**3D Visual Method of Variant Logic Construction for Random Sequence**

Architecture

Sample Results on 3D Maps

Analysis of Results

Conclusions

**OUTLINE**

Keywords: pseudo-random sequence, CA, stream cipher, RC4 keystream, 3D maps

**Random sequences play an important role in cryptography. Both CA (Cellular Automata) and RC4 contain pseudo-random number generators and may have intrinsic properties respectively.**

**Summary of contents**

**Name: Wang Huan**

Background

**Outline**

Variant Logic Construction

Measurement Model

Probability Calculation Model

Cellular Automata is a great discovery in the 20th century, it forms a time series according to a given function in an iterations process by introducing logic function and related calculation methods in natural pattern.Because of the implicated expression of the logic function, the spatial characteristic cannot be directly observed from the function formula

CA

Background

Pseudorandom Number Generators:

Logic Function

RC4

Stream ciphers are a very important class of encryption algorithms. [1] A stream cipher is a symmetric cipher which operates with a time-varying transformation on individual plaintext digits. Typically, in a stream cipher, the keystream is the sequence which is combined, digit-by-digit, to the plaintext sequence for obtaining the ciphertext sequence. The keystream is generated by a finite state automaton called the keystream generator。The RC4 is one of the most popular among state-of-the-art software stream ciphers with varied industrial applications. It is used for encrypting the Internet traffic in network protocols such as Secure Sockets Layer (SSL), Transport Layer Security (TLS), Wired Equivalent Privacy (WEP), Wi-Fi Protected Access (WPA) etc.

RC4 Keystream

Measurement Model

Probability Calculation Model

Measurement Model

Probability Model

CM

Computation Model of CA (CMCA)

CMCA module is used to measure the features of a logic function based on CA. Consider a logic function ƒ: Y= ƒ (X) as a function of CA, the output sequence Y can be generated by given initial input sequence X with 2 states. For a N bits initial input sequence, a total of states are generated under the logic function ƒ: . A pair of vectors (X, Y) could be collected for their correspondences on the pair of input-output relationships. There are groups of this corresponding relationship.

Input Group:

X A 0-1 vector with N elements,

n An integer indicating a 0-1 vector with n elements,

ƒ A function with 2 variables

Intermediate Group:

Y A 0-1 vector with N elements,

Output Group:

Exhaustive set of all states of N bit vectors with elements

For an L bits input keystream K, divided into G segments and W = L/G bits of each segment

with G<L. The value of parameter G determines the amount of points and W determines spatial distribution for the output keystream in the phase space.

Input Group:

A 0-1 vector with L elements generated by RC4 keystream generator,

L An integer indicates the number of elements in an input sequence,

G An integer indicates the number of segments divided,

W An integer indicates the number of elements in a segment

Output Group:

G sets of W bits 0-1 vectors

The CMRC4 component uses an input vector as input, under different segment strategies to divide into several segments. The output of this component is G sets of W bits 0-1 vectors.

Computation Model of RC4 Keystream (RC4KCM)

MM

The MM component is composed of three modules: Variant Measure (VM), Probability Measurement (PM) and Selection Mechanism (SM). Three parameters are listed as input signals; four variant measures are outputted from VM module, six probability measurements are created from variant measures by Probability Measurement (PM), under the Selection Mechanism (SM) module, a set of triples interactive projections selected.

Input Group:

V A symbol is selected from four types of transformations { },

N An integer indicates the number of elements in an input vector,

A 0-1 data vector

Intermediate Group:

A set of four variant measures,

A set of four probability vectors

Output Group:

A set of three interactive projections under the SM condition,

A set of three probability vectors

Variant Measure (VM)

Considering the transformation of every bit between input sequence and output sequence ,

there are a total of 4 types of transformations: 0→0, 0→1, 1→0, and 1→1.

Define the variant representation as follows.

,

For any N bit 0-1 vector X, under 2-variable function

ƒ, N bit 0-1 output vector Y, . Let Δ be the variant measure function.

Probability Measurement (PM)

Selection Mechanism Module

（SM）

3D Visualization Model

Visualization Results of Unified Model

Two sets of six 3D maps based on unified model in different condition;

(a1~a3) for the file CA; (b1~b3) for the file RC4

Visualization Results of Non-Unified Model

Two sets of six 3D maps based on non-unified model in different condition;

(a1~a3) for the file CA; (b1~b3) for the file RC4

Visualization Results of CA With Different Length of Initial Sequence

Three sets of nine 3D maps under different condition;

(a1~a2) for the logic function f=15 and non-unified model; (b1~b2) for the logic function f=100 and non-unified

model;(c1~c2)for the logic function f=170 and non-unified model

Visualization Results of RC4 Keystream With Different Segment Strategies

Three sets of nine 3D maps under different condition;

(a1~a3) for the key=90 and unified model; (b1~b3) for the key=90 and non-unified model;(c1~c3)for the key=123 and non-unified model

Visualization Results of Unified Model

Visualization Results of Non-Unified Model

Visualization Results of CA With Different Length of Initial Sequence

Visualization Results of RC4 Keystream With Different Segment Strategies

Spatial property of random sequence can be detected from the distribution of cluster point in the 3D maps discussed in details. Different spatial distributions are illustrated to show various distributions on each phase space for relevant logic function or keystream. Further explorations are required on this scheme.

Thank You!

**it’s very hard to trace the characteristics of keystream generators, random characteristics of keystream can be investigated on spatial characteristics of keystream generator to test pseudo-random sequences.**

**Background**

**Variant Logic Construction**

**Architecture**

**Sample Results on 3D Maps**

**Analysis of Results**

**Conclusions**

The overall architecture of system

Computation Model(CM)

Measure Mechanism (MM)

3DVM 3D Visualization Model

The SM Module is proposed two models: Non-unified Model and Unified Model. Under different constructions, two models are established respectively as follows.