Irreducible Trinomials Research Articles

Low Complexity Bit-Parallel Squarer for GF(2n) Defined by Irreducible Trinomials

We present a bit-parallel squarer for GF(2n) defined by an irreducible trinomial xn + xk + 1 using a shifted polynomial basis. The proposed squarer requires TX delay and at most ⌈n/2⌉ XOR gates, where TX is the delay of one XOR gate. As a result, the squarer using the shifted polynomial basis is more efficient than one using the polynomial basis except for k = 1 or n/2.

Relationship between GF(2^m) Montgomery and Shifted Polynomial Basis Multiplication Algorithms

Applying the matrix-vector product idea of the Mastrovito multiplier to the GF(2^{m}) Montgomery multiplication algorithm, we present a new parallel multiplier for irreducible trinomials. This multiplier and the corresponding shifted polynomial basis (SPB) multiplier have the same circuit structure for the same set of parameters. Furthermore, by establishing isomorphisms between the Montgomery and the SPB constructions of GF(2^{m}), we show that the Montgomery algorithm can be used to perform the SPB multiplication without any changes and vice versa.

Formulas for cube roots in [formula omitted

We determine the number of nonzero coefficients (called the Hamming weight) in the polynomial representation of x 1 / 3 in F 3 m = F 3 [ x ] / ( f ) , where f ∈ F 3 [ x ] is an irreducible trinomial.

Bit-parallel finite field multipliers for irreducible trinomials

A new formulation for the canonical basis multiplication in the finite fields GF(2/sup m/) based on the use of a triangular basis and on the decomposition of a product matrix is presented. From this algorithm, a new method for multiplication (named transpositional) applicable to general irreducible polynomials is deduced. The transpositional method is based on the computation of 1-cycles and 2-cycles given by a permutation defined by the coordinate of the product to be computed and by the cardinality of the field GF(2/sup m/). The obtained cycles define groups corresponding to subexpressions that can be shared among the different product coordinates. This new multiplication method is applied to five types of irreducible trinomials. These polynomials have been widely studied due to their low-complexity implementations. The theoretical complexity analysis of the corresponding bit-parallel multipliers shows that the space complexities of our multipliers match the best results known to date for similar canonical GF(2/sup m/) multipliers. The most important new result is the reduction, in two of the five studied trinomials, of the time complexity with respect to the best known results.

The Hilbert–Kunz multiplicity of an irreducible trinomial

The Hilbert–Kunz multiplicity, in characteristic p, of the homogeneous co-ordinate ring of the plane curve x 4 + y 4 + z 4 is known to be 3 + 1 p 2 if p ≡ ± 3 ( 8 ) and 3 if p ≡ ± 1 ( 8 ) . We derive similar results for arbitrary irreducible trinomial plane curves, using the fact that these curves have Fermat curves as branched coverings.

ARITHMETICAL PROPERTIES OF POWERS OF ALGEBRAIC NUMBERS

We consider the sequences of fractional parts {ξαn}, n = 1, 2, 3,…, and of integer parts [ξαn], n = 1, 2, 3,…, where ξ is an arbitrary positive number and α > 1 is an algebraic number. We obtain an inequality for the difference between the largest and the smallest limit points of the first sequence. Such an inequality was earlier known for rational α only. It is also shown that for roots of some irreducible trinomials the sequence of integer parts contains infinitely many numbers divisible by either 2 or 3. This is proved, for instance, for [ ξ ( ( 13 − 1 ) / 2 ) n ] , n = 1, 2, 3,…. The fact that there are infinitely many composite numbers in the sequence of integer parts of powers was proved earlier for Pisot numbers, Salem numbers and the three rational numbers 3/2, 4/3, 5/4, but no such algebraic number having several conjugates outside the unit circle was known. 2000 Mathematics Subject Classification 11J71, 11R04, 11R06, 11A41.

On the distribution of irreducible trinomials over [formula omitted

We present some results about irreducible polynomials over finite fields and use them to prove a conjecture of von zur Gathen concerning the distribution of irreducible trinomials over F 3 .

