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30
Handbook of Applied Cryptography
, 1997
"... As we draw near to closing out the twentieth century, we see quite clearly that the informationprocessing and telecommunications revolutions now underway will continue vigorously into the twentyfirst. We interact and transact by directing flocks of digital packets towards each other through cybers ..."
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Cited by 2524 (30 self)
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As we draw near to closing out the twentieth century, we see quite clearly that the informationprocessing and telecommunications revolutions now underway will continue vigorously into the twentyfirst. We interact and transact by directing flocks of digital packets towards each other through cyberspace, carrying love notes, digital cash, and secret corporate documents. Our personal and economic lives rely more and more on our ability to let such ethereal carrier pigeons mediate at a distance what we used to do with facetoface meetings, paper documents, and a firm handshake. Unfortunately, the technical wizardry enabling remote collaborations is founded on broadcasting everything as sequences of zeros and ones that one's own dog wouldn't recognize. What is to distinguish a digital dollar when it is as easily reproducible as the spoken word? How do we converse privately when every syllable is bounced off a satellite and smeared over an entire continent? How should a bank know that it really is Bill Gates requesting from his laptop in Fiji a transfer of $10,000,000,000 to another bank? Fortunately, the magical mathematics of cryptography can help. Cryptography provides techniques for keeping information secret, for determining that information
On the Periods of Generalized Fibonacci Recurrences
, 1992
"... We give a simple condition for a linear recurrence (mod 2 w ) of degree r to have the maximal possible period 2 w 1 (2 r 1). It follows that the period is maximal in the cases of interest for pseudorandom number generation, i.e. for 3term linear recurrences dened by trinomials which are prim ..."
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Cited by 28 (10 self)
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We give a simple condition for a linear recurrence (mod 2 w ) of degree r to have the maximal possible period 2 w 1 (2 r 1). It follows that the period is maximal in the cases of interest for pseudorandom number generation, i.e. for 3term linear recurrences dened by trinomials which are primitive (mod 2) and of degree r > 2. We consider the enumeration of certain exceptional polynomials which do not give maximal period, and list all such polynomials of degree less than 15. 1.
Uniform Random Number Generators for Supercomputers
 Proc. Fifth Australian Supercomputer Conference
, 1992
"... We consider the requirements for uniform pseudorandom number generators on modern vector and parallel supercomputers, consider the pros and cons of various classes of methods, and outline what is currently available. We propose a class of random number generators which have good statistical propert ..."
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Cited by 26 (11 self)
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We consider the requirements for uniform pseudorandom number generators on modern vector and parallel supercomputers, consider the pros and cons of various classes of methods, and outline what is currently available. We propose a class of random number generators which have good statistical properties and can be implemented efficiently on vector processors and parallel machines. A good method for initialization of these generators is described, and an implementation on a Fujitsu VP 2200/10 vector processor is discussed. 1
A stochastic neural architecture that exploits dynamically reconfigurable FPGAs
 Pocek (Eds.), IEEE Workshop on FPGAs for Custom Computing Machines, IEEE Computer Society Press, Los Alamitos, CA
, 1993
"... I n this paper we present an ezpandable digital architecture that provides a n eflcient real time implementation platform for large neural networks. The architecture makes heavy use of the techniques of bit serial stochastic computing to carry out the large number of required parallel synaptic ca ..."
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Cited by 20 (4 self)
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I n this paper we present an ezpandable digital architecture that provides a n eflcient real time implementation platform for large neural networks. The architecture makes heavy use of the techniques of bit serial stochastic computing to carry out the large number of required parallel synaptic calculations. In this design all real valued quantities are encoded on to stochastic bit streams in which the ‘ I ’ density is proportional to the given quantity. The actual digital circuitry is simple and highly regular thus allowing very eficient space usage of fine grained FPGAs. Another feature of the design is that the large number of weights required by a neural network are generated by circuitry tailored to each of their specific values, thus saving valuable cells. Whenever one of these values is required to change, the appropriate circuitry must be dynamically reconfigured. This may always be achieved in a fized and minimum number of cells for a given bit stream resolution. 1
A fast algorithm for testing reducibility of trinomials mod 2 and some new primitive trinomials of degree 3021377
 Math. Comp
, 2003
"... Abstract. The standard algorithm for testing reducibility of a trinomial of prime degree r over GF(2) requires 2r + O(1) bits of memory. We describe a new algorithm which requires only 3r/2+O(1) bits of memory and significantly fewer memory references and bitoperations than the standard algorithm. ..."
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Cited by 20 (14 self)
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Abstract. The standard algorithm for testing reducibility of a trinomial of prime degree r over GF(2) requires 2r + O(1) bits of memory. We describe a new algorithm which requires only 3r/2+O(1) bits of memory and significantly fewer memory references and bitoperations than the standard algorithm. If 2 r − 1 is a Mersenne prime, then an irreducible trinomial of degree r is necessarily primitive. We give primitive trinomials for the Mersenne exponents r = 756839, 859433, and 3021377. The results for r = 859433 extend and correct some computations of Kumada et al. The two results for r = 3021377 are primitive trinomials of the highest known degree. 1.
