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416
Computing Frobenius Maps And Factoring Polynomials
 Comput. Complexity
, 1992
"... . A new probabilistic algorithm for factoring univariate polynomials over finite fields is presented. To factor a polynomial of degree n over F q , the number of arithmetic operations in F q is O((n 2 +n log q) \Delta (log n) 2 loglog n). The main technical innovation is a new way to compute F ..."
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Cited by 28 (2 self)
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. A new probabilistic algorithm for factoring univariate polynomials over finite fields is presented. To factor a polynomial of degree n over F q , the number of arithmetic operations in F q is O((n 2 +n log q) \Delta (log n) 2 loglog n). The main technical innovation is a new way to compute Frobenius and trace maps in the ring of polynomials modulo the polynomial to be factored. Subject classifications. 68Q40; 11Y16, 12Y05. 1. Introduction We consider the problem of factoring a univariate polynomial over a finite field. This problem plays a central role in computational algebra. Indeed, many of the efficient algorithms for factoring univariate and multivariate polynomials over finite fields, the field of rational numbers, and finite extensions of the rationals solve as a subproblem the problem of factoring univariate polynomials over finite fields (Kaltofen 1990). This problem also has important applications in number theory (Buchmann 1990), coding theory (Berlekamp 1968), and ...
Doing more with fewer bits
 Proceedings Asiacrypt99, LNCS 1716, SpringerVerlag
, 1999
"... Abstract. We present a variant of the DiffieHellman scheme in which the number of bits exchanged is one third of what is used in the classical DiffieHellman scheme, while the offered security against attacks known today is the same. We also give applications for this variant and conjecture a exten ..."
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Cited by 27 (4 self)
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Abstract. We present a variant of the DiffieHellman scheme in which the number of bits exchanged is one third of what is used in the classical DiffieHellman scheme, while the offered security against attacks known today is the same. We also give applications for this variant and conjecture a extension of this variant further reducing the size of sent information. 1
On certain exponential sums and the distribution of DiffieHellman triples
 J. London Math. Soc
, 1999
"... Let g be a primitive root modulo a prime p. It is proved that the triples (gx,gy,gxy), x,y�1,…,p�1, are uniformly distributed modulo p in the sense of H. Weyl. This result is based on the following upper bound for double exponential sums. Let ε�0 be fixed. Then p−� x,y=� exp0 2πiagx�bgy�cgxy ..."
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Cited by 26 (14 self)
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Let g be a primitive root modulo a prime p. It is proved that the triples (gx,gy,gxy), x,y�1,…,p�1, are uniformly distributed modulo p in the sense of H. Weyl. This result is based on the following upper bound for double exponential sums. Let ε�0 be fixed. Then p−� x,y=� exp0 2πiagx�bgy�cgxy
Planar functions and planes of LenzBarlotti class
 II, Des. Codes Cryptogr
, 1997
"... Dedicated to Professor Lenz on the occasion of his 80th birthday Abstract. Planar functions were introduced by Dembowski and Ostrom ([4]) to describe projective planes possessing a collineation group with particular properties. Several classes of planar functions over a finite field are described, i ..."
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Cited by 26 (6 self)
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Dedicated to Professor Lenz on the occasion of his 80th birthday Abstract. Planar functions were introduced by Dembowski and Ostrom ([4]) to describe projective planes possessing a collineation group with particular properties. Several classes of planar functions over a finite field are described, including a class whose associated affine planes are not translation planes or dual translation planes. This resolves in the negative a question posed in [4]. These planar functions define at least one such affine plane of order 3 e for every e ≥ 4 and their projective closures are of LenzBarlotti type II. All previously known planes of type II are obtained by derivation or lifting. At least when e is odd, the planes described here cannot be obtained in this manner. 1.
Absolute Irreducibility Of Polynomials Via Newton Polytopes
, 1998
"... A multivariable polynomial is associated with a polytope, called its Newton polytope. A polynomial is absolutely irreducible if its Newton polytope is indecomposable in the sense of Minkowski sum of polytopes. Two general constructions of indecomposable polytopes are given, and they give many simple ..."
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Cited by 24 (9 self)
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A multivariable polynomial is associated with a polytope, called its Newton polytope. A polynomial is absolutely irreducible if its Newton polytope is indecomposable in the sense of Minkowski sum of polytopes. Two general constructions of indecomposable polytopes are given, and they give many simple irreducibility criteria including the wellknown Eisenstein's criterion. Polynomials from these criteria are over any field and have the property of remaining absolutely irreducible when their coefficients are modified arbitrarily in the field, but keeping certain collection of them nonzero.
4phase sequences with nearoptimum correlation properties
 IEEE Trans. Inform. Theory
, 1992
"... AbstractTwo families of 4phase sequences are constructed using irreducible polynomials over Z4. Family d has period L = 2 ' 1, size L +2, and maximum nontrivial correlation magnitude C,,, 5 1 + m, where r is a positive integer. Family has period L = 2(2 ' l), size ( L + 2)/4, and C,,, 5 2 + m. B ..."
