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137
Probability of error in MMSE multiuser detection
 IEEE Trans. Inform. Theory
, 1997
"... Abstract—Performance analysis of the minimummeansquareerror (MMSE) linear multiuser detector is considered in an environment of nonorthogonal signaling and additive white Gaussian noise. In particular, the behavior of the multipleaccess interference (MAI) at the output of the MMSE detector is exa ..."
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Cited by 193 (15 self)
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Abstract—Performance analysis of the minimummeansquareerror (MMSE) linear multiuser detector is considered in an environment of nonorthogonal signaling and additive white Gaussian noise. In particular, the behavior of the multipleaccess interference (MAI) at the output of the MMSE detector is examined under various asymptotic conditions, including: large signaltonoise ratio; large near–far ratios; and large numbers of users. These results suggest that the MAIplusnoise contending with the demodulation of a desired user is approximately Gaussian in many cases of interest. For the particular case of two users, it is shown that the maximum divergence between the output MAIplusnoise and a Gaussian distribution having the same mean and variance is quite small in most cases of interest. It is further proved in this twouser case that the probability of error of the MMSE detector is better than that of the decorrelating linear detector for all values of normalized crosscorrelations not greater than I
The Elliptic Curve Digital Signature Algorithm (ECDSA)
, 1999
"... The Elliptic Curve Digital Signature Algorithm (ECDSA) is the elliptic curve analogue of the Digital Signature Algorithm (DSA). It was accepted in 1999 as an ANSI standard, and was accepted in 2000 as IEEE and NIST standards. It was also accepted in 1998 as an ISO standard, and is under consideratio ..."
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Cited by 173 (5 self)
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The Elliptic Curve Digital Signature Algorithm (ECDSA) is the elliptic curve analogue of the Digital Signature Algorithm (DSA). It was accepted in 1999 as an ANSI standard, and was accepted in 2000 as IEEE and NIST standards. It was also accepted in 1998 as an ISO standard, and is under consideration for inclusion in some other ISO standards. Unlike the ordinary discrete logarithm problem and the integer factorization problem, no subexponentialtime algorithm is known for the elliptic curve discrete logarithm problem. For this reason, the strengthperkeybit is substantially greater in an algorithm that uses elliptic curves. This paper describes the ANSI X9.62 ECDSA, and discusses related security, implementation, and interoperability issues. Keywords: Signature schemes, elliptic curve cryptography, DSA, ECDSA.
Efficient Algorithms for Elliptic Curve Cryptosystems
, 1997
"... Elliptic curves are the basis for a relative new class of publickey schemes. It is predicted that elliptic curves will replace many existing schemes in the near future. It is thus of great interest to develop algorithms which allow efficient implementations of elliptic curve crypto systems. This th ..."
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Cited by 73 (9 self)
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Elliptic curves are the basis for a relative new class of publickey schemes. It is predicted that elliptic curves will replace many existing schemes in the near future. It is thus of great interest to develop algorithms which allow efficient implementations of elliptic curve crypto systems. This thesis deals with such algorithms. Efficient algorithms for elliptic curves can be classified into lowlevel algorithms, which deal with arithmetic in the underlying finite field and highlevel algorithms, which operate with the group operation. This thesis describes three new algorithms for efficient implementations of elliptic curve cryptosystems. The first algorithm describes the application of the KaratsubaOfman Algorithm to multiplication in composite fields GF ((2 n ) m ). The second algorithm deals with efficient inversion in composite Galois fields of the form GF ((2 n ) m ). The third algorithm is an entirely new approach which accelerates the multiplication of points which i...
Algorithms for computing isogenies between elliptic curves
 Math. Comp
, 2000
"... Abstract. The heart of the improvements by Elkies to Schoof’s algorithm for computing the cardinality of elliptic curves over a finite field is the ability to compute isogenies between curves. Elkies ’ approach is well suited for the case where the characteristic of the field is large. Couveignes sh ..."
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Cited by 41 (7 self)
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Abstract. The heart of the improvements by Elkies to Schoof’s algorithm for computing the cardinality of elliptic curves over a finite field is the ability to compute isogenies between curves. Elkies ’ approach is well suited for the case where the characteristic of the field is large. Couveignes showed how to compute isogenies in small characteristic. The aim of this paper is to describe the first successful implementation of Couveignes’s algorithm. In particular, we describe the use of fast algorithms for performing incremental operations on series. We also insist on the particular case of the characteristic 2. 1.
An Overview of Elliptic Curve Cryptography
, 2000
"... Elliptic curve cryptography (ECC) was introduced by Victor Miller and Neal Koblitz in 1985. ECC proposed as an alternative to established publickey systems such as DSA and RSA, have recently gained a lot attention in industry and academia. The main reason for the attractiveness of ECC is the fact t ..."
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Cited by 36 (3 self)
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Elliptic curve cryptography (ECC) was introduced by Victor Miller and Neal Koblitz in 1985. ECC proposed as an alternative to established publickey systems such as DSA and RSA, have recently gained a lot attention in industry and academia. The main reason for the attractiveness of ECC is the fact that there is no subexponential algorithm known to solve the discrete logarithm problem on a properly chosen elliptic curve. This means that significantly smaller parameters can be used in ECC than in other competitive systems such RSA and DSA, but with equivalent levels of security. Some benefits of having smaller key sizes include faster computations, and reductions in processing power, storage space and bandwidth. This makes ECC ideal for constrained environments such as pagers, PDAs, cellular phones and smart cards. The implementation of ECC, on the other hand, requires several choices such as the type of the underlying finite field, algorithms for implementing the finite field arithmetic and so on. In this paper we give we presen an selective overview of the main methods.
