# Polynomials Having Sparse Multiples.

December 2001

## Date of Award

Winter 12-12-2002

## Institute Name (Publisher)

Indian Statistical Institute

## Document Type

Master's Dissertation

## Degree Name

Master of Technology

Computer Science

## Department

Applied Statistics Unit (ASU-Kolkata)

## Supervisor

Roy, Bimal Kumar (ASU-Kolkata; ISI)

## Abstract (Summary of the Work)

Stream ciphers form an important class of secret-key encryption schemes. They are widely used in applications since they present many advantages: they are usually faster than common block ciphers and they have less complex hard- ware circuitry. Moreover, their use is particularly wellsuited when errors may occur during the transmission because they avoid error propagation. In a binary ad ditive stream cipher the ciphertext is obtained by adding bitwise the plaintext to a pseudorandom sequence s, called the key stream (or the running-key). The runningkey is produced by a pseudorandom generator whose initialization is the secret key shared by the users. Most attacks on such ciphers therefore consist in recovering the initialization of the pseudorandom generator from the knowledge of a few cipher text bits (or of some bits of the running-key in known-plaintext attacks).Linear feedback shift registers (LFSRS) are the basic components of most keystream generators since they are appropriate to hardware implementations, produce sequences with good statistical properties and can be easily analyzed.Linear Feedback Shift Register (LFSR) is a system which generates a pseudo- random bit-sequence using a binary recurrence-relation of the forman = C1an-1 + c2an-2 + ..+ Ck-1an-k+1 + Ckan-k (1.1)where Ck = 1 and for 1 â‰¤ i < k, ciÆ{0, 1}. The length of a LFSR correspond to the order k of the linear-recurrence-relation used. The number of taps t of an LFSR is the number of non-zero bits in { c1, c2,...,Ck}. The successive bits of the LFSR are emitted using the chosen recurrence relation after intialising the seed (ao, a1, a2, , ak-1) of LFSR.The (1.1) is related to the following polynomial over GF(2)C(x) = 1+ c1z + c2xÂ² + ..+ Ckxk(1.2The (1.2) is called the Connection Polynomial of the LFSR.The LFSR-generated sequence of the linear-recurrence-relation(Irr) related to a connection polynomial is same as the one for the corresponding Irr of multiple polynomial of the connection polynomial.In the stream-cipher systems, the key-stream is usually generated by com- bining the outputs of more than one LFSR using a nonlinear boolean function. This arrangement significantly increases the robustness of the system against possible attacks. This keystream is bitwise XORed with the message bitstream to produce the cipher. The decryption machinery is identical to the encryption machinery (see Figure 1.1).In such a systen, n bits from n different LFSRS are generated at each clock.These n bits are the input to the boolean function F(X1, X2, X3,..., Xn). The output of the boolean function F is the key-straem K.The cipher stream C is the XORing of K and the message stream M. i.e., C = K eM.Consider the connection polynomial of degree dxd+ad-1xd-1+ad-2xd-2+...+a1x+1where ai Æ {0,1},i, â‰¤ i â‰¤ d-1. We take connection polynomial of size d with least significant bit starts from the right hand side and the most significant bit at the leftmost postion. There is a tap at ith position if and only if a = 1.

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:28843208

ISI-DISS-2001-80