{'fraction_non_alphanumeric': 0.03748776908023484, 'fraction_numerical': 0.006421232876712328, 'mean_word_length': 4.445554445554445, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The physics of no-bit-commitment : Generalized quantum non-locality versus oblivious transfer

We show here that the recent work of Wolf and Wullschleger (quant-ph/0502030) on oblivious transfer apparently opens the possibility that non-local correlations which are stronger than those in quantum mechanics could be used for bit-commitment. This is surprising, because it is the very existence of non-local correlations which in quantum mechanics prevents bit-commitment. We resolve this apparent paradox by stressing the difference between non-local correlations and oblivious transfer, based on the time-ordering of their inputs and outputs, which prevents bit-commitment.

We show here that the recent work of Wolf and Wullschleger (quant-ph/0502030) on oblivious transfer apparently opens the possibility that non-local correlations which are stronger than those in quantum mechanics could be used for bit-commitment. This is surprising, because it is the very existence of non-local correlations which in quantum mechanics prevents bit-commitment. We resolve this apparent paradox by stressing the difference between non-local correlations and oblivious transfer, based on the time-ordering of their inputs and outputs, which prevents bit-commitment.
In 1984, Bennett and Brassard [1] proposed a quantum physics scheme (BB84) by which two parties can establish a secret key, allowing them to communicate with unconditional security against eavesdroppers. This result remains one of the cornerstones of quantum cryptography, and gave rise to the hope that many other cryptographic primitives (which classically rely on unprovable assumptions), could be made unconditionally secure within a quantum framework. Perhaps the best known of these is bit commitment.
A bit-commitment scheme allows one party (Alice) to commit to a decision in such a way as a second party (Bob) will believe her when she later reveals it, but cannot find out her decision until that point (eg. Alice's decision is sealed in a safe which is given to Bob, while Alice keeps the key. Then in the revealing stage, Alice sends the key to Bob).
Using the same encoding scheme as for keydistribution, Bennett and Brassard constructed a quantum coin-tossing protocol [1] that could directly be used for bit-commitment. Their protocol is secure when Alice is limited to using only separable states. However, as noted by the authors, Alice can cheat convincingly in their protocol by using entangled states.
This result was later expanded by Mayers [2], and by Lo and Chow [3], to show that Alice can cheat in any quantum bit-commitment scheme which is secure against Bob. It is therefore impossible to implement an unconditionally secure bit-commitment scheme within quantum physics.
Entangled quantum states are crucial in proving this no-bit-commitment result, allowing correlations to exist between Alice's and Bob's systems which cannot be simulated by any local hidden-variable model [4]. These 'non-local' correlations do not allow for super-luminal signalling, but are nevertheless extremely powerful [5].
Interestingly, Popescu and Rohrlich have shown that quantum states do not provide the strongest possible non-local correlations consistent with relativity [6]. For two parties, who both have a single binary input (their measurement setting) and a binary output (their measurement result), the strongest possible correlations are instead given by systems known as PR-boxes [6,7,8].
PR-boxes are a valuable conceptual tool in understanding non-locality, as they allow us to separate the concept of non-local non-signalling correlations from the details of a particular physical model (eg. Complex Hilbert space).
PR-boxes (and their analogues with more inputs and outputs) are also very general. All bipartite no-signalling boxes with binary inputs and outputs can be constructed from a PR-box and a mixture of local operations [7]. Furthermore, a single PR-box (with shared randomness) can be used to simulate the results of any bipartite measurement on a maximally entangled quantum singlet state [8].
Recently, Wolf and Wullschleger [9] have shown that a PR-box can also be used to simulate a cryptographic primitive known as one-out-of-two oblivious transfer [10,11], in which Bob can secretly learn either (but not both) of two bits submitted by Alice. We will refer to the device which implements this scheme as an OT-box.
Interestingly, it is known that OT-boxes can be used to implement secure bit commitment. If OT-boxes can be built from PR-boxes, then this implies that bit commitment can be achieved using PR-boxes. This would be a surprising result, as PR-box correlations are very similar to those attainable in quantum theory, where, as emphasized previously, no bit-commitment is possible.
