5 research outputs found
A Swan-like note for a family of binary pentanomials
In this note, we employ the techniques of Swan (Pacific J. Math. 12(3):
1099-1106, 1962) with the purpose of studying the parity of the number of the
irreducible factors of the penatomial
, where is even and .
Our results imply that if , then the polynomial in
question is reducible
Low Complexity Cubing and Cube Root Computation over \F_{3^m} in Polynomial Basis
We present low complexity formulae for the computation
of cubing and cube root over \F_{3^m} constructed using special classes of irreducible
trinomials, tetranomials and pentanomials.
We show that for all those special classes of polynomials, field cubing and field cube root operation
have the same computational complexity when implemented in hardware or software platforms.
As one of the main applications of these two field arithmetic operations lies in pairing-based
cryptography, we also give in this paper a selection of irreducible polynomials that lead to low cost
field cubing and field cube root computations for supersingular elliptic curves defined over
\F_{3^m}, where is a prime number in the pairing-based cryptographic range of interest, namely,
XOR-counts and lightweight multiplication with fixed elements in binary finite fields
XOR-metrics measure the efficiency of certain arithmetic operations in binary
finite fields. We prove some new results about two different XOR-metrics that
have been used in the past. In particular, we disprove an existing conjecture
about those XOR-metrics. We consider implementations of multiplication with
one fixed element in a binary finite field. Here we achieve a complete
characterization of all elements whose multiplication matrix can be
implemented using exactly 2 XOR-operations. Further, we provide new results
and examples in more general cases, showing that significant improvements in
implementations are possible
Mathematical aspects of the design and security of block ciphers
Block ciphers constitute a major part of modern symmetric cryptography. A mathematical analysis is necessary to ensure the security of the cipher. In this thesis, I develop several new contributions for the analysis of block ciphers. I determine cryptographic properties of several special cryptographically interesting mappings like almost perfect nonlinear functions. I also give some new results both on the resistance of functions against differential-linear attacks as well as on the efficiency of implementation of certain block ciphers