18,250 research outputs found
Integer factorization as subset-sum problem
This paper elaborates on a sieving technique that has first been applied in
2018 for improving bounds on deterministic integer factorization. We will
generalize the sieve in order to obtain a polynomial-time reduction from
integer factorization to a specific instance of the multiple-choice subset-sum
problem. As an application, we will improve upon special purpose factorization
algorithms for integers composed of divisors with small difference. In
particular, we will refine the runtime complexity of Fermat's factorization
algorithm by a large subexponential factor. Our first procedure is
deterministic, rigorous, easy to implement and has negligible space complexity.
Our second procedure is heuristically faster than the first, but has
non-negligible space complexity.Comment: 22 pages (including appendix
A New Class of Unsafe Primes
In this paper,
a new special-purpose factorization algorithm is presented,
which finds a prime factor of an integer
in polynomial time, if has the form
where
and is an integer.
Hence such primes should be avoided when we select the
RSA secret keys. Some generalizations of the algorithm are
discussed in the paper as well
Note on Integer Factoring Methods IV
This note continues the theoretical development of deterministic integer
factorization algorithms based on systems of polynomials equations. The main
result establishes a new deterministic time complexity bench mark in integer
factorization.Comment: 20 Pages, New Versio
Deterministic elliptic curve primality proving for a special sequence of numbers
We give a deterministic algorithm that very quickly proves the primality or
compositeness of the integers N in a certain sequence, using an elliptic curve
E/Q with complex multiplication by the ring of integers of Q(sqrt(-7)). The
algorithm uses O(log N) arithmetic operations in the ring Z/NZ, implying a bit
complexity that is quasi-quadratic in log N. Notably, neither of the classical
"N-1" or "N+1" primality tests apply to the integers in our sequence. We
discuss how this algorithm may be applied, in combination with sieving
techniques, to efficiently search for very large primes. This has allowed us to
prove the primality of several integers with more than 100,000 decimal digits,
the largest of which has more than a million bits in its binary representation.
At the time it was found, it was the largest proven prime N for which no
significant partial factorization of N-1 or N+1 is known.Comment: 16 pages, corrected a minor sign error in 5.
- …