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Normal Bases over Finite Fields
, 1993
"... 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 repr ..."
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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...
Factoring Dickson polynomials over finite fields
 Finite Fields Appl
, 1999
"... We derive the factorizations of the Dickson polynomials Dn(X, a) and En(X, a), and of the bivariate Dickson polynomials Dn(X, a) − Dn(Y, a), over any finite field. Our proofs are significantly shorter and more elementary than those previously known. ..."
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Cited by 6 (3 self)
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We derive the factorizations of the Dickson polynomials Dn(X, a) and En(X, a), and of the bivariate Dickson polynomials Dn(X, a) − Dn(Y, a), over any finite field. Our proofs are significantly shorter and more elementary than those previously known.
Dickson Bases and Finite Fields
"... Finite fields have been used for many applications in electronic communications. In the case of extension fields, the nature of computation depends heavily on the choice of basis used to represent the extension over the base field. The most common choices of basis are polynomial bases although optim ..."
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Finite fields have been used for many applications in electronic communications. In the case of extension fields, the nature of computation depends heavily on the choice of basis used to represent the extension over the base field. The most common choices of basis are polynomial bases although optimal normal bases or some variant of these have also been used despite the fact that such bases exist in only a limited set of cases. Building on these, we develop an alternative class of bases that exist for any extension field. We provide hardware models based on the notion of shift registers for computing with respect to such bases, and investigate some of the properties of these models.
Constructing Elements Of Large Order In Finite Fields
, 1999
"... An efficient algorithm is presented which for any finite field Fq of small characteristic finds an extension Fq s of polynomially bounded degree and an element # # Fq s of exponentially large multiplicative order. The construction makes use of certain analogues of Gauss periods of a special type. Th ..."
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An efficient algorithm is presented which for any finite field Fq of small characteristic finds an extension Fq s of polynomially bounded degree and an element # # Fq s of exponentially large multiplicative order. The construction makes use of certain analogues of Gauss periods of a special type. This can be considered as another step towards solving the celebrated problem of finding primitive roots in finite fields efficiently.
unknown title
"... The literature of cryptography has a curious history. Secrecy, of course, has always played a central role, but until the First World War, important developments appeared in print in a more or less timely fashion and the field moved forward in much the same way as other specialized disciplines. As l ..."
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The literature of cryptography has a curious history. Secrecy, of course, has always played a central role, but until the First World War, important developments appeared in print in a more or less timely fashion and the field moved forward in much the same way as other specialized disciplines. As late as 1918, one of the most influential cryptanalytic papers of the twentieth century, William F. Friedman’s monograph The Index of Coincidence and Its Applications in Cryptography, appeared as a research report of the private Riverbank Laboratories [577]. And this, despite the fact that the work had been done as part of the war effort. In the same year Edward H. Hebern of Oakland, California filed the first patent for a rotor machine [710], the device destined to be a mainstay of military cryptography for nearly 50 years. After the First World War, however, things began to change. U.S. Army and Navy organizations, working entirely in secret, began to make fundamental advances in cryptography. During the thirties and forties a few basic papers did appear in the open literature and several treatises on the subject were published, but the latter were farther and farther behind the state of the art. By the end of the war the transition was complete. With one notable exception, the public literature had died. That exception was Claude Shannon’s paper “The Communication Theory of Secrecy Systems, ” which
Research Summary
"... Normal bases and efficient arithmetic in finite fields Efficient arithmetic of finite fields is important in implementing cryptosystems, errorcorrecting codes and computer algebra systems. Normal bases offer considerable advantages. Optimal normal bases in finite fields were introduced at the Unive ..."
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Normal bases and efficient arithmetic in finite fields Efficient arithmetic of finite fields is important in implementing cryptosystems, errorcorrecting codes and computer algebra systems. Normal bases offer considerable advantages. Optimal normal bases in finite fields were introduced at the University of Waterloo by Mullin et al., and are used in practical hardware implementation of publickey cryptosystems. In the same paper, Mullin et al. constructed two families of optimal normal bases and, based on a computer experiment, they conjectured no more exist. This conjecture had remained open for several years before H. W. Lenstra, Jr. proved it for finite fields over F2. Lenstra’s method, however, is not applicable to other fields. In [4], we confirmed the conjecture for all finite fields by using a substantially different argument. Together with Lenstra, we proved the conjecture holds even for any finite Galois extension of an arbitrary field and the proof for the general case is again different but simpler. The final result is published in [5]. By our classification in [5], not all finite fields have optimal normal bases. For fields without optimal normal bases, it is desirable to have a normal basis of low complexity. In [10], we construct several families of such bases, which come from an explicit factorization of cxq+1 +