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Page 7263 of Mathematical Reviews Vol. , Issue 96m
[page]
1996
Mathematical Reviews
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The first new ideas for NFS appeared in a manuscript published here as Pollard’s paper on factoring with cubic integers. These ideas suffice to factor numbers of the form x* +k efficiently. ...
Then one looks for pairs of small coprime integers a and b such that both the integer a + bm and the norm of the algebraic integer a + ba have only small prime factors. ...
Improvements to the general number field sieve for discrete logarithms in prime fields. A comparison with the gaussian integer method
2002
Mathematics of Computation
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Our main contribution consists of a new way to compute individual logarithms with the number field sieve without solving a very large linear system for each logarithm. ...
We show that, with these improvements, the number field sieve outperforms the gaussian integer method in the hundred digit range. ...
Introduction Since their introduction, index calculus techniques have become the methods of choice for factoring large integers and for computing discrete logarithms in finite fields. ...
doi:10.1090/s0025-5718-02-01482-5
fatcat:7ueg4tilh5erhdctxkecxvuovu
Page 3055 of Mathematical Reviews Vol. , Issue 2002E
[page]
2002
Mathematical Reviews
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Let ¢ > 0 and let m, k, and a be integers with m suffi- ciently large and k > (logm)****. ...
(If this were not the case, then he shows that the pigeonhole principle would imply the existence of an auxiliary integer with far too many prime factors p; for its size. )
The result established in this ...
Page 4350 of Mathematical Reviews Vol. , Issue 94h
[page]
1994
Mathematical Reviews
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This paper introduces two new ideas for improving the theoret- ical complexity of the algorithm. ...
Summary: “We present a new deterministic factorization algorithm for polynomials over a finite prime field F,. As in other factor- ization algorithms for polynomials over finite fields such as the ...
Algorithm Design and Implementation for a Mathematical Model of Factoring Integers
2017
IOSR Journal of Mathematics
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Based on an approximate formula of factoring an odd composite number, the article deduces a distribution for factors in big odd composite number and designs an algorithm to pick up the factors. ...
Experiment shows that the algorithm is as efficient as the Pullard's Rho algorithm for conventional numbers. ...
For this purpose, this article combines the idea that was raised in article [3] and the theory that was put forward in article [7] and realizes an approach to factorize integers. ...
doi:10.9790/5728-1301063741
fatcat:yizhhntumvbrvl3gqxlouc5nla
An Efficient Method to Factorize the RSA Public Key Encryption
2011
2011 International Conference on Communication Systems and Network Technologies
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The factorization of N is very intricate. In this paper a New Factorization method is proposed to obtain the factor of positive integer N. ...
The proposed work focuses on factorization of all trivial and nontrivial integer numbers and requires fewer steps for factorization process of RSA modulus N. ...
Hence, factoring large primes is a laborious and complex task [2] . A method for factoring algorithm (specially designed) for semi primes based on new mathematical ideas. ...
doi:10.1109/csnt.2011.29
fatcat:vogxz35iu5h3ljth564sfd3k4e
DISTRIBUTION OF INTEGERS WITH PRESCRIBED STRUCTURE AND APPLICATIONS
2020
Bulletin of the Australian Mathematical Society
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In Chapter 4, we derive new lower bounds on the number of smooth square-free integers up to x in residue classes modulo a prime p, relatively large compared to x, which in some ranges of p and x improve ...
We establish estimates for the number of ways to represent any reduced residue class as a product of a prime and an integer free of small prime factors. ...
doi:10.1017/s0004972720001112
fatcat:4o3kpuzuqbf3famtghuwyyf7v4
Prime factorization using square root approximation
2011
Computers and Mathematics with Applications
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Many cryptosystems are based on the factorization of large integers. ...
The security of RSA relies on the difficulty of factoring large integers. ...
