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30
Feedback shift registers, 2adic span, and combiners with memory
 Journal of Cryptology
, 1997
"... Feedback shift registers with carry operation (FCSR’s) are described, implemented, and analyzed with respect to memory requirements, initial loading, period, and distributional properties of their output sequences. Many parallels with the theory of linear feedback shift registers (LFSR’s) are presen ..."
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Cited by 50 (7 self)
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Feedback shift registers with carry operation (FCSR’s) are described, implemented, and analyzed with respect to memory requirements, initial loading, period, and distributional properties of their output sequences. Many parallels with the theory of linear feedback shift registers (LFSR’s) are presented, including a synthesis algorithm (analogous to the BerlekampMassey algorithm for LFSR’s) which, for any pseudorandom sequence, constructs the smallest FCSR which will generate the sequence. These techniques are used to attack the summation cipher. This analysis gives a unified approach to the study of pseudorandom sequences, arithmetic codes, combiners with memory, and the MarsagliaZaman random number generator. Possible variations on the FCSR architecture are indicated at the end. Index Terms – Binary sequence, shift register, stream cipher, combiner with memory, cryptanalysis, 2adic numbers, arithmetic code, 1/q sequence, linear span. 1
Wavelet transforms associated with finite cyclic groups
 IEEE Trans. Inform. Theory
, 1993
"... Abstmct Multiresolution analysis via decomposition on wavelet bases has emerged as an important tool in the analysis of signals and images when these objects are viewed as sequences of complex or real numbers. An important class of multiresolution decompositions are the socalled Laplacian pyramid ..."
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Cited by 15 (0 self)
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Abstmct Multiresolution analysis via decomposition on wavelet bases has emerged as an important tool in the analysis of signals and images when these objects are viewed as sequences of complex or real numbers. An important class of multiresolution decompositions are the socalled Laplacian pyramid schemes, in which the resolution is successively halved by recursively lowpass filtering the signal under analysis and decimating it by a factor of two. Generally speakhg, the principal framework within which multiresolution techniques have been studied and applied is the same as that used in the discretetime Fourier analysis of sequences of complex numbers. An analogous framework is developed for the multiresolution analysis of finitelength sequences of elements €mm arbitrary fields. Attention is restricted to sequences of length 2 " for n a positive iuteger, so that the resolution may be recursively halved to completion. As in finitelength Fourier analysis, a cyclic group structure of the index set of such sequences is exploited to characterize the transforms of interest for the particular cases of complex and finite fields. This development is motivated by potential applications in areas such as digital signal processing and algebraic coding, in which cyclic Fourier analysis has found widespread applications. Index Terms Multiresolution analysis, wavelet transforms, Laplacian pyramid, finite fields, cyclic group, quadrature mimr filters. I.
Large Period Nearly deBruijn FCSR Sequences (Extended Abstract)
 In L.C. Guillou and J.J. Quisquater� editors� Advances in Cryptology � Eurocrypt �95
, 1995
"... Recently, a new class of feedback shift registers (FCSRs) was introduced, based on algebra over the 2adic numbers. The sequences generated by these registers have many algebraic properties similar to those generated by linear feedback shift registers. However, it appears to be significantly more di ..."
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Cited by 9 (4 self)
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Recently, a new class of feedback shift registers (FCSRs) was introduced, based on algebra over the 2adic numbers. The sequences generated by these registers have many algebraic properties similar to those generated by linear feedback shift registers. However, it appears to be significantly more difficult to find maximal period FCSR sequences. In this paper we exhibit a technique for easily finding FCSRs that generate nearly maximal period sequences. We further show that these sequence have excellent distributional properties. They are balanced, and nearly have the deBruijn property for distributions of subsequences.
A New Algorithm for Multiplication in Finite Fields
 IEEE Transactions on Computers
, 1989
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On modulus replication for residue arithmetic computations of complex inner products
 IEEE TRANS. COMP
, 1990
"... Residue Number Systems require the selection of ring moduli whose product is greater than the predicted dynamic range of the computation being performed. The restriction that the moduli be relatively prime usually limits the set of available moduli and hence the maximum dynamic range. This is partic ..."