Fast Bit-Parallel GF(2^n) Multiplier for All Trinomials

Based on a new representation of GF(2/sup n/), we present two multipliers for all irreducible trinomials. Space complexities of the multipliers match the best results. The time complexity of one multiplier is T/sub A/ + (1 + [log/sub 2/ n])T/sub X/ for all irreducible trinomials, where T/sub A/ and T/sub X/ are the delay of one 2-input AND and XOR gates, respectively.

Hybrid multiplier for GF (2 m ) defined by some irreducible trinomials

A trade-off between performance and area is important to design an efficient hardware structure for arithmetic operations in GF(2m). Proposed is a hybrid multiplier for GF(2m) with an irreducible trinomial, which can be constructed in variable structures depending on such a performance area trade-off.

Parallel multipliers based on special irreducible pentanomials

The state-of-the-art Galois field GF(2/sup m/) multipliers offer advantageous space and time complexities when the field is generated by so special irreducible polynomial. To date, the best complexity results have been obtained when the irreducible polynomial is either a trinomial or an equally spaced polynomial (ESP). Unfortunately, there exist only a few irreducible ESPs in the range of interest for most of the applications, e.g., error-correcting codes, computer algebra, and elliptic curve cryptography. Furthermore, it is not always possible to find an irreducible trinomial of degree m in this range. For those cases where neither an irreducible trinomial nor an irreducible ESP exists, the use of irreducible pentanomials has been suggested. Irreducible pentanomials are abundant, and there are several eligible candidates for a given m. We promote the use of two special types of irreducible pentanomials. We propose new Mastrovito and dual basis multiplier architectures based on these special irreducible pentanomials and give rigorous analyses of their space and time complexity.

Low complexity bit-parallel systolic multiplier over GF(2m) using irreducible trinomials

A bit-parallel systolic multiplier in the finite field GF(2m) over the polynomial basis, where irreducible trinomials xm+xn+1 generate the fields GF(2m) is presented. The latency of the proposed multiplier requires only 2m−1 clock cycles. The architecture has the advantage of low latency, low circuit complexity and high throughput, as compared with traditional systolic multipliers. Moreover, the multiplier is highly regular, modular, and therefore, well-suited for VLSI implementation.

A fast algorithm for testing reducibility of trinomials mod~2 and some new primitive trinomials of degree 3021377

The standard algorithm for testing reducibility of a trinomial of prime degree r over GF(2) requires 2r + O(1) bits of memory. We describe a new algorithm which requires only 3r/2+O(1) bits of memory and significantly fewer memory references and bit-operations than the standard algorithm.If 2r - 1 is a Mersenne prime, then an irreducible trinomial of degree r is necessarily primitive. We give primitive trinomials for the Mersenne exponents r = 756839, 859433, and 3021377. The results for r = 859433 extend and correct some computations of Kumada et al. The two results for r = 3021377 are primitive trinomials of the highest known degree.

Bit-parallel finite field multiplier and squarer using polynomial basis

Bit-parallel finite field multiplication using polynomial basis can be realized in two steps: polynomial multiplication and reduction modulo the irreducible polynomial. In this article, we present an upper complexity bound for the modular polynomial reduction. When the field is generated with an irreducible trinomial, closed form expressions for the coefficients of the product are derived in term of the coefficients of the multiplicands. The complexity of the multiplier architectures and their critical path length are evaluated, and they are comparable to the previous proposals for the same class of fields. An analytical form for bit-parallel squaring operation is also presented. The complexities for bit-parallel squarer are also derived when an irreducible trinomial is used. Consequently, it is argued that to solve multiplicative inverse using polynomial basis can be at least as good as using a normal basis.

Montgomery multiplier and squarer for a class of finite fields

Montgomery multiplication in GF(2/sup m/) is defined by a(x)b(x)r/sup -1/(x) mod f(x), where the field is generated by a root of the irreducible polynomial f(x), a(x) and b(x) are two field elements in GF(2/sup m/), and r(x) is a fixed field element in GF(2/sup m/). In this paper, first, a slightly generalized Montgomery multiplication algorithm in GF(2/sup m/) is presented. Then, by choosing r(x) according to f (x), we show that efficient architectures of bit-parallel Montgomery multiplier and squarer can be obtained for the fields generated with an irreducible trinomial. Complexities of the Montgomery multiplier and squarer in terms of gate counts and time delay of the circuits are investigated and found to be as good as or better than that of previous proposals for the same class of fields.

