**Word & Explanation**

- slicing: partition networks
- secure slice: no packet from one slice can be controlled by or read by the other slice
- take the notion of isolation further, and enable the static analysis of networks sliced in more general ways -> each slice has own control plane
- Ψ: network transfer functions
- Γ: topology transfer function
- H: Header space
- N: Network space
- header: both packet headers (points in H) and wildcard expressions (hyper-cubes in H)

Even when individual protocols function correctly, failures can arise from the complex interactions of their aggregate, requiring network administrators to be masters of detail.

Header Space Analysis (HSA) statically check network specifications and configurations

- entire packet header as a concatenation of bits without any associated meaning -> {0, 1}^L space
- networking boxes transform packets from one point in the space to another point or set of points (multicast)

Goal:

- help system administrators statically analyze production networks
- make it easier for system administrators to guarantee isolation between sets of hosts, users or traffic

Key:

- generalization of the geometric approach to packet classification

- jettison the notion of pre-specified fields in favor of a header space of L bits where each packet is represented by a point in {0, 1}^L space, where L is the header length
- model all router and middlebox processing as box transfer functions transforming subspaces of the L-dimensional space to other subspaces
- model a network of boxes using a network transfer function, Ψ and a topology transfer function, Γ

Model:

Φ(.) = Ψ(Γ(.)) -> if a packet with header h enters a network on port p, the header after k hops will be Ψ(Γ(…(Ψ(Γ(h, p)…), or simply Φ^k (h, p)

Algebra:

- set operations
- intersection
- For two headers to have a non-empty intersection, both headers must have the same bit value at every position that is not a wildcard.
- If two headers differ in bit bi, then the two headers will be in different hyper-planes defined by bi = 0 and bi = 1.

z means the bitwise intersection is empty, if any bit returns z, the intersection of all bits is empty

- union
- cannot be simplified

- complementation
- difference(minus)
- A − B = A ∩ B′
- The difference operation can be used to check if one header is a subset of another: A ⊆ B ⇐⇒ A − B = φ

- intersection
- transfer functions
- Domain
- the set of all possible (header, port) pairs that the transfer function accepts

- Range
- the set of all possible (header, port) pairs that the transfer function can output after applying all possible inputs on every port

- Range Inverse

- Domain

Reachability Analysis:

Complexity:

- Linear Fragmentation assumption: each of the input wildcard expressions will match only a few rules in the transfer function and generate at most cR(c << R)
- running time is O(dR^2) where d is the network diameter and R is the maximum number of forwarding rules in a router

Loop Detection (breadth first search)

- generic loop
- inject an
**all-x**test packet header from**each port**in the network and track the packet until:- It leaves the network
- It returns to a port already visited (Pret) -> report twice
- It returns to the port it was injected from (Pinj) ->
**Loop**

- Complexity
- same algorithmic structure as reachability test -> O(dPR^2), P is number of ports that need to inject

- inject an
- infinite loop
- Let hret denote the part of header space that returns to A1, then horig = Φ^−1(Φ^−1(Φ^−1(Φ^−1(hret, A1)))) is the original header space that produces hret
- hret and horig relate in:
- hret ∩ horig = φ -> finite
- hret ⊆ horig -> infinite
- Neither -> redefine hret := hret ∩ horig and calculate the new horig
An example of a generic, but finite loop, is an IP packet that loops until the TTL decrements to zero

- generic loop
Slice Isolation

- create new slices isolated
- require identification of a region of network space that does not overlap with region belonging to existing slices -> no header space in common on any common port

- detect when slices are leaking traffic
- Leakage occurs when a packet in slice a at any switch-port can be rewritten to fall into the network space of another slice
- apply the network transfer function of slice a to its header space reservation -> generate all possible packet headers (output header set)
- If the output header space of slice a at any switch port, overlaps with any other slice, then there is the potential for leaks

- Complexity
- O(W^2 N ), where W is the maximum number of wildcard expressions used to describe any slice’s reservation and N is the total number of slices in the network

- create new slices isolated