Fibonacci and Galois Representations of FeedbackWithCarry Shift Registers
 IEEE Trans. Inform. Theory
, 2002
"... A feedbackwithcarry shift register (FCSR) with "Fibonacci" architecture is a shift register provided with a small amount of memory which is used in the feedback algorithm. Like the linear feedback shift register (LFSR), the FCSR provides a simple and predictable method for the fast gener ..."
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Cited by 34 (2 self)
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A feedbackwithcarry shift register (FCSR) with "Fibonacci" architecture is a shift register provided with a small amount of memory which is used in the feedback algorithm. Like the linear feedback shift register (LFSR), the FCSR provides a simple and predictable method for the fast generation of pseudorandom sequences with good statistical properties and large periods. In this paper, we describe and analyze an alternative architecture for the FCSR which is similar to the "Galois" architecture for the LFSR. The Galois architecture is more efficient than the Fibonacci architecture because the feedback computations are performed in parallel. We also describe the output sequences generated by theFCSR, a slight modification of the (Fibonacci) FCSR architecture in which the feedback bit is delayed for clock cycles before being returned to the first cell of the shift register. We explain how these devices may be configured so as to generate sequences with large periods. We show that the FCSR also admits a more efficient "Galois" architecture.
A generalized method for constructing subquadratic complexity GF(2 k ) multipliers
 IEEE Transactions on Computers
, 2004
"... We introduce a generalized method for constructing subquadratic complexity multipliers for even characteristic field extensions. The construction is obtained by recursively extending short convolution algorithms and nesting them. To obtain the short convolution algorithms the Winograd short convolu ..."
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Cited by 24 (0 self)
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We introduce a generalized method for constructing subquadratic complexity multipliers for even characteristic field extensions. The construction is obtained by recursively extending short convolution algorithms and nesting them. To obtain the short convolution algorithms the Winograd short convolution algorithm is reintroduced and analyzed in the context of polynomial multiplication. We present a recursive construction technique that extends any d point multiplier into an n = d k point multiplier with area that is subquadratic and delay that is logarithmic in the bitlength n. We present a thorough analysis that establishes the exact space and time complexities of these multipliers. Using the recursive construction method we obtain six new constructions, among which one turns out to be identical to the Karatsuba multiplier. All six algorithms have subquadratic space complexities and two of the algorithms have significantly better time complexities than the Karatsuba algorithm. Keywords: Bitparallel multipliers, finite fields, Winograd convolution 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 23 (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.
Hyperbent functions, Kloosterman sums and Dickson polynomials
"... This paper is devoted to the classification of hyperbent functions, i.e., bent functions which are bent up to a primitive root change. We first exhibit an infinite class of monomial functions which are not hyperbent. This result means that Kloosterman sums at point 1 on F2 m cannot be zero, unless m ..."
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Cited by 22 (1 self)
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This paper is devoted to the classification of hyperbent functions, i.e., bent functions which are bent up to a primitive root change. We first exhibit an infinite class of monomial functions which are not hyperbent. This result means that Kloosterman sums at point 1 on F2 m cannot be zero, unless m = 4. For the functions with multiple trace terms, we express their spectrum by means of Dickson polynomials. We then introduce a new tool to describe these hyperbent functions, whose efficiency is proving by the characterization of a class of binomial bent functions.
Cryptographic Properties of the WelchGong Transformation Sequence Generators
 IEEE Transactions on Information Theory
, 2002
"... Abstract—Welch–Gong (WG) transformation sequences are binary sequences of period 2 1 with twolevel autocorrelation. These sequences were discovered by Golomb, Gong, and Gaal in 1998 and they verified the validity of their construction for 5 20. Later, No, Chung, and Yun found another way to constr ..."
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Cited by 21 (11 self)
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Abstract—Welch–Gong (WG) transformation sequences are binary sequences of period 2 1 with twolevel autocorrelation. These sequences were discovered by Golomb, Gong, and Gaal in 1998 and they verified the validity of their construction for 5 20. Later, No, Chung, and Yun found another way to construct the WG sequences and verified their result for 5 23. Dillon first proved this result for odd in 1998, and, finally, Dobbertin and Dillon proved it for even in 1999. In this paper, we investigate a twofaced property of the WG transformation sequences for application in stream ciphers and pseudorandom number generators. One is to present the randomness or unpredictability of the WG transformation sequences. The other is to exhibit the security properties of the WG transformations regarded as Boolean functions. In particular, we prove that the WG transformation sequences, in addition to the known twolevel autocorrelation and threelevel cross correlation withsequences, have the ideal 2tuple distribution, and large linear span increasing exponentially with. Moreover, it can be implemented efficiently. This is the first type of pseudorandom sequences with good correlation, statistic properties, large linear span, and efficient implementation. When WG transformations are regarded as Boolean functions, they have high nonlinearity. We derive a criterion for the Boolean representation of WG transformations to beresilient and show that they are at least 1resilient under some basis of the finite field GF (2). An algorithm to find such bases is given. The degree and linear span of WG transformations are presented as well. Index Terms—Auto/cross correlation, Boolean function, linear span, nonlinearity, pseudorandom sequence (number) generator,