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Quantum cryptography: Public key distribution and coin tossing
Open accessAbstract
When elementary quantum systems, such as polarized photons, are used to transmit digital information, the uncertainty principle gives rise to novel cryptographic phenomena unachievable with traditional transmission media, e.g. a communications channel on which it is impossible in principle to eavesdrop without a high probability of disturbing the transmission in such a way as to be detected. Such a quantum channel can be used in conjunction with ordinary insecure classical channels to distribute random key information between two users with the assurance that it remains unknown to anyone else, even when the users share no secret information initially. We also present a protocol for coin-tossing by exchange of quantum messages, which is secure against traditional kinds of cheating, even by an opponent with unlimited computing power, but ironically can be subverted by use of a still subtler quantum phenomenon, the Einstein-Podolsky-Rosen paradox.
Overview
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Written from the paper’s full text.
TL;DR
The paper introduces quantum cryptography using non-orthogonal polarized photon states to distribute a secret random key between users who initially share no secret, with eavesdropping detectable in principle. It also presents a quantum coin-tossing protocol secure against traditional cheating even with unlimited computing power, except for a theoretical EPR-based attack requiring perfect photon storage and detection.
Key findings
- In quantum key distribution, an eavesdropper informed of a photon’s original basis only after measuring it can obtain at most 1/2 expected bits about the key bit; any measurement yielding \(b\) bits (\(b \le 1/2\)) must induce disagreement with probability at least \(b/2\) when the photon is later re-measured in its original basis. The optimum tradeoff example: measuring and retransmitting all intercepted photons rectilinearly learns the correct polarizations of half the photons and induces disagreements in 1/4 of those later re-measured in the original basis.
- Alice and Bob can test for eavesdropping by publicly comparing a random subset—say one third—of correctly received bits; if comparisons agree, the remaining same-basis bits can be used as a one-time pad.
- For coin tossing, Alice sends about one thousand random bits encoded in one basis; Bob guesses the basis; Alice reveals the basis and her original bit string; Bob verifies against his rectilinear and diagonal tables. The honest protocol gives each party exactly 50 per cent chance of winning.
- Cheating by Bob would require guessing Alice’s basis with probability greater than 1/2, which would imply distinguishing randomly polarized photon trains and enable faster-than-light communication via the Einstein-Podolsky-Rosen effect. Alice can cheat using EPR photon pairs, but this requires perfect efficiency of storage and detection of photons, beyond current technology.
Methods
- Theoretical protocol design and security analysis based on quantum mechanics of single polarized photons, Hilbert-space formalism, conjugate rectilinear and diagonal bases, the uncertainty principle, and the no-cloning theorem.
- Quantum key distribution protocol: Alice sends random bits in randomly chosen bases; Bob measures randomly in one of the two bases; public classical discussion identifies successful same-basis bits; optional Wegman-Carter authentication can relax the need for an active-eavesdropping-immune public channel if a small secret key is shared.
- Coin-tossing protocol: Alice sends a photon train in one basis, Bob guesses the basis, Alice reports the actual basis and certifies by revealing her original bit sequence, and Bob verifies against his two measurement tables. The paper provides illustrative examples but no empirical data.
Limitations
- Stated: quantum transmissions are necessarily very weak and cannot be amplified in transit; quantum cryptography does not provide digital signatures, certified mail, or the ability to settle disputes before a judge.
- Stated: the public channel must be immune to active eavesdropping unless Alice and Bob share a small secret key for Wegman-Carter authentication; an eavesdropper can still suppress communication, though users can detect suppression with high probability.
- Stated: the EPR cheating attack is merely theoretical because it requires perfect efficiency of photon storage and detection, beyond current technology; the honestly followed protocol could be realized with current technology. Evident from the text: no experimental implementation or empirical data are reported, and the security arguments are theoretical.
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25 Own interpretation · 8 No basis given · 6 Not checked yet
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“Thus Bob obtains mean ingful data from only half the photons he detects- those for which he guessed the corre'ct polarization basis.”
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“and that any such measurement yielding b bits of expected information (b s 1/2) must induce a disagreement with probability at least b/2 if the measured photon, or an attempted forgery of it, is later re-measured in its original basis.”
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“when the ea vesdropper measures and retransmits all intercepted photons in the rectilinear basis, thereby learning the correct polarizations of half the photons and inducing disagreements in 1/4 of those that are lat er re-measured in the original basis.”
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“such cloning can be shown to be inconsistent with the foundations of quant . um mechanics [WZ)”
What the sources say
- Not checked yetcites [WZ)
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In the paper
“Even in the presence of active eavesdropping, the two parties can still distribute key securely if they share some secret information initially, provided the eavesdropping is not so active as to suppress connunications completely.”
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- The authors state this as an assumption: The public classical channel used in the protocol is susceptible to eavesdropping but not to injection or alteration of messages.
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In the paper
“the transmitted photons, in passing through the P polarizer, enoerge with exactly P polariza tion, having lost a,11 memory of their previous po larization a.”
