1. Secure Error-Correcting (SEC) Network Codes Raymond W. Yeung Institute of Network Coding & Department of Information Engineering The Chinese University of Hong Kong Joint work with Chi-kin Ngai, Kenneth Shum, and Shenghao Yang

2. Outline Network error-correcting codes Secure network codes Secure error-correcting (SEC) network codes Concluding Remarks

3. Network Error-Correcting Codes

4. Point-to-Point Error Correction in a Network Classical error-correcting codes are devised for point-to- point communications. Such codes are applied to networks on a link-by-link basis.

5. ChannelDecoderChannelDecoder NetworkEncoder ChannelEncoder

6. A Motivation for Network Error Correction Observation Only the receiving nodes have to know the message transmitted; the immediate nodes don’t. In general, channel coding and network coding do not need to be separated  Network Error Correction Network error correction generalizes classical point-to-point error correction.

7. NetworkCodec

8. Singleton Bound for Network Error Correction (Cai and Y 02, 06) A d-error-correcting network code can correct any d errors in the networks at all the sink nodes. m = min T maxflow(T) r = rate of a network error-correcting code Singleton bound: r ≤ m – 2d The tightness of the Network Singleton bound is achievable by linear network error-correcting codes! This implies that for large base fields, only linear transformations need to be performed at the intermediate nodes! No decoding needed.

9. Sphere Packing    d min

10. Malicious Injection of Errors Malicious nodes in the network may inject errors deliberately to disturb data transmission. Classical error correction does not help because redundancy is injected only in time. Linear network error-correcting code is a natural solution because redundancy is injected in both time and space.

11. Secure Network Codes (Cai and Y, 2002, 2008)

12. Problem Formulation A message is multicast on a network. Some sets of channels can be wiretapped. A wiretap set is a subset of the edge set E. Let A be the collection of all possible wiretap sets. Each wiretapper chooses to access one wiretap set in A. No wiretapper may access more than one wiretap set. The network code needs to be information-theoretically secure. The model is a network generalization of secret sharing (Blakley, Shamir, 78) and wiretap channel II (Ozarow and Wyner 84).

13. A Example of a Secure Network Code

14. s-ws+w s-w s+w w ww  Any single channel can be wiretapped  s is the secure message  w is the randomness

15. A Construction of Secure Network Codes The wiretapper may access any k channels. m = min T maxflow(T) Our construction builds on any linear network code that can multicast m symbols. s: message; w: key Let |s| = m - k and |w| = k. Our scheme – Maximizes the amount of information that can be multicast securely. – Uses the minimum possible amount of randomness.

16. NETWORK linear network code, rate = m Q -1 rate = m - k s s w w m-kk

17. Secure Error-Correcting (SEC) Network Codes

18. A Construction of SEC Network Codes The code can correct any d errors. The wiretapper may access any k channels. m = min T maxflow(T) Our construction concatenates – a secure network code (SNC) – a network error-correcting code (NEC) Rate = m - 2d - k

19. NETWORK NEC s s w w m-2d-kk SEC m m-2d m-2d-k

20. Proof of Error Correction and Security Error correction of the SEC code inherits from the NEC inner code. Proof of security when there is no error – algebraic Proof of security in the presence of errors – need to set up a model for noise and eavesdropper random errors and key are statistically independent eavesdropper can inject errors causally – information theoretic

21. + eavesdropper random error injection error

22. Parallel noiseless transmission Parallel noiseless transmission Network coding Network coding Secret sharing Secret sharing Point-to-point error correction Point-to-point error correction Network error correction Network error correction Secure network coding Secure network coding SEC coding