# Research

### Network Coding

Network coding is a paradigm where the nodes, instead of simply forwarding the incoming flows through the outgoing paths, distribute a function of the inputs. Network codes allow one to improve on several aspects of networks, in particular, their security.

In [9] we introduced weakly trusted repeaters (WTR) for QKD networks. The objective is to overcome the distance limitation of QKD. WTR are formalized using a network coding approach and we constructed information theoretically secure scenarios, assuming specific trust structures, in particular, that there is at least one nonmalicious path among the several disjoint paths used.

Securing networks with WTR reduces the strong full trust dependence that is assumed for traditional trusted repeaters that, in turn, improves the security of applications and services relying on them.

[9]. **D Elkouss**, J Martinez-Mateo, A Ciurana, V Martin. JOCN 5 (4), 316-328 (2013)

**Communications between emitters and receivers are represented using colored circles (from Tx1) and triangles (from Tx2). The color indicates the wavelength. There are two different paths joining Tx1 with Tx2, e.g. via Rx1 or Rx2. **

### Error Correcting Codes

LDPC codes are a capacity approaching family of error correcting codes. We became interested in LDPC codes for information reconciliation. In order to reduce the impact on the secret key we designed ensembles of codes for the binary symmetric channel (BSC) with thresholds closed to the capacity of the BSC [2]. For these ensembles -that have both, check and node, irregular distributions- we proposed a code construction algorithm in [8].

Puncturing is a well-known coding technique widely used for constructing rate-compatible codes. With the motivation of reducing the leakage for the rate-adaptive protocol in [4] we designed a puncturing algorithm (blue [7]). For moderate amounts of puncturing (<10%) the simulation results show that its performance in terms of the FER is better than previous proposals (solid black [5], red [6]).

[5]. J Ha, J Kim, D Klinc, S McLaughlin. IEEE TIT 52 (2) 728-738 (2006)

[6]. B Vellambi, F Fekri. IEEE TC 57 (2) 297-301 (2009)

[7]. **D Elkouss**, J Martinez-Mateo, V Martin. IEEE WCL 1 (6) 585-588 (2013)

[8]. J Martinez-Mateo, **D Elkouss**, V Martin. IEEE CL 14 (12) 1155-1157 (2010)

### Information Reconciliation

Quantum key distribution protocols provide the legitimate parties with correlated but different keys. In order to obtain a common key they are allowed to discuss over a public channel. This process leaks information which in turn reduces the distillable secret key.

Cascade was one of the first protocols proposed for reconciliation [1], it is simple and rate-adaptive, but also highly interactive and not very efficient (solid black). In [2] we applied good error correcting codes and reduced the leakage for fixed error rates (red). The solution was not very practical and in [3] we proposed a rate adaptive solution (blue, green). In [4] we studied of the achievable secret key with this method and the effect of finite sizes.

[1]. G Brassard, L Salvail. pp. 410-423 Eurocrypt (1993)

[2]. **D Elkouss**, A Leverrier, R Alleaume, J Boutros. pp. 1879-1883 ISIT (2009)

[3]. **D Elkouss**, J Martinez-Mateo, V Martin. QIC 11 (3) 226-238 (2010)

[4]. **D Elkouss**, J Martinez-Mateo, V Martin. PRA 87 (4), 042334 (2013)

### Quantum Capacities

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[10]. T Cubitt, **D Elkouss**, W Matthews, M Ozols, D Pérez-García, S Strelchuk. (2015)