[PPT] - Header Space Analysis Hauptseminar: Innovative Internettechnologien PowerPoint Presentation

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Lehrstuhl Netzarchitekturen und Netzdienste

Institut für Informatik Technische Universität München

Formal Methods: Header Space Analysis

Hauptseminar: Innovative Internettechnologien und Mobilkommunikation Benedikt Jaeger 23.5.2014

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Content

 Header in Messages

Composition of Messages
Different header

 Header Space Analysis

Headerspace, Networkspace
Transfer functions
Reachability
Example

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Composition of Messages

 Header :

information about the packet

 Payload :

actual Data

Header Payload

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Header

 IPv4-Header  IPv6-Header  MAC-Header  ARP Request / Reply  ICMP Echo Request / Reply, TTL Exceeded  TCP- / UDP-Header

(Headercontent is from the script of „Grundlagen: Rechnernetze und Verteilte Systeme“ in the SS 2013, TUM)

Version IHL TOS Total Length Identification Flags Fragment Offset TTL Protocol Header Checksum Source Address Destination Address Options / Padding (optional)

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Header

 IPv4-Header  IPv6-Header  MAC-Header  ARP Request / Reply  ICMP Echo Request / Reply, TTL Exceeded  TCP- / UDP-Header

(Headercontent is from the script of „Grundlagen: Rechnernetze und Verteilte Systeme“ in the SS 2013, TUM)

Version Traffic Class Flow Label Payload Length Next Header Hop Limit Source Address Destination Address

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Header

 IPv4-Header  IPv6-Header  MAC-Header  ARP Request / Reply  ICMP Echo Request / Reply, TTL Exceeded  TCP- / UDP-Header

(Headercontent is from the script of „Grundlagen: Rechnernetze und Verteilte Systeme“ in the SS 2013, TUM)

Destination Hardware Address Destination Hardware Address Source Hardware Address Source Hardware Address Type

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Header

 IPv4-Header  IPv6-Header  MAC-Header  ARP Request / Reply  ICMP Echo Request / Reply, TTL Exceeded  TCP- / UDP-Header

(Headercontent is from the script of „Grundlagen: Rechnernetze und Verteilte Systeme“ in the SS 2013, TUM)

Hardware Type Protocol Type Hw Addr. Length

P. Addr. Length

Operation Sender Hardware Address Sender Hardware Address Sender Protocol Address Sender Protocol Address Target Hardware Address Target Hardware Address Target Protocol Address

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Header

 IPv4-Header  IPv6-Header  MAC-Header  ARP Request / Reply  ICMP Echo Request / Reply, TTL Exceeded  TCP- / UDP-Header

(Headercontent is from the script of „Grundlagen: Rechnernetze und Verteilte Systeme“ in the SS 2013, TUM)

Type Code Checksum Identifier Sequence Number Data (optional)

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Header

 IPv4-Header  IPv6-Header  MAC-Header  ARP Request / Reply  ICMP Echo Request / Reply, TTL Exceeded  TCP- / UDP-Header

(Headercontent is from the script of „Grundlagen: Rechnernetze und Verteilte Systeme“ in the SS 2013, TUM)

Source Port Destination Port Sequence Number Acknowledgement Number Offset Reserved Window Checksum Urgent Pointer Options Source Port Destination Port Length Checksum

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Header Space Analysis

Represent reality in a model Reality Model Router, switch, NAT, firewall, … Middlebox (or simply switch) Packet Headerbits Position in Network Unique Ports

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Header Space Analysis

 Headerspace

𝐼 ⊆ 0,1 𝑀

 Point in Headerspace ℎ  Networkspace

𝑂 ⊆ 0,1 𝑀 × 0, … , 𝑄

 Point in Networkspace (ℎ, 𝑞)

00110010101010100101001010010010 00110010101010100101001010010010 12

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Header Space Analysis

 Box Transfer Function 𝑈 ℎ, 𝑞 : ℎ, 𝑞 →

ℎ1, 𝑞1 , ℎ2, 𝑞2 , …

𝐂𝐩𝐲 𝑼𝒔𝒃𝒐𝒕𝒈𝒇𝒔 𝑮𝒗𝒐𝒅𝒖𝒋𝒑𝒐 𝑼𝑺 𝒊 𝒒 𝑼(𝒊, 𝒒) 1001xxxx 𝑆1 { ℎ, 𝑆2 } 1011xxxx 𝑆2 { ℎ, 𝑆1 } 1111xxxx 𝑆1 { ℎ, 𝑆2 , ℎ, 𝑆3 } … … …

