WO2008140290A2

WO2008140290A2 - Qudits in two way deterministic quantum key distribution

Info

Publication number: WO2008140290A2
Application number: PCT/MY2008/000038
Authority: WO
Inventors: Mohamed Ridza Wahidin; Jesni Bin Shamsul Shaari; Marco Lucamarni
Original assignee: Mimos, Berhad; International Islamic University Malaysia
Priority date: 2007-05-11
Filing date: 2008-05-09
Publication date: 2008-11-20
Also published as: WO2008140290A3; MY153146A

Abstract

The present invention relates to a qudits in two way deterministic quantum key distribution wherein it involved a two way transmission where qubits are sent to and from between communicating parties while information is encoded by unitary transformers on the qubits. A protocol sees U1 sending to U2 a state that U1 prepared and wherein U2 would then encode the said received protocol with unitary transformation. U2 would then thereafter send the encoded state back to U1 who would then measure or decode the sent state deterministically. In order to ensure the security of the protocol, a control mode used and is randomly done with a certain probability wherein instead of encoding. U2 would do a projective measurements on the qudit received whilst U1 measures the qudit sent by U2.

Description

OUDΓΓS IN TWO WAY DETERMINISTIC QUANTUM KEY DISTRIBUTION

FIEI J> OF THF, INVENTION

The present invention relates to a qudits in two way deterministic quantum key distribution.

BACKGROUND OF THE INVENTION

In the prior art the use of qubit was basically the leader in quantum key distribution (QKD) in the BB84 protocol. A higher dimensional system was used wherein the protocol was extendable to the use of qutrits, exhibiting a higher level of security. The two way deterministic protocols emerged with development in various protocols. This involved a two way transmission where qubits are sent to and from between communicating parties while information is encoded by unitary transformers on the qubits.

Generalization to exhaustive use of all the MUB in both the qubit and qutrit case involves nontrivial extensions due to he nonexistence of the unitary universal NOT gate as well as a non go theorem.

In the present invention, the inventors have introduced the use of d (d being a prime number) dimensional quantum system (qudit) in a two way deterministic scheme.

Further to this, a generalized protocol is considered using an encoding recipe for unitary efficiency. The security was compared and efficiencies of the protocols both in the context of theoretical as well as practical efficiencies were recorded.

SUMMARY OF THE INVENTION

The present invention relates to a qudits in two way deterministic quantum key distribution wherein it involved a two way transmission where qubits are sent to and from between communicating parties while information is encoded by unitary transformers on the qubits. A protocol sees Ul sending to U2 a state that Ul prepared and wherein U2 would then encode the said received protocol with unitary transformation. U2 would then thereafter send the encoded state back to Ul who would then measure or decode the sent state deterministically and wherein in order to ensure the security of the protocol, a control mode used and is randomly done with a certain probability wherein instead of encoding, U2 would do a projective measurements on the qudit received whilst Ul measures the qudit sent by U2. In a noisy channel, the measure of security is essentially translated into the robustness of the protocol.

A secure key can be distilled with the use of standard error correction and privacy amplification procedures only when the information shared between U2 and Ul is greater than the lesser of the information shared between U2 and U3. U3 substitute the noisy channel between Ul and U2 and attack only certain fractions of the qudits to glean some information whilst inducing a lesser amount of error in the control mode. In a full fraction attack by U3, U3 would destroy the information in the qudit when U3 measures it in a different basis that that which Ul has prepared.

BRIEF DESCRIPTION OF THE FIGURE

Figure 1 shows a graph plotted to show the probability of detection for the generalized protocol (square points) and non-generalized protocol (triangular points) against the number of dimensions.

DETAILED DESCRPTION OF Tlfe PRESENT INVENTION

In the present invention two users would be used as to better describe the protocol. Wherein the said two users would be referred as User 1 or Ul and User 2 or U2. Subsequent users would be referred as U3, U4 and etc.

In the present invention it would be notice that a protocol sees Ul sending to U2 a state that Ul prepared (also referred as forward path). U2 would then encode the said received protocol with unitary transformation. U2 would then thereafter send the encoded state back to Ul (also referred as backward path) who would then measure or decode the sent state deterministically.

According to the present invention, in order to ensure the security of the protocol, a control mode is necessary. This is randomly done with a certain probability where instead of encoding U2 would do a projective measurements on the qudit in the forward path whilst Ul measures the qudit in the backward path. In the instances wherein U2 choice of basis coincides with Ul's , the measurements should correlate. If this is not the case, an eavesdropper would be detected.

In the case the present invention only use d basis, a U3 may correctly guess only Md times and avoids detection. However, in the case that U3 makes a mistake, she gets to evade detection in both the forward and backward path with probability 1/d². on average U3 would be detected with probability Pr(Eve) =[(d-l)/d](d²-l)/(d²).