In order to investigate this argument in more detail, this letter is organized as follows. First, we introduce the requirements for a bit commitment scheme and give an explicit example of such a scheme using OT-boxes. Next, we recall Wolf and Wullschleger's connection between the PR-and the OT-boxes. Finally, we show that the analogous scheme involving PR-boxes does not in fact allow secure bit-commitment and elaborate on its significance.
Formally, a Bit commitment scheme consists of two protocols (COMMIT and REVEAL) which satisfy the following requirements: 1. Correctness: If Alice and Bob are both honest, then during the COMMIT protocol, Alice selects a value for her committed bit α ∈ {0, 1}, and this value is learnt by Bob during the REVEAL protocol.
2. Privacy: If Alice is honest, then Bob can learn nothing about α until the REVEAL protocol is enacted.
3. Binding: If Bob is honest, then after the COM-MIT protocol has finished, there is (at most) only one specific value of α (e.g. α = 0) which Bob will accept during the REVEAL protocol. He will never accept the other possible value of α. This prevents Alice from 'changing her mind' about which value of α she committed.
A bit-commitment scheme satisfying these three conditions would be perfectly secure. However, for cryptographic purposes it is interesting to consider the slightly weaker case of secure bit commitment, in which a bitcommitment scheme can be made arbitrarily close to perfectly secure.
In particular, we consider a weakening of the binding requirement to the following: 3. Secure binding: The scheme must permit arbitrarily large values of a security parameter N ǫ ∈ N, such that if Bob is honest, then after the COMMIT process has finished there is at most one value of α (e.g. α = 0) which Bob will accept with probability greater than 2 −Nǫ in the REVEAL protocol. This is Alice's committed value of α.
Note that given a secure bit-commitment protocol with secure binding parameter N ǫ = 1, it is possible to obtain a protocol with an arbitrarily large security parameter N ′ ǫ by running 2N ′ ǫ − 1 copies of the N ǫ = 1 protocol in parallel (with Bob only accepting Alice's commitment if she is not caught cheating in any of the parallel runs). As the schemes we consider have perfect privacy, this will not help Bob gain any further information about Alice's committed bit. Now, it is possible to construct a secure bitcommitment scheme using oblivious transfer. Following Wolf and Wullschleger [9], we consider OT-boxes which implement one-out-of-two oblivious transfer, in which Alice inputs two bits, x 0 and x 1 , into the system and Bob inputs a single choice bit c. Finally, the system outputs where ⊕ denotes addition modulo 2. This allows Bob to learn one of Alice's two input bits, without Alice knowing which bit he has obtained (see figure 1).
A well-known way to implement bit-commitment from one-out-of-two oblivious transfer is to use this to simulate ordinary oblivious transfer [12] and then use the latter for bit commitment [13]. Here we give an explicit protocol which uses one-out-of-two oblivious transfer (our OT-box) to directly implement bit-commitment.
Consider first a scheme in which Alice and Bob share a single OT-box, which is as follows: COMMIT : 1. Alice selects a random bit r, and her committed bit α. 2. Alice inputs x 0 = r and x 1 = r ⊕ α into the OT-box, where ⊕ denotes addition modulo 2. 3. Alice sends a message to Bob telling him it is his turn. 4. Bob selects a random bit s and inputs c = s into the OT-box 5. Bob records the output bit x c . If the OT-box fails to produce an output (as Alice has not yet made her inputs), Bob knows that Alice is cheating and will not accept her revealed bit.
REVEAL : 1. Alice sends α and r to Bob. 2. Bob checks to see if x c = r ⊕ αs. If this relation is true, Bob accepts α as the revealed bit, otherwise he knows that Alice has cheated and rejects Alice's revelation.
COMMIT and REVEAL constitute a secure bitcommitment scheme with security parameter N ǫ = 1, as can easily be checked. Using 2N ′ ǫ − 1 OT-boxes in parallel, it is therefore possible to obtain arbitrarily secure bit commitment.
We now consider whether one can generate an analogous bit-commitment scheme to that given above using PR-boxes (figure 1). Recall that a PR-box can be thought of as an abstraction and generalization of a Belltype experiment [4], in which Alice and Bob share an entangled system on which they can each perform one of two different dichotomic measurements. If we denote Alice's and Bob's binary inputs (their measurement settings) by x and y respectively, and their binary outputs (their measurement results) by a and b respectively, the extremal non-local correlations given by a PR-box are of the form where all outcomes consistent with (1) are equally likely, and each party obtains their output bit immediately after entering their input. Although such correlations are stronger than anything attainable in quantum theory, Alice and Bob cannot use the PR-box to signal to one another.
In order to simulate a PR-box using an OT-box and vice-versa, Wolf and Wullschleger propose the following protocols [9]: PR-box from OT-box : 1. Alice chooses a random bit a.