Algorithms for Finding Almost Irreducible and Almost Primitive Trinomials
 in Primes and Misdemeanours: Lectures in Honour of the Sixtieth Birthday of Hugh Cowie Williams, Fields Institute
, 2003
"... Consider polynomials over GF(2). We describe ecient algorithms for nding trinomials with large irreducible (and possibly primitive) factors, and give examples of trinomials having a primitive factor of degree r for all Mersenne exponents r = 3 mod 8 in the range 5 < r < 10 , although t ..."
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Cited by 17 (6 self)
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Consider polynomials over GF(2). We describe ecient algorithms for nding trinomials with large irreducible (and possibly primitive) factors, and give examples of trinomials having a primitive factor of degree r for all Mersenne exponents r = 3 mod 8 in the range 5 < r < 10 , although there is no irreducible trinomial of degree r.
Random Number Generators with Period Divisible by a Mersenne Prime
 Proc. ICCSA 2003
, 2003
"... Pseudorandom numbers with long periods and good statistical properties are often required for applications in computational finance. We consider the requirements for good uniform random number generators, and describe a class of generators whose period is a Mersenne prime or a small multiple of ..."
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Cited by 14 (5 self)
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Pseudorandom numbers with long periods and good statistical properties are often required for applications in computational finance. We consider the requirements for good uniform random number generators, and describe a class of generators whose period is a Mersenne prime or a small multiple of a Mersenne prime. These generators are based on "almost primitive" trinomials, that is trinomials having a large primitive factor. They enable very fast vector/parallel implementations with excellent statistical properties.
A Device For Generating Binary Sequences For Stochastic Computing
 Electronics Letters
, 1992
"... A novel technique for the generation of high speed stochastic bit streams in which the `1' density is proportional to a given value. Bit streams of this type are particularly useful in bit serial stochastic computing systems, such as digital stochastic neural networks. The proposed circuitry is ..."
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Cited by 13 (5 self)
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A novel technique for the generation of high speed stochastic bit streams in which the `1' density is proportional to a given value. Bit streams of this type are particularly useful in bit serial stochastic computing systems, such as digital stochastic neural networks. The proposed circuitry is highly suitable for VLSI fabrication. INTRODUCTION The use of stochastic bit streams allows a dramatic simplification of the circuitry required to implement many devices [1], since the multiplication of two values may be performed by computing the bitwise conjunction of two corresponding streams. The highly pipelined digital design described in this paper was developed for use with high speed digital stochastic neural networks, as described in [2, 3, 4]. The proposed stochastic bit stream generator is highly pipelined and thus potentially extremely fast. The individual pipeline stages, known as modulators are simple, and the number used defines the overall resolution of the generated bit strea...
A fast algorithm for testing irreducibility of trinomials mod 2
 pub199.html
, 2000
"... The standard algorithm for testing reducibility of a trinomial of prime degree r over GF(2) requires 2r+O(1) bits of memory and Θ(r 2) bitoperations. We describe an algorithm which requires only 3r/2 + O(1) bits of memory and significantly fewer bitoperations than the standard algorithm. Using the ..."
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Cited by 9 (7 self)
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The standard algorithm for testing reducibility of a trinomial of prime degree r over GF(2) requires 2r+O(1) bits of memory and Θ(r 2) bitoperations. We describe an algorithm which requires only 3r/2 + O(1) bits of memory and significantly fewer bitoperations than the standard algorithm. Using the algorithm, we have found 18 new irreducible trinomials of degree r in the range 100151 ≤ r ≤ 700057. If r is a Mersenne exponent (i.e. 2 r −1 is a Mersenne prime), then an irreducible trinomial is primitive. Primitive trinomials are of interest because they can be used to give pseudorandom number generators with period at least 2 r − 1. We give examples of primitive trinomials for r = 756839, 859433, and 3021377. The three results for r = 756839 are new. The results for r = 859433 extend and correct some computations of Kumada et al. [Math. Comp. 69 (2000), 811–814]. The two results for r = 3021377 are primitive trinomials of the highest known degree. 1 Copyright c○2000, the authors. rpb199tr typeset using L ATEX 1 1
Uniform Random Number Generators for Vector and Parallel Computers
 REVISION APPEARED IN PROC. FIFTH AUSTRALIAN SUPERCOMPUTER CONFERENCE
, 1992
"... We consider the requirements for uniform pseudorandom number generators on modern vector and parallel machines; consider the pros and cons of various popular classes of methods and some new methods; and outline what is currently available. We then make a proposal for a class of random number gen ..."
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Cited by 8 (1 self)
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We consider the requirements for uniform pseudorandom number generators on modern vector and parallel machines; consider the pros and cons of various popular classes of methods and some new methods; and outline what is currently available. We then make a proposal for a class of random number generators which have good statistical properties and can be implemented efficiently on vector processors and parallel machines. A proposal regarding initialization of these generators is made. We also discuss the results of a trial implementation on a Fujitsu VP 2200/10 vector processor.