However, in a realistic setup with a noisy channel, the measure of security is essentially translated into the robustness of the protocol; i.e., how much of error could be tolerated before communication becomes insecure. A secure key can be distilled with the use of standard error correction and privacy amplification procedures only when the information shared between U2 and Ul , I_AB. is greater than the lesser of the information shared between U2 and U3, IAE , or U 1 and U3, I_BE, (IAB > min(lAB, IBE)).

In such instance, it could be considered that U3 substitute the noisy channel between them with a perfect one and attack only certain fractions of the qudits to glean some information whilst inducing a lesser amount of error in the control mode. A full fraction attack would induce an error amounting to e = ((d-l)/d)² . In other words U3 needs to attack and achieve IAB ≤ min (IAE, IBE). For the purpose of describing the present invention, it is considered that the information shared between the parties, wherein for U2 and U3, the information shared is given as:

IAB = 1 - (2d - 1) / (d²) log* (2d - 1) / d²) - ((d - I)/ (d))² log,, (d - 1) / d² (1 Wherein, I_AE = 1 for a full fraction attack by U3, the case is quite different for

I_BE as U3 would destroy the information in the qudit when U3 measures it in a different basis that that which Ul has prepared. U3's chances of having equal results with Ul about U2's encoding is only (2d-l)/d² and U3's uncertainty of Ul's result becomes as shown below:-

H (B) = (2d -l)/(d²) log_d (2d - l)/(d²) + (d -I)² /(d²) log* (d - 1) / (d²) (2)

It is straightforward to see that with IB_E = 1 - H (B), only when U3 attacks all the qudits would be I_BE = U_B- In considering the case when we generalize to d+ 1 basis we refer to for a reliable recipe for blind encoding to achieve a protocol of unitary efficiency. Ul would send to U3 d qudits of differing bases and U2 would encode with any of the d* possible transformations.

Wherein a, b, ..... k are different values to encode onto the relevant qudits, Uj is the unitary transformation to shift a state in any basis except yth and φ¹ belongs to d different MUB. In an IR attack, U3 has the chances of d / (d+1) making a mistake and the probability to avoid detection in both the forward and backward path is 1/d². The probability on the whole to detect U3 would be Pr (Evβgcnerauzed) = (d-l)/d.

In term of efficiency of the protocol, it could be considered that only the transmission mode as opposed to the control mode. In the theoretical framework where a channel is considered to be noiseless and lossless, this is not unjustifiable as one may minimize greatly the probability of the control mod, leading to a negligible number of qudits to be discarded. With respect to the practical efficiency, we consider the transmittance of the channel to be T. With the distance between U2 and Ul being L, every qudit would have to travel a distance of 2L. For the generalized protocol, considering every round of communication, the practical efficiency amounts to the overall probability that all the qudits arrive at the end of the round trip: ε = T²"¹. As in a realistic setup, wherein

r <l,limf' =0

The practical efficiency is literally zero. Given the fact that probability of detecting U3 for both the protocol asymptotically approaches 1 as d increases.

Hm Pτ(Eve) = Hm ?r(Eve _gmmllied ) = 1

It is therefore concluded that the generalized two way protocols become less efficient with increasing dimensions in realistic cases where the transmittance is less than perfect.

Claims

OCLAIMS

1. A qudits in two way deterministic quantum key distribution wherein it involved a two way transmission where qubits are sent to and from between communicating parties while information is encoded by unitary transformers on the qubits characterized in that wherein a protocol sees Ul sending to U2 a state that Ul prepared and wherein U2 would then encode the said received protocol with unitary transformation and wherein U2 would then thereafter send the encoded state back to Ul who would then measure or decode the sent state deteπninistically and wherein in order to ensure the security of the protocol, a control mode used and is randomly done with a certain probability wherein instead of encoding, U2 would do a projective measurements on the qudit received whilst Ul measures the qudit sent by U2.

2. A qudits in two way deterministic quantum key distribution as claimed in Claim

1 wherein with a noisy channel, the measure of security is essentially translated into the robustness of the protocol.

3. A qudits in two way deterministic quantum key distribution as claimed in Claim 1 wherein a secure key can be distilled with the use of standard error correction and privacy amplification procedures only when the information shared between

U2 and Ul is greater than the lesser of the information shared between U2 and

U3.

4. A qudits in two way deterministic quantum key distribution as claimed in Claim

3 wherein U3 substitute the noisy channel between Ul and U2 and attack only certain fractions of the qudits to glean some information whilst inducing a lesser amount of error in the control mode.

5. A qudits in two way deterministic quantum key distribution as claimed in Claim

1 wherein in a full fraction attack by U3, U3 would destroy the information in the qudit when U3 measures it in a different basis that that which Ul has prepared.