her corresponding quantum signal if his measured quantum signal is lower than the threshold[5] . A quantum signal that is lost in transit registers a null signal at Bob. Neglecting null signals, Bob holds 2No quadrature values at the end of this process. Step 2: Heterodyne Detection. • S Bob randomly selects a subset 2Ne from the 2No quadrature values and sends this estimation subset, along with the corresponding time information, to Alice via classical communications (we adopt Ne = 1 2No, unless otherwise stated). Alice uses the timing information to best pair the signals in this subset (and therefore the corresponding quadrature values) sent by her and then estimates the covariance matrix between the shared states. Based on the estimated covariance matrix, Alice determines the channel transmissivity, T , excess noise, ξ, Bob’s SNR, γ, the Holevo Information, χBE, between Bob and the eavesdropper (Eve), and the mutual information between Alice and Bob, IAB. Finally, for a given target reconciliation efficiency β, Alice compares χBE with βIAB. Alice aborts the protocol if χBE ≥ βIAB. Otherwise, Alice informs Bob of the estimation results, i.e. T , ξ, γ, χBE and IAB. tep 3: Parameter Estimation. • Using Gray Labelling, Alice and Bob represent each of the quadrature values embedded in each signal with m bits. Then, for quadrature values selected in the estimation subset, Alice forms a Ne-by-m bit matrix and Bob does the same. Next, Alice and Bob exchange their matrices and compare the j th column of the two matrices to estimate the Bit Error Ratio[6] (BER), pj , j ∈ {0, 1, · · · , m−1}, for all the digits in the j th column. The estimated pj will be used in SR. Finally, Alice and Bob discard all the quadrature values in the estimation subset. At the end of this step, Alice and Bob each hold a mN-bit string. Step 4: Bit Error Estimation for SR. • For each column, Alice and Bob agree on an LDPC code with block length NR that is closest to the capacity determined by pj. Bob forms a new NR-bit string (referred as a “slice” in SR) by selecting the j th digit (bit) of each of the NR quadrature values, encodes the new bit string (the slice) into syndrome bits, and sends those bits to Alice (see III.B for details). Alice then initiates SR to obtain her best estimate of Bob’s string. Alice repeats this process until all her mN bits are reconciled. Finally, Alice and Bob obtain two hashed strings by applying the same hash function to their reconciled strings and exchange the hash results to check whether SR is successful. If successful, Alice holds a mN-bit string identical to Bob’s mN-bit string. Otherwise, they abort the protocol and restart from Step 1. Step 5: Reverse Reconciliation. • Based on Eq. 19, Alice and Bob compute the length of the secret key that can be extracted and then apply a 2-universal hashing function on their reconciled string to obtain two identical and shorter secret key strings about which Eve has effectively no knowledge. Step 6: Privacy Amplification. [4] The slice reconciliation can be implemented with 2 other variants: Bit Interleaved Coded Modulation (BICM) [30] and Multi-Level Coding/MultiStage Decoding (MLC/MSD) [31], [32]. We note that the MLC/MSD takes advantage of the dependence between slices to select the optimal LDPC code rates [30], [32]. However, as a special case of MLC/MSD, MSHD assumes that the slices are independent [30], [33], [34]. Using MSHD leads to a tractable analysis at the expense of sub-optimal selection of LDPC code rates, but such expense is negligible if Gray Labelling [35] is adopted and the number of slices is at least 5 [30], [34], as is the case in this work. [5] The Gaussian post-selection technique at Bob’s side effectively improves the channel conditions between Alice and Bob [38] so that SR is preferred for reconciliation (rather than multidimensional reconciliation). [6] At this step, sources of bit errors include the channel transmission, heterodyne detection, and quantisation.

Optimised Multithreaded CV-QKD Reconciliation for Global Quantum Networks: System Overview

