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Page 7263 of Mathematical Reviews Vol. , Issue 96m
[page]

1996
*
Mathematical Reviews
*

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*. ...##
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Improvements to the general number field sieve for discrete logarithms in prime fields. A comparison with the gaussian integer method

2002
*
Mathematics of Computation
*

Our main contribution consists of a

doi:10.1090/s0025-5718-02-01482-5
fatcat:7ueg4tilh5erhdctxkecxvuovu
*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. ...##
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Page 3055 of Mathematical Reviews Vol. , Issue 2002E
[page]

2002
*
Mathematical Reviews
*

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 ...##
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Page 4350 of Mathematical Reviews Vol. , Issue 94h
[page]

1994
*
Mathematical Reviews
*

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 ...##
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Algorithm Design and Implementation for a Mathematical Model of Factoring Integers

2017
*
IOSR Journal of Mathematics
*

Based on an approximate formula of

doi:10.9790/5728-1301063741
fatcat:yizhhntumvbrvl3gqxlouc5nla
*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*. ...##
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An Efficient Method to Factorize the RSA Public Key Encryption

2011
*
2011 International Conference on Communication Systems and Network Technologies
*

The

doi:10.1109/csnt.2011.29
fatcat:vogxz35iu5h3ljth564sfd3k4e
*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*. ...##
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DISTRIBUTION OF INTEGERS WITH PRESCRIBED STRUCTURE AND APPLICATIONS

2020
*
Bulletin of the Australian Mathematical Society
*

In Chapter 4, we derive

doi:10.1017/s0004972720001112
fatcat:4o3kpuzuqbf3famtghuwyyf7v4
*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*. ...##
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Prime factorization using square root approximation

2011
*
Computers and Mathematics with Applications
*

Many cryptosystems are based on the

doi:10.1016/j.camwa.2011.02.027
fatcat:hdjphnv6hncl7k5niuwwuyroae
*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*. ...##
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Carmichael's Conjecture on the Euler Function is Valid Below 10^{10,000, 000}

1994
*
Mathematics of Computation
*

The main

doi:10.2307/2153585
fatcat:6mrx75nlhbhu7gyst4wtgdq4xm
*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]. ...##
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A Realization Scheme for the Identity-Based Cryptosystem
[chapter]

1988
*
Lecture Notes in Computer Science
*

The basic

doi:10.1007/3-540-48184-2_29
fatcat:ndrc557kw5bulkfa2h3drn6h4i
*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. ...##
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New Results on Primes from an Old Proof of Euler's
[article]

2003
*
arXiv
*
pre-print

Our short paper uses a simple modification of Euler's argument to obtain

arXiv:math/0210282v2
fatcat:borjs4x3ivc7vp2zfjhdoxgznq
*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*. ...##
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Page 240 of Mathematics Teacher Vol. 32, Issue 5
[page]

1939
*
Mathematics Teacher
*

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. ...##
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A Comparison of Practical Public-Key Cryptosystems based on Integer Factorization and Discrete Logarithms
[chapter]

1991
*
Lecture Notes in Computer Science
*

The security of the RSA and E1Gama.l cryptosystems is generally equated to the difficulty of

doi:10.1007/3-540-38424-3_40
fatcat:we4bib4mwzagpjco34alhoqfhi
*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. ...##
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Page 4559 of Mathematical Reviews Vol. , Issue 88i
[page]

1988
*
Mathematical Reviews
*

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*. ...##
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Page 88 of None Vol. 6, Issue 9
[page]

1985
*
None
*

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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