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Cited by 22 (6 self)
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AbstractTwo families of 4phase sequences are constructed using irreducible polynomials over Z4. Family d has period L = 2 ' 1, size L +2, and maximum nontrivial correlation magnitude C,,, 5 1 + m, where r is a positive integer. Family has period L = 2(2 ' l), size ( L + 2)/4, and C,,, 5 2 + m. Both families are asymptotically optimal with respect to the Welch lower bound on C,,, for complexvalued sequences. Of particular interest, Family d has the same size and period as the family of binary Gold sequences, but its maximum nontrivial correlation is smaller by a factor of 4. Since the Gold family for r odd is optimal with respect to the Welch bound restricted to binary sequences, Family d is thus superior to the best possible binary design of the same family size. Unlike the Gold design, Families d and!t? are asymptotically optimal whether r is odd or even. Both families are suitable for achieving codedivision multipleaccess and are easily implemented using shift registers. The exact distribution of correlation values is given for both families. Index TermsSequence design, pseudorandom sequences, nonbinary sequences, quadriphase sequences, periodic correlation, codedivision multipleaccess. I.
Finite Field Multiplier Using Redundant Representation
 IEEE Transactions on Computers
, 2002
"... This article presents simple and highly regular architectures for finite field multipliers using a redundant representation. The basic idea is to embed a finite field into a cyclotomic ring which has a basis with the elegant multiplicative structure of a cyclic group. One important feature of our ar ..."
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Cited by 21 (1 self)
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This article presents simple and highly regular architectures for finite field multipliers using a redundant representation. The basic idea is to embed a finite field into a cyclotomic ring which has a basis with the elegant multiplicative structure of a cyclic group. One important feature of our architectures is that they provide areatime tradeoffs which enable us to implement the multipliers in a partialparallel/hybrid fashion. This hybrid architecture has great significance in its VLSI implementation in very large fields. The squaring operation using the redundant representation is simply a permutation of the coordinates. It is shown that when there is an optimal normal basis, the proposed bitserial and hybrid multiplier architectures have very low space complexity. Constant multiplication is also considered and is shown to have advantage in using the redundant representation. Index terms: Finite field arithmetic, cyclotomic ring, redundant set, normal basis, multiplier, squaring.
On a problem of Byrnes concerning polynomials with restricted coefficients
 Math. Comp
, 1997
"... Abstract. We consider a question of Byrnes concerning the minimal degree n of a polynomial with all coefficients in {−1, 1} which has a zero of a given order m at x =1. Form≤5, we prove his conjecture that the monic polynomial of this type of minimal degree is given by ∏m−1 k=0 (x2k − 1), but we dis ..."
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Cited by 21 (0 self)
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Abstract. We consider a question of Byrnes concerning the minimal degree n of a polynomial with all coefficients in {−1, 1} which has a zero of a given order m at x =1. Form≤5, we prove his conjecture that the monic polynomial of this type of minimal degree is given by ∏m−1 k=0 (x2k − 1), but we disprove this for m ≥ 6. We prove that a polynomial of this type must have n ≥ e √ m(1+o(1)) , which is in sharp contrast with the situation when one allows coefficients in {−1, 0, 1}. The proofs use simple number theoretic ideas and depend ultimately on the fact that −1 ≡ 1(mod2). 1.
Crosscorrelations of linearly and quadratically related geometric
 DISCRETE APPLIED MATHEMATICS
, 1993
"... In this paper we study the crosscorrelation function values of geometric sequences obtained from qary msequences whose underlying msequences are linearly or quadratically related. These values are determined by counting the points of intersection of pairs of hyperplanes or of hyperplanes and qua ..."
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Cited by 20 (8 self)
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In this paper we study the crosscorrelation function values of geometric sequences obtained from qary msequences whose underlying msequences are linearly or quadratically related. These values are determined by counting the points of intersection of pairs of hyperplanes or of hyperplanes and quadric hypersurfaces of a finite geometry. The results are applied to obtain the crosscorrelations of msequences and GMW sequences with different primitive polynomials.
Fast Polynomial Factorization Over High Algebraic Extensions of Finite Fields
 In Kuchlin [1997
, 1997
"... New algorithms are presented for factoring polynomials of degree n over the finite field of q elements, where q is a power of a fixed prime number. When log q = n 1+a , where a ? 0 is constant, these algorithms are asymptotically faster than previous known algorithms, the fastest of which require ..."
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Cited by 20 (5 self)
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New algorithms are presented for factoring polynomials of degree n over the finite field of q elements, where q is a power of a fixed prime number. When log q = n 1+a , where a ? 0 is constant, these algorithms are asymptotically faster than previous known algorithms, the fastest of which required time \Omega\Gamma n(log q) 2 ), y or \Omega\Gamma n 3+2a ) in this case, which corresponds to the cost of computing x q modulo an n degree polynomial. The new algorithms factor an arbitrary polynomial in time O(n 3+a+o(1) +n 2:69+1:69a ). All measures are in fixed precision operations, that is in bit complexity. Moreover, in the special case where all the irreducible factors have the same degree, the new algorithms run in time O(n 2:69+1:69a ). In particular, one may test a polynomial for irreducibility in O(n 2:69+1:69a ) bit operations. These results generalize to the case where q = p k , where p is a small prime number relative to q. 1 Introduction The expected run...