Perfect Secrecy Over Binary Erasure Wiretap Channel of Type II

We introduce a binary erasure wiretap channel of type II in which the number of eavesdropped bits μ becomes available a posteriori. We aim at achieving perfect secrecy over such a channel model. The most appropriate application is a secret key agreement scheme. We present a secret key agreement scheme that adopts the formulation S = HX of Wyner-Ozarows's linear coset coding. The scheme is based on the following simple observation: even if some information on a secret message leaked out, I(S; X<sub>μ</sub>) >; 0, where X<sub>μ</sub> is a binary sequence of length μ, it is still possible to have perfect secrecy I(S<sub>J</sub>; X<sub>μ</sub>) = 0 for some subsequence S<sub>J</sub> of S. Our secret key agreement scheme achieves perfect secrecy by taking only those subsequences S<sub>J</sub> that are independent of the eavesdropped bits X<sub>μ</sub>. Our secret key agreement scheme naturally leads to defining a security measure D<sub>H</sub>(μ) for parity-check matrices such that the eavesdropper gets zero information on S<sub>J</sub> as long as the length of S<sub>J</sub> is less than D<sub>H</sub>(μ). We study basic properties of D<sub>H</sub>(μ) and prove the perfect secrecy of our key agreement scheme. For parity-check matrices of small sizes, we perform an exhaustive search for matrices maximizing D<sub>H</sub>(μ).

[1]  Andrew Thangaraj,et al.  Strong Secrecy on the Binary Erasure Wiretap Channel Using Large-Girth LDPC Codes , 2010, IEEE Transactions on Information Forensics and Security.

[2]  Lawrence H. Ozarow,et al.  Wire-tap channel II , 1984, AT&T Bell Laboratories Technical Journal.

[3]  W. Wootters,et al.  A single quantum cannot be cloned , 1982, Nature.

[4]  Frank R. Kschischang,et al.  Universal weakly secure network coding , 2009, 2009 IEEE Information Theory Workshop on Networking and Information Theory.

[5]  Gilles Brassard,et al.  Quantum cryptography: Public key distribution and coin tossing , 2014, Theor. Comput. Sci..

[6]  Krishna R. Narayanan,et al.  Weakly Secure Network Coding , 2005 .

[7]  Jean Cardinal,et al.  Reconciliation of a quantum-distributed Gaussian key , 2001, IEEE Transactions on Information Theory.

[8]  Whitfield Diffie,et al.  New Directions in Cryptography , 1976, IEEE Trans. Inf. Theory.

[9]  A. D. Wyner,et al.  The wire-tap channel , 1975, The Bell System Technical Journal.

[10]  Alexander Vardy,et al.  Achieving the secrecy capacity of wiretap channels using Polar codes , 2010, ISIT.

[11]  I. G. Núñez,et al.  Generalized Hamming Weights for Linear Codes , 2001 .

[12]  H. Vincent Poor,et al.  Secure Nested Codes for Type II Wiretap Channels , 2007, 2007 IEEE Information Theory Workshop.

[13]  Onur Ozan Koyluoglu,et al.  Polar coding for secure transmission and key agreement , 2010, 21st Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications.

[14]  Byung-Jae Kwak,et al.  LDPC Codes for the Gaussian Wiretap Channel , 2009, IEEE Transactions on Information Forensics and Security.

[15]  A. Robert Calderbank,et al.  Applications of LDPC Codes to the Wiretap Channel , 2004, IEEE Transactions on Information Theory.

[16]  Ueli Maurer,et al.  Information-Theoretic Key Agreement: From Weak to Strong Secrecy for Free , 2000, EUROCRYPT.

[17]  Shlomo Shamai,et al.  Secrecy-achieving polar-coding , 2010, 2010 IEEE Information Theory Workshop.

[18]  Mikael Skoglund,et al.  Nested Polar Codes for Wiretap and Relay Channels , 2010, IEEE Communications Letters.

[19]  U. Maurer,et al.  Secret key agreement by public discussion from common information , 1993, IEEE Trans. Inf. Theory.

[20]  Andrew Thangaraj,et al.  Strong secrecy for erasure wiretap channels , 2010, 2010 IEEE Information Theory Workshop.

[21]  Gilles Brassard,et al.  Secret-Key Reconciliation by Public Discussion , 1994, EUROCRYPT.

[22]  Ekert,et al.  Quantum cryptography based on Bell's theorem. , 1991, Physical review letters.

[23]  Mikael Skoglund,et al.  Performance Analysis and Design of Two Edge-Type LDPC Codes for the BEC Wiretap Channel , 2013, IEEE Transactions on Information Theory.

[24]  Claude E. Shannon,et al.  Communication theory of secrecy systems , 1949, Bell Syst. Tech. J..

[25]  Martin E. Hellman,et al.  A note on Wyner's wiretap channel (Corresp.) , 1977, IEEE Trans. Inf. Theory.