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Cited by 6 (5 self)
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Residue Number Systems require the selection of ring moduli whose product is greater than the predicted dynamic range of the computation being performed. The restriction that the moduli be relatively prime usually limits the set of available moduli and hence the maximum dynamic range. This is particularly the case when small moduli are to be considered for efficient hardware implementation. Severe restrictions occur when algebraic constraints, such as those posed by the necessity to implement quadratic residue rings, are a factor. This paper presents a technique for coding weighted magnitude components (e.g. bits) of numbers directly into polynomial residue rings, such that repeated use may be made of the same set of moduli to effectively increase the dynamic range of the computation. This effectively limits the requirement for large sets of relatively prime moduli. For practical computations over quadratic residue rings, at least 6bit moduli have to be considered; we show, in this paper, that 5bit moduli can be effectively used for large dynamic range computations.
Finite field polynomial multiplication in the frequency domain with application to elliptic curve cryptography
 In Proceedings of the 21th International Symposium on Computer and Information Sciences (ISCIS 2006), volume 4263 of Lecture Notes in Computer Science (LNCS
, 2006
"... Abstract. We introduce an efficient method for computing Montgomery products of polynomials in the frequency domain. The discrete Fourier transform (DFT) based method originally proposed for integer multiplication provides an extremely efficient method with the best asymptotic complexity, i.e. O(m l ..."
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Cited by 5 (3 self)
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Abstract. We introduce an efficient method for computing Montgomery products of polynomials in the frequency domain. The discrete Fourier transform (DFT) based method originally proposed for integer multiplication provides an extremely efficient method with the best asymptotic complexity, i.e. O(m log m log log m), for multiplication of mbit integers or (m − 1) st degree polynomials. However, the original DFT method bears significant overhead due to the conversions between the time and the frequency domains which makes it impractical for short operands as used in many applications. In this work, we introduce DFT modular multiplication which performs the entire modular multiplication (including the reduction step) in the frequency domain, and thus eliminates costly back and forth conversions. We show that, especially in computationally constrained platforms, multiplication of finite field elements may be achieved more efficiently in the frequency domain than in the time domain for operand sizes relevant to elliptic curve cryptography (ECC). To the best of our knowledge, this is the first work that proposes the use of frequency domain arithmetic for ECC and shows that it can be efficient. Key Words: Finite field multiplication, DFT, elliptic curve cryptography. 1
Computation of convolutions and discrete Fourier transforms by polynomial transforms
 IBM J. Res. Develop
, 1978
"... Abstract: Discrete transforms are introduced and are defined in a ring of polynomials. These polynomial transforms are shown to have the convolution property and can be computed in ordinary arithmetic, without multiplications. Polynomial transforms are particularly well suited for computing discrete ..."
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Cited by 3 (0 self)
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Abstract: Discrete transforms are introduced and are defined in a ring of polynomials. These polynomial transforms are shown to have the convolution property and can be computed in ordinary arithmetic, without multiplications. Polynomial transforms are particularly well suited for computing discrete twodimensional convolutions with a minimum number of operations. Efficient algorithms for computing onedimensional convolutions and Discrete Fourier Transforms are then derived from polynomial transforms.
On the design and implementation of efficient zeroknowledge proofs of knowledge
 In Software Performance Enhancements for Encryption and Decryption and Cryptographic Compilers – SPEEDCC 09
"... Abstract. Zeroknowledge proofs of knowledge (ZKPoK) play an important role in many cryptographic applications. Direct anonymous attestation (DAA) and the identity mixer anonymous authentication system are first real world applications using ZKPoK as building blocks. But although being used for ma ..."