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Header Space Analysis

 Box Transfer Function 𝑈 ℎ, 𝑞 : ℎ, 𝑞 →

ℎ1, 𝑞1 , ℎ2, 𝑞2 , …

 Transform Subspace 𝑇 to another Subspace 𝑇∗

𝑼𝟐 𝑼𝟑

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Header Space Analysis

 Box Transfer Function 𝑈 ℎ, 𝑞 : ℎ, 𝑞 →

ℎ1, 𝑞1 , ℎ2, 𝑞2 , …

 Reversible: 𝑈−1

𝑼𝟐

−𝟐

𝑼𝟑

−𝟐

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Example: IPv4-Router

 What must an IPv4-Router do?  Rewrite source and destination Mac: 𝑈

𝑛𝑏𝑑( )

 Decrement TTL:

𝑈𝑢𝑢𝑚( )

 Update checksum:

𝑈𝑑ℎ𝑙𝑡𝑣𝑛( )

 Forward packet :

𝑈

𝑔𝑥𝑒

𝑈

𝐽𝑄𝑤4 .

= 𝑈

𝑔𝑥𝑒 𝑈 𝑑ℎ𝑙𝑡𝑣𝑛 𝑈 𝑢𝑢𝑚 𝑈 𝑛𝑏𝑑 .

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Example: IPv4-Router

 Rewrite source and destination Mac: 𝑈

𝑛𝑏𝑑( )

ignored

 Decrement TTL:

𝑈𝑢𝑢𝑚( ) ignored

 Update checksum:

𝑈𝑑ℎ𝑙𝑡𝑣𝑛( ) ignored

 Forward packet:

𝑈

𝑔𝑥𝑒

𝑈

𝐽𝑄𝑤4 .

= 𝑈

𝑔𝑥𝑒 .

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Example: IPv4-Router 𝑈

𝐽𝑄𝑤4 ℎ, 𝑞 =

ℎ, 𝑞1 𝑗𝑔 𝑗𝑞_𝑒𝑡𝑢(ℎ) ∈ 𝑇1 ℎ, 𝑞2 𝑗𝑔 𝑗𝑞_𝑒𝑡𝑢(ℎ) ∈ 𝑇2 ℎ, 𝑞3 𝑗𝑔 𝑗𝑞_𝑒𝑡𝑢(ℎ) ∈ 𝑇3 𝑝𝑢ℎ𝑓𝑠𝑥𝑗𝑡𝑓 𝑺𝑱𝑸𝒘𝟓 𝑻𝟐 𝑻𝟑 𝑻𝟒 𝒒𝟒 𝒒𝟑 𝒒𝟐

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Header Space Analysis

 Box Transfer Function 𝑈 ℎ, 𝑞 : ℎ, 𝑞 →

ℎ1, 𝑞1 , ℎ2, 𝑞2 , …

 Network Transfer Function Ψ( )

Ψ ℎ, 𝑞 = 𝑈

1 ℎ, 𝑞 𝑗𝑔 𝑞 ∈ 𝑡𝑥𝑗𝑢𝑑ℎ1

… … 𝑈

𝑜 ℎ, 𝑞 𝑗𝑔 𝑞 ∈ 𝑡𝑥𝑗𝑢𝑑ℎ𝑜

 Topology Transfer Function Γ

Γ ℎ, 𝑞 = (ℎ, 𝑞∗) 𝑗𝑔 𝑞 𝑑𝑝𝑜𝑜𝑓𝑑𝑢𝑓𝑒 𝑢𝑝 𝑞∗ {} 𝑗𝑔 𝑞 𝑗𝑡 𝑜𝑝𝑢 𝑑𝑝𝑜𝑜𝑓𝑑𝑢𝑓𝑒

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Header Space Analysis

 Multihop Packet Traversal

Φ . = Ψ(Γ(.))

 Position(s) of (h,p) after k steps:

Φ𝑙(ℎ, 𝑞) = Ψ(Γ(… (Ψ Γ ℎ, 𝑞 … = ℎ1, 𝑞1 , …

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Reachability

 “Can Host A talk to Host B?“  Wildcard expression ∈ 0, 1, 𝑦 𝑀

E. g. 1001xxxx like 192.168.x.x

 Send Wildcard header from A and check if it ever reaches B

𝑩 𝑪 …

𝒚𝒚𝒚𝒚𝒚𝒚𝒚𝒚

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Reachability

 Send Wildcard header from A and check if it ever reaches B

𝑏 → 𝑇1 → … → 𝑇𝑜−1 → 𝑇𝑜 → 𝑐 𝑆𝑏→𝑐 = {𝑈

𝑜(Γ(𝑈 𝑜−1 … Γ 𝑈 1 ℎ, 𝑞

… }

𝑏→𝑐 𝑞𝑏𝑢ℎ𝑡

𝑩 𝑪 …

𝒚𝒚𝒚𝒚𝒚𝒚𝒚𝒚 {𝟐𝟏𝟏𝟐𝟏𝒚𝟐𝟏, 𝟏𝟐𝟏𝟐𝟐𝒚𝟐𝟏}

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Reachability

 Use 𝑈−1 and Γ−1 = Γ to get the original header

ℎ𝑏 = 𝑈

1 −1(Γ(… (𝑈 𝑜−1 −1 (Γ 𝑈 𝑜 −1 ℎ, 𝑐 … )

