## Date of Submission

2-28-2004

## Date of Award

2-28-2005

## Institute Name (Publisher)

Indian Statistical Institute

## Document Type

Doctoral Thesis

## Degree Name

Doctor of Philosophy

## Subject Name

Cryptology

## Department

Applied Statistics Unit (ASU-Kolkata)

## Supervisor

Sarkar, Palash (ASU-Kolkata; ISI)

## Abstract (Summary of the Work)

In this thesis we study combinatorial aspects of Boolean functions and S-boxes with impor- tant cryptographic properties and construct new functions possesing such properties. These have possible applications in the design of private key (symmetric key) cryptosystems.Symmetric key cryptosystems are broadly divided into two classes.1. Stream Ciphers,2. Block Ciphers.Some recent proposals of stream ciphers are SNOW [37], SCREAM [52], TURING (98], MUGI (117), HBB (102], RABBIT (9), HELIX (38] and some proposals of block ciphers are DES, AES, RC6 [97), MARS (12], SERPENT (6], TWOFISH (104].In stream cipher cryptography a pseudorandom sequence of bits of length cqual to the message length is generated. This sequence is then bitwise XOR-ed (addition modulo 2) with the message sequence and the resulting sequence is transmitted. At the receiving end, deciphering is done by generating the same pseudorandom sequence and again bitwise XOR- ing the cipher bits with the random bits. The seed of the pseudorandom bit generator is obtained from the secret key.Linear Fecdback Shift Registers (LFSRS) are important building blocks in stream cipher systems. A standard model (see Figure 1) of stream cipher (109, 110, 34] combines the out- puts of several independent LFSR sequences using a nonlinear Boolean function to produce the keystream. Design and analysis of practical stream cipher was kept confidential for a long time. An important boost occurred in the 1970's, when several research papers on the design of LFSR-based stream cipher occurred. As LFSRS are linear, some form of nonlinearity is introduced by using nonlinear Boolean functions (see (100]).Properties of the nonlinear combining Boolean function received a lot of attention in literature for the last two decades and it is now possible to get good Boolean functions which resist many of the known attacks. In this thesis we have not considered algebraic attacks as this class of attacks have become known only very recently. We consider balancedness, nonlinearity, algebraic degree, correlation immunity and resiliency of Boolean functions and S-boxes for use in stream ciphers model based on Figure 1.A Boolean function used in stream cipher should be balanced, which is required for the pseudorandomness of generated keystream. In the stream cipher model, the combining Boolean function is so chosen that it increases the linear complexity [100] of the resulting key stream. High algebraic degree provides high linear complexity [101, 34). Therefore high algebraic degree is desirable in stream ciphers. A Boolean function should have high nonlinearity to be used in stream ciphers. A function with low nonlinearity is prone to linear approximation attack. Linear approximation means approximating the combining function by a linear function. To resist divide-and-conquer attack a Boolean function in stream cipher should be correlation immune of higher order (109, 110). See Chapter 2.3.1 for a more detailed discussion of these properties.We can not achieve all the desirable properties of our liking, so there will be some trade of between these properties. Depending on the application we have to decide which properties are more important.In block cipher eryptography, the message bits are divided into blocks and each block is separately enciphered using the same key and transmitted.

## Control Number

ISILib-TH159

## Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

## DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

## Recommended Citation

Gupta, Kishan Chand Dr., "Cryptographic and Combinatorial Properties of Boolean Functions and s-Boxes." (2005). *Doctoral Theses*. 121.

https://digitalcommons.isical.ac.in/doctoral-theses/121

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842897