2.
Alice inputs x 0 = a and x 1 = x ⊕ a into the OT-box.
3. Bob inputs c = y into the OT-box and obtains output b = x c . Note that b = a ⊕ xc = a ⊕ xy as required.
OT-box from PR-box : 1. Alice inputs x = (x 0 ⊕ x 1 ) into the PR-box and obtains output a.
2. Alice sends m = x 0 ⊕ a to Bob 3. Bob inputs y = c into the PR-box and obtains output b.

Bob computes the output
Directly applying the latter simulation procedure to the COMMIT protocol for the OT-box, we find the analogous procedure COMMIT ′ for the PR-box. The RE-VEAL protocol is the same in both cases: 1. Alice selects a random bit r, and her committed bit α.

2.
Alice inputs x = α into the PR-box, and records her output bit a 3. Alice sends Bob the message m = r ⊕ a 4. Bob selects a random bit s and inputs y = s into his PR-box. 5. Bob records his output bit-string b, and computes Apparently the correctness, privacy and secure binding of this protocol COMMIT ′ should be just as good as for the protocol COMMIT (i.e. secure bit commitment with N ǫ = 1), and should therefore yield arbitrarily secure bit-commitment if enough PR-boxes are used. However, this strongly suggests that a similar protocol based on quantum entanglement instead of the PR-boxes should work equally well, something proven impossible! In fact, the PR-box protocol does not provide secure bit commitment, as we will now show. As is the case when proving the impossibility of quantum bit commitment, it is the secure binding that fails. Indeed, unlike the OT case, Bob has no way of telling if Alice has applied her inputs during the COMMIT ′ protocol, as the PR-box will give him an output as soon as he applies his input, regardless of Alice's actions. Interestingly, this allows Alice to successfully cheat by adopting the following strategy: During the COMMIT ′ protocol, Alice does not select her committed bit α or apply any input to her PR-box, but instead sends a random bit m to Bob. Then, during the REVEAL protocol, Alice chooses which α she wishes to reveal, enters x = α into her PR-box, and transmits r = a ⊕ m (as well as her selected α) to Bob. When he checks whether he should accept α, Bob will find that and will therefore accept Alice's revealed α with certainty, despite the fact that it was selected after the COMMIT ′ protocol had finished. This violates the secure binding condition, which requires that only one value of α will be accepted with probability greater than 1/2 during the REVEAL protocol (for N ǫ = 1). Even with multiple PR-boxes in parallel, Alice can use the same trick independently on each to cheat with certainty. COMMIT ′ and REVEAL do not therefore form a secure bit-commitment protocol.
Let us stress that all PR-boxes must yield an output for Bob even if Alice has not yet entered her inputs, as this is essential to prevent signalling from Alice to Bob. This is in full analogy with the case of quantum entanglement. For instance, if Alice and Bob share a singlet, then Bob can measure his half independently of Alice, and viceversa. There is thus no chance of Bob detecting Alice cheating during the COMMIT ′ protocol.
Consequently, despite the fact that it is possible to simulate the outputs of an OT-box using a PR-box and 1 bit of communication, this result shows that the simulation is not universally composable. In particular, the simulation cannot be used to implement a secure bitcommitment protocol in the same way as the original OT-box.
The crucial difference between the OT-box and the PR-box lies in the time ordering of the inputs. Bob can only obtain an output from the OT-box after Alice has entered her inputs, and can therefore check to see whether Alice has entered her inputs during the COM-MIT protocol. In the case of the PR-box, the output is independent of the time-ordering of the two measurements, and no such check exists. [16] In conclusion, in [9] Wolf and Wullschleger showed that there is a deep connection between a fundamental primitive of non-local correlations, the PR-box, and a fundamental cryptographic primitive, oblivious transfer. In particular they show that a PR-box can be used to simulate the correlations produced by an OT-box and viceversa.
In this letter, we have shown that such simulation is not the whole story. In addition to recreating the correct probability distributions, it is important to incorporate any restrictions on the timings of the box inputs. Such restrictions play a crucial role in any attempt to implement secure bit-commitment schemes using the two primitives.
We have shown that the protocols COMMIT and RE-VEAL form a secure bit-commitment scheme using OTboxes. However, the analogous protocols COMMIT ′ and REVEAL, in which the OT-boxes are simulated using PR-boxes, do not constitute a secure bit-commitment scheme as they are not binding.
The main issue is that, due to the non-signalling character of the PR box, Alice can postpone her measurements, and actually perform them during the REVEAL protocol, rather than the COMMIT ′ protocol. We emphasize that it is this liberty (which is in common with quantum mechanics) that is the main element in the nobit-commitment property of nature, rather than any of the particular characteristics of quantum entanglement.
The above results were obtained for a particular bitcommitment scheme based on the OT-box protocol. However, we conjecture that no-bit-commitment is a general feature of nature in the presence of non-signalling non-local correlations, since any such correlations allow Alice the liberty of postponing her measurements. Note that in order for this conjecture to be true, we need to allow for the existence of all possible non-local correlationboxes (not only PR boxes) [7], and the corresponding dynamics of such boxes (which generalize quantum evolutions) [14]. Any limitations could allow bit-commitment. This is similar to the quantum mechanical case: If Alice can only use limited entanglement, it is easy to construct a reliable protocol based on states for which cheating would require more entanglement than Alice can produce.
To conclude, no signalling -no obligation to perform the measurement until the end-no commitment! Note added: After the completion of this work, we became aware of a work by Buhrman et al. [15] that addresses the same problem from a different perspective.