Galois Groups of Trinomials

Galois groups of irreducible trinomials Xn+aXs+b∈Z[X] are investigated assuming the classification of finite simple groups. We show that under some simple yet general hypotheses bearing on the integers n, s, a and b only very specific groups can occur. For instance, if the two integers nb and as(n−s) are coprime and if s is a prime number, then already the Galois group of f(X) is either the alternating group An or the symmetric group Sn. This significantly extends work of Osada.

Mastrovito multiplier for all trinomials

An efficient algorithm for the multiplication in GF(2/sup m/) was introduced by Mastrovito. The space complexity of the Mastrovito multiplier for the irreducible trinomial x/sup m/+x+1 was given as m/sup 2/-1 XOR and m/sup 2/ AND gales. In this paper, we describe an architecture based on a new formulation of the multiplication matrix and show that the Mastrovito multiplier for the generating trinomial x/sup m/+x/sup n/+1, where m/spl ne/2n, also requires m/sup 2/-1 XOR and m/sup 2/ AND gates, However, m/sup 2/-x/sup m/2/ XOR gates are sufficient when the generating trinomial is of the form x/sup m/+x/sup m/2/+1 for an even m. We also calculate the time complexity of the proposed Mastrovito multiplier and give design examples for the irreducible trinomials x/sup 7/+x/sup 4/+1 and x/sup 6/+x/sup 3/+1.

GFSR pseudorandom number generation using APL

This paper demonstrates the effectiveness of APL for implementing the generalized feedback shift register (GFSR) approach to producing a random number stream. Various approaches to generating streams of pseudorandom numbers computationally have been devised, dating at least as far back as Norbert Wiener. These are normally discussed in the context of a course in modeling and simulation, since there may be practical implications to consider when building simulation models in APL or any other high level language. This is particularly the case for those circumstances when a (pseudo) random number sequence needs to be repeated, or when multiple streams are needed. In such cases a pseudorandom number generator other than that supplied in the programming language command set needs to be implemented. The linear congruential method for pseudorandom number generation is still commonly used, and is easily implemented in APL or any other high level language; however, it is known to have undesirable n-space uniformity characteristics. Moreover, the period length of its random number stream is limited by the underlying machine's word size. This is a serious issue, since at present day computer speeds, a simulation run could exhaust such a random number stream in a few hours. Very long period (VLP) pseudorandom number generators remedy this deficiency.One class of these, generalized feedback shift register (GFSR) pseudorandom number generators, is based on algebraic manipulation of irreducible trinomials of high order. The manner in which this is accomplished particularly lends itself to elegant APL implementation.

New bit-serial systolic multiplier for GF (2 m ) using irreducible trinomials

A bit-serial systolic architecture is presented for the product-sum computation P = AB + C in a finite field GF(2m) = GF(2)[x]/[F(x)], such that F(x) = xm + x + 1 is an irreducible trinomial over GF(2). It has a low complexity and does not require connections for the serial transfer of F(x).

On bit-serial multiplication and dual bases in GF(2/sup m/)

The existence of certain types of dual bases in finite fields GF(2/sup m/) is discussed. These special types of dual bases are needed for efficient implementation of (generalized) bit-serial multiplication in GF(2/sup m/). In particular, the question of choosing a polynomial basis of GF(2/sup m/), for example (1, alpha , alpha /sup 2/, alpha /sup 3/, . . ., alpha /sup m-1/), such that the change of basis matrix from the dual basis to a scalar multiple of the original basis has as few '1' entries as possible, is studied. It was previously shown by M. Wang and I. F. Blake (1990) that the optimal situation occurs when the minimal polynomial of alpha is an irreducible trinomial of degree m; then, an appropriate scalar multiple, beta , yields a change of basis matrix that is a permutation matrix. A construction is presented that often yields bases where the change of basis matrix has low weight, in the case where no irreducible trinomial of degree m exists. A simple formula can be used to compute beta and the weight of the change of basis matrix, given the minimal polynomial of alpha .< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Decomposition of primes in number fields defined by trinomials

In this paper we deal with the problem of finding the prime-ideal decomposition of a prime integer in a number field K defined by an irreducible trinomial of the type X p m +AX+B∈ℤ[X], in terms of A and B. We also compute effectively the discriminant of K.