#### Date of Submission

2-28-2000

#### Date of Award

2-28-2001

#### 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

Roy, Bimal Kumar (ASU-Kolkata; ISI)

#### Abstract (Summary of the Work)

In this thesis we concentrate on properties of cryptographically significant Boolean functions.The techniques are mainly combinstorial and provide new resulta on enumeration and construction of such functions. Initially we concentrate on a partieular subset of Boolean functions called the symmetric Boolean functions. A closed form expression for the Walsh transform of an arbitrary symmetric Boolean function is presented. We completely characterize the symmetric functions with maximum nonlinearity and show that the maximum nonlinearity of n-variable symmetrie function can be 2n-1-2[n-1l2], Moreover, new classes of symmetric balanced and symmetric correlation immune functions are considered.We provide a randomised heuristic to construct balanced Boolean functions on n variables (n â‰¥15 and odd) with nonlinearity strictly greater than - - 2n-1-2[n-1l2]. For such function a the algebraic degree is also maintained at its maximum, nâ‰¤1. For odd n â‰¤13, we construct balanced functionS with nonlinearity 2n-1-2and algebruie degree n- I. Moreover, we desiga optimised 1-resilient functions with currently best known nanlinearity. We alao consider propagation characteristics and strict avalanche eriteria. Our constructions provide balanced functions with these properties which maintain very high nonlinearity.The set of correlatian immune Boolean functions can be partitioned into several dieÄ¼oint subseta with respect to the Hamming weights of their output column. It is shown that the number of n variable correlation immune functions of Hamming weight 2Î± + 2 is strictly greater than the number of such functions of weight 2Î± for 2Î± < 2n-1, We also relate the caumeration problem of correlatios immune functions to the enumeration problem of balanced correlation immune functions and provide a closed form expression for the number of corvrelation immune functions.We then identify some small but interesting subseta of correlation immune Boolena functions and provide some estimates on the cardinality of those subsets. We also consider a subset of correlation immune functions which satisty one or more of a few other conditions e.g. balancedness, nondegeneracy and nonaffinity.Moreover, we provide a new construction method using a senall set of recursive operations for a large class of highly monlinesr, resllient Boolean functions with maximum possible algehraic degroe. Comparisons to previous constructions show that better nonlinearity can be obtnined by this method. Our technique can be used to construct functions on largo number of input variables with simple hardware implementation. The architecture is programmable and can be dynamically reconfigured to compute different functions of this class.

#### Control Number

ISILib-TH236

#### 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

Maitra, Subhamoy Dr., "Boolean Functions with Important Cryptographic Properties." (2001). *Doctoral Theses*. 389.

https://digitalcommons.isical.ac.in/doctoral-theses/389

## 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:28843746