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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.  ... 

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

Antoine Joux, Reynald Lercier
2002 Mathematics of Computation  
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  
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  
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

Jianhui LI
2017 IOSR Journal of Mathematics  
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

B.R. Ambedkar, Ashwani Gupta, Pratiksha Gautam, S.S. Bedi
2011 2011 International Conference on Communication Systems and Network Technologies  
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

KAM HUNG YAU
2020 Bulletin of the Australian Mathematical Society  
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

Joseph Zalaket, Joseph Hajj-Boutros
2011 Computers and Mathematics with Applications  
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}

Aaron Schlafly, Stan Wagon
1994 Mathematics of Computation  
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]

Hatsukazu Tanaka
1988 Lecture Notes in Computer Science  
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]

Charles W. Neville
2003 arXiv   pre-print
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  
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]

Paul C. van Oorschot
1991 Lecture Notes in Computer Science  
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  
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  
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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