Basis Of GF Research Articles

Design of Two-times Faster Bit-Serial MultiplierUsing Normal Basis of GF(2m)

Design of Two-times Faster Bit-Serial MultiplierUsing Normal Basis of GF(2m)

Hamiltonians of quantum systems with positions and momenta in GF(pℓ)

A quantum system with positions and momenta in GF(pℓ) is considered. Such a system can be constructed from ℓ smaller systems, in which the positions and momenta take values in Zp, if the Hamiltonian of this ℓ-partite system is compatible with GF(pℓ). The concept of compatibility of a Hamiltonian with GF(pℓ) allows the quantum formalism in the ℓ-partite system to be expressed in terms of Galois arithmetic. Transformations of the basis in GF(pℓ) produce unitary transformations of the quantum states, which form a representation of GL(ℓ,Zp). They are used to define which subset of the general set of Hamiltonians in the ℓ-partite system is compatible with GF(pℓ).

Concurrent Error Detection in Multiplexer-Based Multipliers for Normal Basis of GF(2 m ) Using Double Parity Prediction Scheme

Successful implementation of elliptic curve cryptographic systems primarily depends on the efficient and reliable arithmetic circuits for finite fields with very large orders. Thus, the robust encryption/decryption algorithms are elegantly needed. Multiplication would be the most important finite field arithmetic operation. It is much more complex compared to the finite field addition. It is also frequently used in performing point operations in elliptic curve groups. The hardware implementation of a multiplication operation may require millions of logic gates and may thus lead to erroneous outputs. To obtain reliable cryptographic applications, a novel concurrent error detection (CED) architecture to detect erroneous outputs in multiplexer-based normal basis (NB) multiplier over GF(2 m ) using the parity prediction scheme is proposed in this article. Although various NB multipliers, depending on $$ \alpha \alpha^{{2^i }} = \sum\limits_{j = 0}^{m - 1} {t_{i,j} } \alpha^{{2^j }} $$ , have different time and space complexities, NB multipliers will have the same structure if they use a parity prediction function. By using the structure of the proposed CED NB multiplier, a CED scalable multiplier over composite fields with 100% error detection rate is also presented.

Low-Complexity Bit-Parallel Multiplier over GF(2m) Using Dual Basis Representation

Recently, cryptographic applications based on finite fields have attracted much attention. The most demanding finite field arithmetic operation is multiplication. This investigation proposes a new multiplication algorithm over GF(2m) using the dual basis representation. Based on the proposed algorithm, a parallel-in parallel-out systolic multiplier is presented. The architecture is optimized in order to minimize the silicon covered area (transistor count). The experimental results reveal that the proposed bit-parallel multiplier saves about 65% space complexity and 33% time complexity as compared to the traditional multipliers for a general polynomial and dual basis of GF(2m).

Theory and applications of q-ary interleaved sequences

A new class of q-ary sequences, called interleaved sequence, is introduced. Their periods, shift equivalence, linear spans, and autocorrelation functions are derived. The interleaved sequences include a large number of popular sequences, such as multiplexed sequences, clock-controlled sequences, Kasami (1966) sequences, GMW sequences, geometric sequences, and No (1989) sequences. A special class of the interleaved sequences is constructed by mapping GF(q/sup m/) sequences into GF(q) sequences in terms of different bases of GF(q/sup m/) over GF(q). As an application of the theory of interleaved sequences, some new families of binary pseudo-random sequences are constructed, which have large linear spans, optimal periodic cross/autocorrelation functions, balance, and the rapidly hopped properties. A complete comparison of the new family of sequences with the Gold sequence family, the Kasami (small and large set) sequence families, the Bent-function sequence family, and the No sequence family is discussed. This shows that the new sequence families have important advantages for use in spread-spectrum multiple-access communication systems. >

Low complexity normal bases

A normal basis in GF( q m ) is a basis of the form { β, β q , β q 2,…, β q m−1 }, i.e., a basis of conjugate elements in the field. In GF(2 m ) squaring with respect to a normal basis representation becomes simply a cyclic shift of the vector. For hardware design this is one of the very attractive features of these bases. Multiplication with respect to a normal basis can be defined in terms of a certain bilinear form. Define the complexity of the normal basis to the number of nonzero terms in this form. Again, for hardware design, it is important to find normal bases with low complexity. In this paper we investigate low complexity normal bases, give a construction for such bases and apply it to a number of cases of interest.

The binary weight distribution of the extended (2 m,2 m − 4) code of the Reed-Solomon code over GF(2 m) with generator polynomial ( x − α)( x − α2)( x − α3)

Consider an ( n, k) linear code with symbols from GF(2 m ). If each code symbol is represented by a binary m-tuple using a certain basis for GF(2 m ), we obtain a binary ( nm, km) linear code, called a binary image of the original code. In this paper, we present a lower bound on the minimum weight of a binary image of a cyclic code over GF(2 m ) and the weight enumerator for a binary image of the extended (2 m ,2 m − 4) code of the Reed-Solomon code over GF(2 m ) with generator polynomial ( x − α)( x − α 2) ( x − α 3) and its dual code, where α is a primitive element in GF(2 m ).

On the sharpness of a theorem of B. segre

The theorem of B. Segre mentioned in the title states that a complete arc of PG(2,q),q even which is not a hyperoval consists of at mostq−√q+1 points. In the first part of our paper we prove this theorem to be sharp forq=s 2 by constructing completeq−√q+1-arcs. Our construction is based on the cyclic partition of PG(2,q) into disjoint Baer-subplanes. (See Bruck [1]). In his paper [5] Kestenband constructed a class of (q−√q+1)-arcs but he did not prove their completeness. In the second part of our paper we discuss the connections between Kestenband’s and our constructions. We prove that these constructions result in isomorphic (q−√q+1)-arcs. The proof of this isomorphism is based on the existence of a traceorthogonal normal basis in GF(q 3) over GF(q), and on a representation of GF(q)3 in GF(q 3)3 indicated in Jamison [4].