1. Bhupendra Singh Bhupendra Singh Scientist ‘B’ Scientist ‘B’ scientistbsingh@gmail.com scientistbsingh@gmail.comscientistbsingh@gmail.com Centre for Artificial Intelligence and Robotics (CAIR) Centre for Artificial Intelligence and Robotics (CAIR) Defence Research and Development Organization Bangalore Bangalore

2. CAIR Centre for Artificial Intelligence and Robotics Defence Research and Development Organization Bangalore 2 1.Problems in area of finite fields 1.Problems in area of finite fields Linear Feedback Shift Resister (LFSR): 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 polynomial. LFSR is an essential part of many stream ciphers, but LFSR itself is not secure 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 polynomial. LFSR is an essential part of many stream ciphers, but LFSR itself is not secure Jump Linear Feedback Shift Resister (JLFSR): JLFSR is LFSR in which multiple shifting is achieved by modifying the transition matrix from A to A+I. when A=A+I, with this the LFSR shift through J steps. J is called Jump index. JLFSR is LFSR in which multiple shifting is achieved by modifying the transition matrix from A to A+I. when A J =A+I, with this the LFSR shift through J steps. J is called Jump index. Jump Index (JI): Let f(x) be an irreducible polynomial over GF(2). If x J ≡ x+1(mod f(x)) for some integer J, then J is called the JI of f(x). If x J ≡ x+1(mod f(x)) for some integer J, then J is called the JI of f(x). Jump index is an important parameter for analysis of JLFSR. Jump index is an important parameter for analysis of JLFSR. PROBLEM: How to find jump index efficiently and analyze JLFSR with respect to security. We are also interested jump index for irreducible (non-primitive) polynomials.

3. CAIR Centre for Artificial Intelligence and Robotics Defence Research and Development Organization Bangalore 3 Problems in area of finite fields cont… Primitive polynomial: A polynomial of degree n over GF(2) is said to be primitive if it is irreducible and period is 2 n -1. Weight: weight of polynomial is number of terms in the polynomial. PROBLEM: General formula for finding number of primitive polynomials of given degree and given weight.

4. CAIR Centre for Artificial Intelligence and Robotics Defence Research and Development Organization Bangalore 4 2.Problem in Sequences: 2.Problem in Sequences: Pseudo Randomness : Must meet NIST STANDARDS. Period : When Sequence is going to repeat. Linear Complexity : Shortest length of LFSR which can generate that sequence. Autocorrelation test: correlations between the sequence and its non-cyclic shifted versions of it. Cross correlation: correlation between any pair of sequences. PROBLEM: How to Design Pseudo Random Binary Sequence of large period and large linear complexity such that they have good Autocorrelation and simultaneously good cross correlation property.

5. CAIR Centre for Artificial Intelligence and Robotics Defence Research and Development Organization Bangalore 5 3.Problem related to functions: Let f be function from {0,1} n to {0,1} m Case1: when n>m=1 (Boolean function), Case2:when n>m>1(S-Box), Case3: when n=m (Permutation), Boolean function properties :degree, non- linearity, resilience, algebraic immunity. S- Box property: Non-linearity (Max). Permutation properties : DP,LP. PROBLEM: How to design these functions which have optimal cryptography property.

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