We propose a new heuristic method based on the square root approximation that allows factoring large integers. ...
doi:10.1016/j.camwa.2011.02.027
fatcat:hdjphnv6hncl7k5niuwwuyroae
Carmichael's Conjecture on the Euler Function is Valid Below 10^{10,000, 000}
1994
Mathematics of Computation
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The main new idea is the application of a prime-certification technique that allows us to very quickly certify the primality of the thousands of large numbers that must divide a counterexample. ...
Carmichael's conjecture states that if 0(x) = n , then +(y) = n for some y :& x (q is Euler's totient function). We show that the conjecture is valid for all x under 1010,900,000 . ...
The idea for generating large lower bounds on x goes back to a theorem of Carmichael [3], later refined by Klee [5, 10]. ...
doi:10.2307/2153585
fatcat:6mrx75nlhbhu7gyst4wtgdq4xm
A Realization Scheme for the Identity-Based Cryptosystem
[chapter]
1988
Lecture Notes in Computer Science
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The basic idea of the scheme is based on the discrete logarithm problem and the difficulty of factoring a large integer composed of two large primes. ...
At the Crypto'84, Shamir has presented a new concept of the identity-based cryptosystem, but no idea is presented on the realization scheme. ...
The basic idea of the scheme is based on the two well-known one-way functions, i.e. a factorization of a large integer composed of two large primes, and a discrete logarithm. ...
doi:10.1007/3-540-48184-2_29
fatcat:ndrc557kw5bulkfa2h3drn6h4i
New Results on Primes from an Old Proof of Euler's
[article]
2003
arXiv
pre-print
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Our short paper uses a simple modification of Euler's argument to obtain new results about the distribution of prime factors of sets of integers, including a weak one-sided Tschebyshev inequality. ...
In 1737 Leonard Euler gave what we often now think of as a new proof, based on infinite series, of Euclid's theorem that there are infinitely many prime numbers. ...
This short paper uses a simple modification of Euler's argument to obtain new results about the distribution of prime factors of sets of integers. ...
arXiv:math/0210282v2
fatcat:borjs4x3ivc7vp2zfjhdoxgznq
Page 240 of Mathematics Teacher Vol. 32, Issue 5
[page]
1939
Mathematics Teacher
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kW El ments of the T heor / of Integers By Joseph Published by New York,
Joseph Bowden,
1931. x +268 pp. ...
Factors
Division (Juotient
Factors
. Greatest Common Factor.
9. Least Common Multiple.
LO. ...
A Comparison of Practical Public-Key Cryptosystems based on Integer Factorization and Discrete Logarithms
[chapter]
1991
Lecture Notes in Computer Science
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The security of the RSA and E1Gama.l cryptosystems is generally equated to the difficulty of integer factorization and that of the computation of discrete logarithms in finite fields, respectively. ...
Based on the current literature, this survey considers a detailed analysis of a version of the multiple polynomial quadratic sieve integer factorization algorithm [261, and a variation of the Coppersmith ...
TWO important new ideas that apply to the quadratic sieve have been demonstrated by A.K. Lenstra and Manasse. ...
doi:10.1007/3-540-38424-3_40
fatcat:we4bib4mwzagpjco34alhoqfhi
Page 4559 of Mathematical Reviews Vol. , Issue 88i
[page]
1988
Mathematical Reviews
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Then, the authors describe the special processor that they built for factoring large integers via the continued fraction algorithm: they name it EPOC, for Extended Precision Operand Computer; their main ...
{For the entire collection see MR 88h:68002.}
88i:11098 11Y05 11-04 11A51
Wagstaff, Samuel S., Jr. (1-PURD-C);
Smith, J. W. [Smith, Jeffrey W.] (1-GA-S)
Methods of factoring large integers. ...
Page 88 of None Vol. 6, Issue 9
[page]
1985
None
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There are no methods currently known for efficiently determin- ing factors of a large nonprime integer. ...
This cooperative spirit can be seen in the current state of the art in factoring large integers. It is easy to tell, at least with very high probability, whether or not a large integer is a prime. ...
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