𝑩 𝑪 …

𝒚𝒚𝒚𝒚𝒚𝒚𝒚𝒚 {𝟐𝟏𝟏𝟐𝟏𝒚𝟐𝟏, 𝟏𝟐𝟏𝟐𝟐𝒚𝟐𝟏} {𝟐𝟏𝟏𝟐𝟏𝒚𝟐𝟏, 𝟐𝟏𝟏𝟐𝟐𝒚𝟐𝟏}

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Complexity

 Worst Case:  Single box:

𝑃 𝑆1𝑆2

𝑆1: Number of wildcard expressions
𝑆2: Number of rules of Transfer function

 Overall network:

𝑃 𝑆1𝑆2 … 𝑆𝑜 = 𝑃 𝑆𝑒

𝑒 ∶ Diameter of Network
𝑆 ∶ Maximum number of forwarding rules

𝑆𝑏→𝑐 = {𝑈

𝑜(Γ(𝑈 𝑜−1 … Γ 𝑈 1 ℎ, 𝑞

… }

𝑏→𝑐 𝑞𝑏𝑢ℎ𝑡

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Complexity

 But: Linear Fragmentation Assumption:

Not 𝑆1𝑆2 only 𝑑𝑆1 and 𝑑 ≪ 𝑆2

𝑒 ∗ 𝑃 𝑆2 = 𝑃(𝑒𝑆2)

 𝑒 ∶ Diameter of Network  𝑆 ∶ Maximum number of forwarding rules

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Reachability Example

 Which header can reach 𝑐 from 𝑏?

𝒃 𝑩 𝑪 𝑫 𝑬 𝒄

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Reachability Example

 2 different paths

𝒃 𝑩 𝑪 𝑫 𝑬 𝒄

𝑩𝟏 𝑩𝟐 𝑩𝟑 𝑪𝟏 𝑪𝟐 𝑫𝟏 𝑫𝟐 𝑬𝟏 𝑬𝟐 𝑬𝟑

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Reachability Example 𝒃 𝑩 𝑪 𝑬 𝒄

𝑩𝟏 𝑩𝟐 𝑪𝟏 𝑪𝟐 𝑫𝟏 𝑫𝟐 𝑬𝟏 𝑬𝟐 𝑼𝑩 𝒊, 𝒒 = 𝒊, 𝑩𝟏 𝒋𝒈 𝒊 = 𝟐𝟏𝟏𝟐𝒚𝒚𝒚𝒚 𝒊, 𝑩𝟐 𝒋𝒈 𝒊 = 𝟐𝟏𝟐𝟏𝒚𝒚𝒚𝒚 𝒊, 𝑩𝟑 𝒋𝒈 𝒊 = 𝟐𝟐𝟏𝟏𝒚𝒚𝒚𝒚

𝒚𝒚𝒚𝒚 𝒚𝒚𝒚𝒚 𝟐𝟏𝟐𝟏 𝒚𝒚𝒚𝒚

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Reachability Example 𝒃 𝑩 𝑪 𝑬 𝒄

𝑩𝟏 𝑩𝟐 𝑪𝟏 𝑪𝟐 𝑫𝟏 𝑫𝟐 𝑬𝟏 𝑬𝟐 𝑼𝑪 𝒊, 𝒒 = 𝒊, 𝑪𝟐 𝒋𝒈 𝒊 = 𝟐𝟏𝒚𝒚𝒚𝒚𝟏𝟐 𝒊, 𝑪𝟏 𝒋𝒈 𝒊 = 𝟐𝟏𝒚𝒚𝒚𝒚𝟏𝟏

𝒚𝒚𝒚𝒚 𝒚𝒚𝒚𝒚 𝟐𝟏𝟐𝟏 𝒚𝒚𝒚𝒚 𝟐𝟏𝟐𝟏 𝒚𝒚𝟏𝟐

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Reachability Example 𝒃 𝑩 𝑪 𝑬 𝒄

