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Parity-Checked Strassen Algorithm
[article]

2022
*
arXiv
*
pre-print

Nesting the

arXiv:2011.15082v3
fatcat:bfvzvx6qsnhyzhhj5bfpehbcoi
*parity*-*checked**algorithms*, we weave a product code flavor protection. ... With empirical frequency 99.8%, 729 workers suffice, wherein 729 is the complexity of the schoolbook*algorithm*. ... To calculate A B, the manager might repeat a*parity*-*checked**Strassen**algorithm*twice, or introduce checksums to the squared*Strassen**algorithm*. It might as well do both. ...##
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Page 978 of Mathematical Reviews Vol. 48, Issue 3
[page]

1974
*
Mathematical Reviews
*

He also defines an rs xn

*parity*-*check*matrix whose jth column contains the rs elements of the form 2 Iba (t;—Yxy)~ 1, K=1, 2, «++, 8; $=1, 2, ---, 7. ... The author defines a linear code of length n over the field GF(q) by means of the s x*parity*-*check*matrix H, where Hy; =Y9;(%;), Xj, yy € GF(q"), the x, are distinct, and g, is a polynomial of degree < ...##
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Experimental evidence of quantum randomness incomputability

2010
*
Physical Review A. Atomic, Molecular, and Optical Physics
*

.\ it is not exactly reproducible by any

doi:10.1103/physreva.82.022102
fatcat:tyhhrfgiunc4xhb7c3yh6z6rci
*algorithm*. ... We provide experimental evidence of incomputability --- an asymptotic property --- of quantum randomness by performing finite tests of randomness inspired by*algorithmic*information theory. ... sequences "00" and "11" are discarded, and only the odd*parity*ones "01" and "10" are kept. ...##
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Efficient Hardware Implementation of the LEDAcrypt Decoder

2021
*
IEEE Access
*

an efficient implementation of the iterative decoder that is the main part of the decryption stage in the LEDAcrypt cryptosystem, recently proposed for post-quantum cryptography based on low-density

doi:10.1109/access.2021.3076245
fatcat:tfus2ptqrbflhigu6eyjrstkma
*parity*-*check*... is another valid*parity*-*check*matrix for the same code. ... Crucial quantities in a generic BF*algorithm*are the Unsatisfied*Parity**Check*(UPC) counters, which correspond to the number of unsatisfied*parity*equations for each element of the unknown error vector ...##
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An Efficient Maximum-Likelihood Decoding of LDPC Codes Over the Binary Erasure Channel

2004
*
IEEE Transactions on Information Theory
*

We propose an efficient maximum likelihood decoding

doi:10.1109/tit.2004.836694
fatcat:gt6r54gcp5d5za3epgxvek2jx4
*algorithm*for decoding low-density*parity*-*check*codes over the binary erasure channel. ... We also analyze the computational complexity of the proposed*algorithm*. ... degeneracy in the*parity**check*equations. ...##
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Relative class number of imaginary Abelian fields of prime conductor below 10000

1999
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Mathematics of Computation
*

This problem can be solved efficiently with the following strategy: we use the Schönhage-

doi:10.1090/s0025-5718-99-01139-4
fatcat:iuk4awfbpnadbp4eipr36xq7bm
*Strassen**algorithm*for multiplying integers. ...*Parity*of the class number. The*parity*of the relative class number h − (p) of the field Q(ζ p ) has attracted a lot of attention ever since Kummer's introduction of cyclotomic class numbers. ...##
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Evaluation and comparison of two efficient probabilistic primality testing algorithms

1980
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Theoretical Computer Science
*

We analyse two recent probabiktic primality testing

doi:10.1016/0304-3975(80)90007-9
fatcat:auxd53ib4bbnjku2cxien6xp5m
*algorithms*; the first one is derived from Miller [6] in a formulation given uy Rabin [7], and the second one is from Solovay and*Strassen*[9]. ... Our comparison shows that the first*algorithm*is always more efficient than the second, both in probabilistic and*algorithmic*terms. ... By M(n) we denote the cost of multiplying two integers sn ; it is known by Schonhage-*Strassen**algorithm*[8] that M(n) = O(log n llog n lllog n). ...##
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Page 71 of Mathematical Reviews Vol. , Issue 95a
[page]

1995
*
Mathematical Reviews
*

The authors describe the probabilistic prime number tests due to Rabin and Solovay-