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Cited by 3 (2 self)
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Abstract. Zeroknowledge proofs of knowledge (ZKPoK) play an important role in many cryptographic applications. Direct anonymous attestation (DAA) and the identity mixer anonymous authentication system are first real world applications using ZKPoK as building blocks. But although being used for many years now, design and implementation of sound ZKPoK remains challenging. In fact, there are security flaws in various protocols found in literatur. Especially for nonexperts in the field it is often hard to design ZKPoK, since a unified and easy to use theoretical framework on ZKPoK is missing. With this paper we overcome important challenges and facilitate the design and implementation of efficient and sound ZKPoK in practice. First, Camenisch et al. have presented at EUROCRYPT 2009 a first unified and modular theoretical framework for ZKPoK. This is compelling, but makes use of a rather inefficient 6move protocol. We extend and improve their framework in terms of efficiency and show how to realize it using efficient 3move Σprotocols. Second, we perform an exact security and efficiency analysis for our new protocol and various protocols found in the literature. The analysis yields novel and perhaps surprising results and insights. It reveals for instance that using a 2048 bit RSA modulus, as specified in the DAA standard, only guarantees an upper bound on the success probability of a malicious prover between 1/2 4 and 1/2 24. Also, based on that analysis we show how to select the most efficient protocol to realize a given proof goal. Finally, we also provide lowlevel support to a designer by presenting a compiler realizing our framework and optimization techniques, allowing easy implementation of efficient and sound protocols.
Achieving efficient polynomial multiplication in Fermat fields using the fast Fourier Transform
 ACM Southeast Regional Conference Proceedings of the 44th annual Southeast regional conference, ACM Press
, 2006
"... We introduce an efficient way of performing polynomial multiplication in a class of finite fields GF (p m) in the frequency domain. The Fast Fourier Transform (FFT) based frequency domain multiplication technique, originally proposed for integer multiplication, provides an extremely efficient method ..."
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Cited by 2 (1 self)
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We introduce an efficient way of performing polynomial multiplication in a class of finite fields GF (p m) in the frequency domain. The Fast Fourier Transform (FFT) based frequency domain multiplication technique, originally proposed for integer multiplication, provides an extremely efficient method for multiplication with the best known asymptotic complexity, i.e. O(n log n log log n). Unfortunately, the original FFT method bears significant overhead due to the conversions between the time and the frequency domains, which makes it impractical to perform multiplication of relatively short (160 − 1024 bits) integer operands as used in many applications. In this work, we introduce an efficient way of performing polynomial multiplication in finite fields using the FFT. We show that, with careful selection of parameters, all the multiplications required for the FFT computations can be avoided and polynomial multiplication in finite fields can be achieved with only O(m) multiplications in addition to O(m log m) simple shift, addition and subtraction operations. We show that, especially in constrained devices where multiplication is expensive, polynomial multiplication in the suggested finite fields using the FFT outperforms both the schoolbook and Karatsuba methods for practically small finite fields, e.g., relevant to elliptic curve cryptography.
An implementation of Schonhage's multiplication algorithm (or how to compute the square of a number with one million digits on your workstation in less than one minute)
"... This report describes an implementation of a fast multiplication algorithm proposed by A. Schonhage [5]. The algorithm performs the multiplication of two integers modulo a number of the form 2 N + 1 in O(N log N log log N) operations. Using the BigNum package [2], we wrote a C program of less than ..."
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Cited by 2 (0 self)
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This report describes an implementation of a fast multiplication algorithm proposed by A. Schonhage [5]. The algorithm performs the multiplication of two integers modulo a number of the form 2 N + 1 in O(N log N log log N) operations. Using the BigNum package [2], we wrote a C program of less than 350 lines that performs both the modular and integer multiplication. We give detailed timings and comparisons with the naive method and Karatsuba's algorithm on two particular machines, a DecStation 5000 and an IBM RS 6000. 1 Introduction The Fast Fourier Transform (FFT) is a wellknown tool to reduce the theoretical complexity of algorithms, typically to transform a O(n 2 ) algorithm into a O(n log n) one. But so far, only a few people published an implementation of an algorithm using the FFT, and studied values of n for which the "fast" algorithm was really better. In the field of integer multiplication, two papers describe such implementations. In 1986, A. Schonhage encoded his algori...