𝑩𝟏 𝑩𝟐 𝑪𝟏 𝑪𝟐 𝑫𝟏 𝑫𝟐 𝑬𝟏 𝑬𝟐 𝑼𝑬 𝒊, 𝒒 = 𝒊, 𝑬𝟏 𝒋𝒈 𝒊 = 𝒚𝒚𝒚𝒚𝟐𝒚𝒚𝒚 𝒊, 𝑬𝟐 𝒋𝒈 𝒊 = 𝒚𝒚𝒚𝒚𝟏𝒚𝒚𝒚

𝒚𝒚𝒚𝒚 𝒚𝒚𝒚𝒚 𝟐𝟏𝟐𝟏 𝒚𝒚𝒚𝒚 𝟐𝟏𝟐𝟏 𝒚𝒚𝟏𝟐 𝟐𝟏𝟐𝟏 𝟏𝒚𝟏𝟐

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𝒃 𝑩 𝑫 𝑬 𝒄

𝑩𝟏 𝑩𝟑 𝑫𝟏 𝑫𝟐 𝑬𝟐 𝑬𝟑

Reachability Example

𝑼𝑩 𝒊, 𝒒 = 𝒊, 𝑩𝟏 𝒋𝒈 𝒊 = 𝟐𝟏𝟏𝟐𝒚𝒚𝒚𝒚 𝒊, 𝑩𝟐 𝒋𝒈 𝒊 = 𝟐𝟏𝟐𝟏𝒚𝒚𝒚𝒚 𝒊, 𝑩𝟑 𝒋𝒈 𝒊 = 𝟐𝟐𝟏𝟏𝒚𝒚𝒚𝒚

𝒚𝒚𝒚𝒚 𝒚𝒚𝒚𝒚 𝟐𝟐𝟏𝟏 𝒚𝒚𝒚𝒚

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𝒃 𝑩 𝑫 𝑬 𝒄

𝑩𝟏 𝑩𝟑 𝑫𝟏 𝑫𝟐 𝑬𝟐 𝑬𝟑

Reachability Example

𝑼𝑫 𝒊, 𝒒 = 𝒊&𝟏𝟏𝟏𝟏𝟐𝟐𝟐𝟐 𝟏𝟏𝟐𝟐𝟏𝟏𝟏𝟏, 𝑫𝟐 𝒋𝒈 𝒊 = 𝟐𝟐𝟏𝟏𝒚𝒚𝒚𝒚 𝒊&𝟏𝟏𝟏𝟏𝟐𝟐𝟐𝟐 𝟐𝟐𝟏𝟏𝟏𝟏𝟏𝟏, 𝑫𝟏 𝒋𝒈 𝒊 = 𝟏𝟏𝟐𝟐𝒚𝒚𝒚𝒚

𝒚𝒚𝒚𝒚 𝒚𝒚𝒚𝒚 𝟐𝟐𝟏𝟏 𝒚𝒚𝒚𝒚 𝟏𝟏𝟐𝟐 𝒚𝒚𝒚𝒚

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𝒃 𝑩 𝑫 𝑬 𝒄

𝑩𝟏 𝑩𝟑 𝑫𝟏 𝑫𝟐 𝑬𝟐 𝑬𝟑

Reachability Example 𝒚𝒚𝒚𝒚 𝒚𝒚𝒚𝒚 𝟐𝟐𝟏𝟏 𝒚𝒚𝒚𝒚 𝟏𝟏𝟐𝟐 𝒚𝒚𝒚𝒚 𝟏𝟏𝟐𝟐 𝟏𝒚𝒚𝒚

𝑼𝑬 𝒊, 𝒒 = 𝒊, 𝑬𝟏 𝒋𝒈 𝒊 = 𝒚𝒚𝒚𝒚𝟐𝒚𝒚𝒚 𝒊, 𝑬𝟐 𝒋𝒈 𝒊 = 𝒚𝒚𝒚𝒚𝟏𝒚𝒚𝒚

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Reachability Example 𝒃 𝑩 𝑪 𝑫 𝑬 𝒄

𝑩𝟏 𝑩𝟐 𝑩𝟑 𝑪𝟏 𝑪𝟐 𝑫𝟏 𝑫𝟐 𝑬𝟏 𝑬𝟐 𝑬𝟑

𝟐𝟏𝟐𝟏 𝟏𝒚𝟏𝟐 𝟏𝟏𝟐𝟐 𝟏𝒚𝒚𝒚 𝟐𝟏𝟐𝟏 𝟏𝒚𝟏𝟐 𝟐𝟐𝟏𝟏 𝟏𝒚𝒚𝒚

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Sources

 Kazemian P., Varghese G., McKeow N.: Header space analysis: static

checking for networks. In: Proceedings of the 9th USENIX Conference

n Networked Systems Design and Implementation, USENIX

Association, 2012

 Shuyuan Zhang, Sharad Malik, Rick McGeer: Verification of Compuer

Switching Networks: An Overview. In Automated Technology for Verification and Analysis, Springer-Verlag, 2012

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