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“we also present a protocol for coin-tossing by exchange of quantWII noes sages, which is secure against traditional kinds of cheating, even by an opponent· with unlinited computing power”
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“being transmit ted with probability cos 2 (a-P) and absorbed with the comple11entary probability sin 7 («-P).”
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“The eavesdropper cannot even gain partial information about such a trans11is sion without altering it a random and uncontrollable way likely to be detected by the channel's legiti mate users.”
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“Except where otherwise noted the protocols are provably secure even against an opponent with superior technology and unli11ited COftputing power, barring fundamental violations of accepted physical laws.”
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In the paper
“Here we show that quantum coding by itself achieves one of the main advantages of public key cryptography by per mitting secure distribution of random key informa tion between parties who share .no secret information”
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In the paper
“Moreover, quantum cryptography does not provide di gital signatures, or applications such as certified 11ail or the ability to settle disputes before a judge.”
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In the paper
“This cheat requ.i. 1:.-s that Alice be able to store the twin photons fc-r 3 considerable time and then measure them wit!, hcqh detection efficien cy”
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In the paper
“However, in either case, Alice and Bob will conclude with high proba bility that their secret communications are being suppressed,. and will not be fooled into thinking their communicaticns are secure when in fact they're not.”
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“By contrast, in traditional coin-tossing schemes, analogous attempts to seize a lucky victory frOft the jaws of defeat, though unlikely to succeeed, are unaccompanied by any danger of detection.”
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In the paper
“The eavesdropper can still prevent communication by suppressing messages in the public channel, as of course he can by sup pressing or excessively perturbing the photons sent through the quantum channel.”
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It may state two separate claims as one.
- The authors state this as an assumption: The public classical channel used in the protocol is susceptible to eavesdropping but not to injection or alteration of messages.
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In the paper
“The bit positions used in this comparison should be a random subset (say one third) of the correctly received bits, so that eavesdropping on more than a few pho tons is unlikely to escape detection.”
plainconditional- Asserted: passes — The authors put it forward as true.
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“This iaplausible-sounding explanation is supported by formal quantum mechanics, which represents the state of a pair of photons as a vector in a 4-dimensional Hilbert space obtained by taking the tensor product of two 2-dimensional Hilbert spaces.”
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In the paper
“Bob woulr: nc,,d to guess Alice's basis ·:-th i:robability greate: than 1/2.”
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In the paper
“If all the comparisons agree, Alice and Bob can conclude that the quantum transmission has been free of signifi cant eavesdropping; and those of the remaining bits that were sent and received with the same basis also agree, and can safely be used as a one time pad for subsequent secure communications over the public channel.”
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It may state two separate claims as one.
- The authors state this as an assumption: The public classical channel used in the protocol is susceptible to eavesdropping but not to injection or alteration of messages.
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In the paper
“she would need to convince Bob that her photons were :Hagonally polarized, which she can only do by prod ucing a sequence of bits in perfect agreement with Bob's diagonal table.”
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“Any pholons lost by Alice during storage or measurement would result in holes in her pretended bit sequence, which she would have to fill by guessing, and these guesse3 would risk detection by Bob if they failed to agr·?c, with his tables.”
hedgeddirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
The authors state this as an assumption.
The public classical channel used in the protocol is susceptible to eavesdropping but not to injection or alteration of messages.
“susceptible to eavesdropping but not to the injection or alteration of messages.”
Claims other papers make about this one
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In the citing paper
“QKD protocols such as BB84 and E91 have been successfully implemented since 2003.”
From Quantum-resistance in blockchain networks (2023) · cites this paper as [31]
Experimental — accuracy not yet measured
Citations
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Self-citation
Research works printed but not linked: 1
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Metadata
- Authors
- Bennett, C. H., & Brassard, G.
- Affiliations
- IBM (United States) (US)
- Corresponding author
- Bennett, Charles H.
- ORCID
- Bennett, Charles H.
- Venue
- Theoretical Computer Science
- Publisher
- Elsevier BV
- Published
- 2014-12
- Volume
- 560
- Pages
- 7-11
- ISSN
- 0304-3975 (ISSN-L)
- OpenAlex
- W2015179126
- Semantic Scholar
- 17c16c133ab46e66ea0a08f40d19b3308733c348
- Cited by
- 8,299
- References
- 6
- Citations by year
- 2026: 479 · 2025: 773 · 2024: 554 · 2023: 518 · 2022: 463 · 2021: 616 · 2020: 519 · 2019: 512 · 2018: 374 · 2017: 377 · 2016: 326 · 2015: 343 · 2014: 308
- FWCI
- 449.98
- Type
- journal article
- Language
- en
- Open access
- hybrid
- doi.org
- Licence
- Locations
- Theoretical Computer Science (publishedVersion, cc-by-nc-nd)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Physical Sciences › Computer Science › Computer Networks and Communications › Cryptography and Data Security
- Physical Sciences › Engineering › Electrical and Electronic Engineering › Wireless Communication Security Techniques
- Keywords
- quantum cryptography, public key distribution, quantum key distribution, polarized photons, uncertainty principle, eavesdropping detection, quantum channel security, EPR paradox
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