Bhupendra SinghBhupendra Singh Scientist ‘B’Scientist ‘B’ scientistbsingh@gmail.comscientistbsingh@gmail.com

Centre for Artificial Intelligence and Centre for Artificial Intelligence and Robotics (CAIR)Robotics (CAIR)

Defence Research and Development Defence Research and Development OrganizationOrganization

BangaloreBangalore

CAIR

Centre for Artificial Intelligence and RoboticsDefence Research and Development

OrganizationBangalore

2

1.Problems in area of finite fields 1.Problems in area of finite fields

LLinear inear FFeedback eedback SShift hift RResister (LFSR):esister (LFSR): LFSR is a finite state machine in which states are shifting regularly and LFSR is a finite state machine in which states are shifting regularly and

feedback for next state is calculated from the present state using linear feedback for next state is calculated from the present state using linear feedback polynomial. LFSR is an essential part of many stream ciphers, but feedback polynomial. LFSR is an essential part of many stream ciphers, but LFSR itself is not secureLFSR itself is not secure

JJump ump LLinear inear FFeedback eedback SShift hift RResister (JLFSR):esister (JLFSR): JLFSR is LFSR in which multiple shifting is achieved by modifying the JLFSR is LFSR in which multiple shifting is achieved by modifying the

transition matrix from transition matrix from AA to to A+IA+I. when A. when AJ=A+I, with this the LFSR shift =A+I, with this the LFSR shift through J steps. J is called Jump index.through J steps. J is called Jump index.

JJump ump IIndex (JI): Let f(x) be an irreducible polynomial over GF(2). ndex (JI): Let f(x) be an irreducible polynomial over GF(2). If xIf xJJ ≡ x+1(mod f(x)) for some integer J, then J is called the JI of ≡ x+1(mod f(x)) for some integer J, then J is called the JI of

f(x).f(x).

Jump index is an Jump index is an importantimportant parameter for analysis of JLFSR. parameter for analysis of JLFSR.

PROBLEM:PROBLEM: How to find jump index efficiently and analyze JLFSR How to find jump index efficiently and analyze JLFSR with respect to security. We are also interested jump index with respect to security. We are also interested jump index for irreducible (non-primitive) polynomials.for irreducible (non-primitive) polynomials.

CAIR

Centre for Artificial Intelligence and RoboticsDefence Research and Development

OrganizationBangalore

3

Problems in area of finite fields cont…Problems in area of finite fields cont…

Primitive polynomial: A polynomial of degree n Primitive polynomial: A polynomial of degree n over GF(2) is said to be primitive if it is over GF(2) is said to be primitive if it is irreducible and period is 2irreducible and period is 2nn-1.-1.

Weight: weight of polynomial is number of terms Weight: weight of polynomial is number of terms in the polynomial. in the polynomial.

PROBLEM: General formula for finding PROBLEM: General formula for finding number of primitive polynomials of number of primitive polynomials of given degree and given weight. given degree and given weight.

CAIR

Centre for Artificial Intelligence and RoboticsDefence Research and Development

OrganizationBangalore

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2.Problem in Sequences: 2.Problem in Sequences:

Pseudo Randomness : Must meet NIST STANDARDS.Pseudo Randomness : Must meet NIST STANDARDS.

Period : When Sequence is going to repeat.Period : When Sequence is going to repeat.

Linear Complexity : Shortest length of LFSR which can Linear Complexity : Shortest length of LFSR which can generate that sequence.generate that sequence.

Autocorrelation test: correlations between the sequence Autocorrelation test: correlations between the sequence and its non-cyclic shifted versions of it.and its non-cyclic shifted versions of it.

Cross correlation: correlation between any pair of Cross correlation: correlation between any pair of sequences. sequences.

PROBLEM: How to Design Pseudo Random PROBLEM: How to Design Pseudo Random Binary Sequence of large period and large Binary Sequence of large period and large linear complexity such that they have good linear complexity such that they have good Autocorrelation and simultaneously good Autocorrelation and simultaneously good cross correlation property. cross correlation property.

CAIR

Centre for Artificial Intelligence and RoboticsDefence Research and Development

OrganizationBangalore

5

3.Problem related to functions:3.Problem related to functions:

Let f be function from {0,1}Let f be function from {0,1}nn to {0,1} to {0,1}mm

Case1: when n>m=1 (Boolean function),Case1: when n>m=1 (Boolean function),

Case2:when n>m>1(S-Box),Case2:when n>m>1(S-Box),

Case3: when n=m (Permutation) ,Case3: when n=m (Permutation) ,

Boolean function properties :degree, Boolean function properties :degree, non-linearity, resilience, algebraic non-linearity, resilience, algebraic immunity. immunity.

S- Box property: Non-linearity (Max). S- Box property: Non-linearity (Max).

Permutation properties : DP,LP.Permutation properties : DP,LP.

PROBLEM: How to design these PROBLEM: How to design these functions which have optimal functions which have optimal cryptography property . cryptography property .

Thank YouThank You