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Parity-Checked Strassen Algorithm [article]

Hsin-Po Wang, Iwan Duursma
2022 arXiv   pre-print
Nesting the 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.  ... 
arXiv:2011.15082v3 fatcat:bfvzvx6qsnhyzhhj5bfpehbcoi

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 <  ... 

Experimental evidence of quantum randomness incomputability

Cristian S. Calude, Michael J. Dinneen, Monica Dumitrescu, Karl Svozil
2010 Physical Review A. Atomic, Molecular, and Optical Physics  
.\ it is not exactly reproducible by any 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.  ... 
doi:10.1103/physreva.82.022102 fatcat:tyhhrfgiunc4xhb7c3yh6z6rci

Efficient Hardware Implementation of the LEDAcrypt Decoder

Kristjane Koleci, Paolo Santini, Marco Baldi, Franco Chiaraluce, Maurizio Martina, Guido Masera
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 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  ... 
doi:10.1109/access.2021.3076245 fatcat:tfus2ptqrbflhigu6eyjrstkma

An Efficient Maximum-Likelihood Decoding of LDPC Codes Over the Binary Erasure Channel

D. Burshtein, G. Miller
2004 IEEE Transactions on Information Theory  
We propose an efficient maximum likelihood decoding 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.  ... 
doi:10.1109/tit.2004.836694 fatcat:gt6r54gcp5d5za3epgxvek2jx4

Relative class number of imaginary Abelian fields of prime conductor below 10000

M. A. Shokrollahi
1999 Mathematics of Computation  
This problem can be solved efficiently with the following strategy: we use the Schönhage-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.  ... 
doi:10.1090/s0025-5718-99-01139-4 fatcat:iuk4awfbpnadbp4eipr36xq7bm

Evaluation and comparison of two efficient probabilistic primality testing algorithms

Louis Monier
1980 Theoretical Computer Science  
We analyse two recent probabiktic primality testing 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).  ... 
doi:10.1016/0304-3975(80)90007-9 fatcat:auxd53ib4bbnjku2cxien6xp5m

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.  ... 

An Optimized Quantum Implementation of ISD on Scalable Quantum Resources [article]

Andre Esser, Sergi Ramos-Calderer, Emanuele Bellini, José I. Latorre, Marc Manzano
2021 arXiv   pre-print
The security of code based constructions is usually assessed by Information Set Decoding (ISD) 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.  ... 
arXiv:2112.06157v1 fatcat:lggamxkp4fbj3bpvpwyslk4uui

Low-Complexity Implementation of RAID Based on Reed-Solomon Codes

P. Trifonov
2015 ACM Transactions on Storage  
The performance results show that the new 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.  ... 
doi:10.1145/2700308 fatcat:oqxquanplnhrxd2vv47c6pr6mm

Lower bounds for algebraic computation trees

Michael Ben-Or
1983 Proceedings of the fifteenth annual ACM symposium on Theory of computing - STOC '83  
This extends the result due to 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.  ... 
doi:10.1145/800061.808735 dblp:conf/stoc/Ben-Or83 fatcat:lwhhw5siejedjagvnnvadm2oeq

Infinitely Many Carmichael Numbers for a Modified Miller-Rabin Prime Test [article]

Eric Bach, Rex Fernando
2015 arXiv   pre-print
We show that the test can also be thought of as a variant of the Solovay-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.  ... 
arXiv:1512.00444v2 fatcat:ylsztuwesffktf5y5co6ggznqy

A Simple Unpredictable Pseudo-Random Number Generator

L. Blum, M. Blum, M. Shub
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 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).  ... 
doi:10.1137/0215025 fatcat:gueghdrpfveonc7mkgaxjfqovm

How Random Is Quantum Randomness? An Experimental Approach [article]

Cristian S. Calude, Michael J. Dinneen, Monica Dumitrescu, Karl Svozil
2009 arXiv   pre-print
We base our experimental approach on 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.  ... 
arXiv:0912.4379v1 fatcat:y3523r33nffrpemh3awnbbxkgu

Congruent Number Theta Coefficients to 1012 [chapter]

William B. Hart, Gonzalo Tornaría, Mark Watkins
2010 Lecture Notes in Computer Science  
In the computation using the second algorithm, a different check was performed.  ...  In 1971 Schönhage and Strassen presented two algorithms for multiplication of large integers based on the FFT [33] .  ... 
doi:10.1007/978-3-642-14518-6_17 fatcat:ciztklmtmrbezbekcwohpic2b4
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