*Strassen*. ... Using FORTRAN, the tests are implemented and*checked*for some numbers of size between 10° and 10%; the running times are less than 45 minutes. W. ...##
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An Optimized Quantum Implementation of ISD on Scalable Quantum Resources
[article]

2021
*
arXiv
*
pre-print

The security of code based constructions is usually assessed by Information Set Decoding (ISD)

arXiv:2112.06157v1
fatcat:lggamxkp4fbj3bpvpwyslk4uui
*algorithms*. ... We show that against common belief, Prange's ISD*algorithm*can be implemented rather efficiently on a quantum computer, namely with only a logarithmic overhead in circuit depth compared to a classical ... In these cases our Punctured-Hybrid*algorithm*can disregard*parity*-*check*equations without introducing multiple solutions to the reduced instance. ...##
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Low-Complexity Implementation of RAID Based on Reed-Solomon Codes

2015
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ACM Transactions on Storage
*

The performance results show that the new

doi:10.1145/2700308
fatcat:oqxquanplnhrxd2vv47c6pr6mm
*algorithms*provide substantially better performance compared with the standard*algorithm*. ... The complexity of the obtained*algorithms*is much lower than those for existing MDS array codes. Software implementation of the proposed*algorithms*is discussed. ... Furthermore, this approach relies on the Schönhage-*Strassen*fast polynomial multiplication*algorithm*, which becomes practical only for very large problems. ...##
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Lower bounds for algebraic computation trees

1983
*
Proceedings of the fifteenth annual ACM symposium on Theory of computing - STOC '83
*

This extends the result due to

doi:10.1145/800061.808735
dblp:conf/stoc/Ben-Or83
fatcat:lwhhw5siejedjagvnnvadm2oeq
*Strassen*[21] , since*checking*whether ai(x) = ai may actually be easier than computing the values of ai(x). ... So by example 1 the*algorithm*must make fl(n logn) steps. This extends the result due to Baur and*Strassen*[1]. Example 10. Resultant. ...##
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Infinitely Many Carmichael Numbers for a Modified Miller-Rabin Prime Test
[article]

2015
*
arXiv
*
pre-print

We show that the test can also be thought of as a variant of the Solovay-

arXiv:1512.00444v2
fatcat:ylsztuwesffktf5y5co6ggznqy
*Strassen*test. ... In the case where z = 1, the test can be thought of as the following variant of Solovay-*Strassen*: after generating a,*check*whether a (n−1)/2 ≡ ±1 (mod n). ... Instead of*checking*the whole sequence (1), only*check*the last z + 1 numbers. ...##
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A Simple Unpredictable Pseudo-Random Number Generator

1986
*
SIAM journal on computing (Print)
*

Q is a product of distinct primes, each congruent to 3 mod 4, and x 0 is a quadratic residue mod N), outputs bob1 b2" where bi

doi:10.1137/0215025
fatcat:gueghdrpfveonc7mkgaxjfqovm
*parity*(xi) and xi+ x mod N. ... It follows that there is an efficient, i.e., polynomial (n), probabilistic*algorithm*to find special n-bit primes: simply generate n bit numbers at random and use a probabilistic primality test [*Strassen*-Solovay ... Test each of the resulting numbers, x. r 2 mod N, for quadratic residuacity by*checking*if PIN, xr2] 1 (using sequences of coin tosses as may be required by P). ...##
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How Random Is Quantum Randomness? An Experimental Approach
[article]

2009
*
arXiv
*
pre-print

We base our experimental approach on

arXiv:0912.4379v1
fatcat:y3523r33nffrpemh3awnbbxkgu
*algorithmic*information theory which provides characterizations of*algorithmic*random sequences in terms of the degrees of incompressibility of their finite prefixes ...*Algorithmic*random sequences are incomputable, but the converse implication is false. ... sequences "00" and "11" are discarded, and only the odd*parity*ones "01" and "10" are kept. ...##
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Congruent Number Theta Coefficients to 1012
[chapter]

2010
*
Lecture Notes in Computer Science
*

In the computation using the second

doi:10.1007/978-3-642-14518-6_17
fatcat:ciztklmtmrbezbekcwohpic2b4
*algorithm*, a different*check*was performed. ... In 1971 Schönhage and*Strassen*presented two*algorithms*for multiplication of large integers based on the FFT [33] . ...
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