This paper is available on arxiv under CC 4.0 license. Authors: (1) Xiaoyu Ai, School of Electrical Engineering & Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia; (2) Robert Malaney, School of Electrical Engineering & Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia. Table of Links Abstract & Introduction System Overview Overcoming the limitation of key reconciliation Final Key Rate Experimental Results Discussion Conclusion & References Appendix Abstract Designing a practical Continuous Variable (CV) Quantum Key Distribution (QKD) system requires an estimation of the quantum channel characteristics and the extraction of secure key bits based on a large number of distributed quantum signals. Meeting this requirement in short timescales is difficult. On standard processors, it can take several hours to reconcile the required number of quantum signals. This problem is exacerbated in the context of Low Earth Orbit (LEO) satellite CV-QKD, in which the satellite flyover time is constrained to be less than a few minutes. A potential solution to this problem is massive parallelisation of the classical reconciliation process in which a large-code block is subdivided into many shorter blocks for individual decoding. However, the penalty of this procedure on the important final secured key rate is non-trivial to determine and hitherto has not been formally analysed. Ideally, a determination of the optimal reduced block size, maximising the final key rate, would be forthcoming in such an analysis. In this work, we fill this important knowledge gap via detailed analyses and experimental verification of a CV-QKD sliced reconciliation protocol that uses large block-length low-density parity-check decoders. Our new solution results in a significant increase in the final key rate relative to non-optimised reconciliation. In addition, it allows for the acquisition of quantum secured messages between terrestrial stations and LEO satellites within a flyover timescale even using off-the-shelf processors. Our work points the way to optimised global quantum networks secured via fundamental physics. I. INTRODUCTION Continuous Variable (CV) Quantum Key Distribution (QKD) has been intensively studied and significant breakthroughs have been achieved in both theory and experiment (see [1] for review). Compared to Discrete Variable (DV) QKD [2]–[5], CV-QKD can be implemented with well-developed technologies (e.g., homodyne detectors) in commercial fibreoptic networks [6], [7] and free-space optical communications [8], [9], providing it a potential advantage in practical deployments [10]–[14]. Considering the finite-key security of CV-QKD and DV-QKD, there are three critical parameters. These are, No, the number of original quantum signals sent by the transmitter (Alice) that are collected by the receiver (Bob); Ne, the number of quantum signals from which the protocol parameters are estimated;[1] and, ǫ, the probability that a QKD protocol fails to generate secret keys [15], [16]. To satisfy an upper limit on the failure probability of parameter estimation, Alice and Bob set Ne to a large value, which in turn implies a larger No. Despite the advantages in deployment, CV-QKD systems tend to demand a larger No to reach the same ǫ relative to DVQKD protocols. For example, to achieve a final key rate of 0.1 bits per pulse with ǫ = 10−9 , a CV-QKD protocol studied in [17] required No ≈ 109 signals. However, to achieve the same final key rate with ǫ = 10−14, the DV-QKD protocol in [18] required No ≈ 104 signals. This higher number of required signals in CV-QKD can render the classical post-processing (i.e. key reconciliation and privacy amplification[2] ) slow - possibly failing to meet target timescales for reconciliation. The end-users of a CV-QKD system expect the system to deliver two identical and secure keys under a limited time interval. For example, for satellite-based deployments, we would hope that the reconciliation is completed while maintaining a line-of-sight connection with the ground station. For a CV-QKD-enabled satellite with orbital parameters similar to Micius [20], this would mean the reconciliation should be completed in less than a few minutes. For the protocol we use in this work (see later), and for ǫ = 10−9 , this, in turn, would require the data rate of reconciliation to be at least 3.6 × 106 bits per second. For real-time reconciliation (say in subsecond timescales), two orders of magnitude increases in the reconciliation rates would be required. Demands for smaller ǫ will exacerbate the issue. Ideally, the rate of reconciliation should always be faster than the rate of quantum signalling. This all raises the question as to whether current CV-QKD reconciliation schemes are optimised for the highest possible key rates in bits per second. As we show here, this is not the case. Further optimisation is possible on all current schemes. To understand the issue better, we define reconciliation in the context of CV-QKD as a two-step scheme where the inputs to the reconciliation are non-identical N = 2No − 2Ne quadrature values[3] held by Alice and Bob (after parameter estimation), and the output is an identical bit string held by Alice and Bob [21]–[23]. Assuming a reverse reconciliation scheme, Bob first converts the quadrature values encoded by Alice in each signal to m bits. Alice, after converting each of her encoded real numbers also to m bits, then initiates some discrepancy-correction algorithms based on pre-defined errorcorrection codes to ensure her mN bits are identical to Bob’s. In this work, we will adopt Low-Density Parity-Check (LDPC) codes for the error correction. However, as alluded to above, reconciling mN bits within a limited time frame can be challenging. State-of-the-art LDPCbased reconciliation schemes for CV-QKD systems involve parallelised computation on a Graphics Processing Unit (GPU) [13], [24] or Field-Programmable Gate Arrays (FPGAs) [25], [26]. Reconciliation schemes implemented on FPGAs offer more programmable flexibility, but sometimes at the cost of reduced memory access relative to GPUs. For our purposes, both hardware architectures are useful - both offer massive parallelisation opportunities. These parallelisation solutions generally take the following two-step approach: 1) The mN bits are organised as m N-bit blocks to be reconciled. Each N-bit block is divided into multiple shorter blocks of size, say, NR. This is usually just set to a block size that can be processed within some timescale. 2) Then the m NR-bit blocks are reconciled in parallel (via independent processors) using optimally-designed LDPC decoders. However, what is missing in this approach is a proper optimisation analysis as to what the optimal value of NR is. As we show below, simply reducing NR at the cost of additional processing units is not an optimal solution. It transpires that in QKD the “penalty” cost of reducing the code rate (implicit in the use of small block lengths) significantly influences the bit per second final key rate. A more sophisticated analysis is required to determine the optimal reduced block length. Such an analysis is the key contribution of this work. Although we will adopt a specific CV-QKD protocol for our analysis, the key steps of our scheme will apply to any CV-QKD protocol. Our reconciliation scheme will deliver the highest reconciliation rate for a given processor speed - thus allowing for the optimal solution to CV-QKD reconciliation. [1] More precisely, in a CV-QKD protocol, Alice and Bob randomly select a Ne-signal subset from the No signals to estimate the parameters. [2] In this work, we focus on the key reconciliation step because it is the more time-consuming part in the post-processing steps while the privacy amplification involving only bit-wise operations can be easily implemented faster than the reconciliation [19]. [3] No and Ne are multiplied by 2 since Alice and Bob utilise both quadratures from heterodyne detection - the detection process we assume in this work.

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