this paper we generalize the well-known Schonhage-Strassen algorithm for multiplying large integers to an algorithm for multiplying polynomials with coefficients from an arbitrary, not necessarily commutative, not necessarily associative, algebra A. Our main result is an algorithm to multiply polynomials of degree ! n in

In this paper we give a randomized O(n log n)-time algorithm for the string matching with don't cares problem. This improves the Fischer-Paterson bound [10] from 1974 and answers the open problem posed (among others) by Weiner [30] and Galil [11]. Using the same technique, we give an O(n log n)-time algorithm for other problems, including subset matching and tree pattern matching [15, 21, 9, 7, 17] and (general) approximate threshold matching [28, 17]. As this bound essentially matches the complexity of computing of the Fast Fourier Transform which is the only known technique for solving problems of this type, it is likely that the algorithms are in fact optimal. Additionally, the technique used for the threshold matching problem can be applied to the on-line version of this problem, in which we are allowed to preprocess the text and require to process the pattern in time sublinear in the text length. This result involves an interesting variant of the Karp-Rabin fingerprint m...

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A new algorithm is developed for decoding Reed-Solomon codes. It uses fast Fourier transforms and computes the message symbols directly without explicitly finding error locations or error magnitudes. In the decoding radius (up to half of the minimum distance), the new method is easily adapted for error and erasure decoding. It can also detect all errors outside the decoding radius. Compared with the Berlekamp-Massey algorithm, discovered in the late 1960's, the new method seems simpler and more natural yet it has a similar time complexity.

Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to represent finite fields was noted by Hensel in 1888. With the introduction of optimal normal bases, large finite fields, that can be used in secure and e#cient implementation of several cryptosystems, have recently been realized in hardware. The present thesis studies various theoretical and practical aspects of normal bases in finite fields. We first give some characterizations of normal bases. Then by using linear algebra, we prove that F q n has a basis over F q such that any element in F q represented in this basis generates a normal basis if and only if some groups of coordinates are not simultaneously zero. We show how to construct an irreducible polynomial of degree 2 n with linearly i...

Abstract. A C implementation of Niederreiter’s algorithm for factoring polynomials over F2 is described. The most time-consuming part of this algorithm, which consists of setting up and solving a certain system of linear equations, is performed in parallel. Once a basis for the solution space is found, all irreducible factors of the polynomial can be extracted by suitable gcdcomputations. For this purpose, asymptotically fast polynomial arithmetic algorithms are implemented. These include Karatsuba & Ofman multiplication, Cantor multiplication and Newton inversion. In addition, a new efficient version of the half-gcd algorithm is presented. Sequential run times for the polynomial arithmetic and parallel run times for the factorization are given. A new “world record ” for polynomial factorization over the binary field is set by showing that a pseudo-randomly selected polynomial of degree 300000 can be factored in about 10 hours on 256 nodes of the IBM SP2 at the Cornell Theory Center. 1.

Abstract. We improve our proposal of a new variant of the McEliece cryptosystem based on QC-LDPC codes. The original McEliece cryptosystem, based on Goppa codes, is still unbroken up to now, but has two major drawbacks: long key and low transmission rate. Our variant is based on QC-LDPC codes and is able to overcome such drawbacks, while avoiding the known attacks. Recently, however, a new attack has been discovered that can recover the private key with limited complexity. We show that such attack can be avoided by changing the form of some constituent matrices, without altering the remaining system parameters. We also propose another variant that exhibits an overall increased security level. We analyze the complexity of the encryption and decryption stages by adopting efficient algorithms for processing large circulant matrices. The Toom-Cook algorithm and the short Winograd convolution are considered, that give a significant speed-up in the cryptosystem operations.

Abstract. In this paper, we discuss an implementation of various algorithms for multiplying polynomials in GF(2)[x]: variants of the window methods, Karatsuba’s, Toom-Cook’s, Schönhage’s and Cantor’s algorithms. For most of them, we propose improvements that lead to practical speedups.

We describe algorithms for polynomial multiplication and polynomial factorization over the binary field F2 and their implementation. They allow polynomials of degree up to 100,000 to be factored in about one day of CPU time.

Abstract. The most time-consuming part of the Niederreiter algorithm for factoring univariate polynomials over finite fields is the computation of elements of the nullspace of a certain matrix. This paper describes the so-called “black-box ” Niederreiter algorithm, in which these elements are found by using a method developed by Wiedemann. The main advantages over an approach based on Gaussian elimination are that the matrix does not have to be stored in memory and that the computational complexity of this approach is lower. The black-box Niederreiter algorithm for factoring polynomials over the binary field was implemented in the C programming language, and benchmarks for factoring high-degree polynomials over this field are presented. These benchmarks include timings for both a sequential implementation and a parallel implementation running on a small cluster of workstations. In addition, the Wan algorithm, which was recently introduced, is described, and connections between (implementation aspects of) Wan’s and Niederreiter’s algorithm are given. 1.