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On The Oracle Complexity Of Factoring Integers
 COMPUTATIONAL COMPLEXITY
, 1996
"... The problem of factoring integers in polynomial time with the help of an (infinitely powerful) oracle who answers arbitrary questions with yes or no is considered. The goal is to minimize the number of oracle questions. Let N be a given composite nbit integer to be factored, where n = dlog 2 ..."
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The problem of factoring integers in polynomial time with the help of an (infinitely powerful) oracle who answers arbitrary questions with yes or no is considered. The goal is to minimize the number of oracle questions. Let N be a given composite nbit integer to be factored, where n = dlog 2 Ne. The trivial method of asking for the bits of the smallest prime factor of N requires n/2 questions in the worst case. A nontrivial algorithm of Rivest and Shamir requires only n/3 questions for the special case where N is the product of two n/2bit primes. In this paper, a polynomialtime oracle factoring algorithm for general integers is presented which, for any ffl ? 0, asks at most ffln oracle questions for sufficiently large N , thus solving an open problem posed by Rivest and Shamir. Based on a plausible conjecture related to Lenstra's conjecture on the running time of the elliptic curve factoring algorithm it is shown that the algorithm fails with probability at most N ...
COMPRESSION IN FINITE FIELDS AND TORUSBASED CRYPTOGRAPHY
"... This paper is dedicated to the memory of the cat Ceilidh. Abstract. We present efficient compression algorithms for subgroups of multiplicative groups of finite fields, we use our compression algorithms to construct efficient public key cryptosystems called T2 and CEILIDH, we disprove some conjectur ..."
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This paper is dedicated to the memory of the cat Ceilidh. Abstract. We present efficient compression algorithms for subgroups of multiplicative groups of finite fields, we use our compression algorithms to construct efficient public key cryptosystems called T2 and CEILIDH, we disprove some conjectures, and we use the theory of algebraic tori to give a better understanding of our cryptosystems, the Lucasbased, XTR and GongHarn cryptosystems, and conjectured generalizations. 1.
A New Class of Unsafe Primes
"... In this paper, a new specialpurpose factorization algorithm is presented, which finds a prime factor p of an integer n in polynomial time, if 4p − 1 has the form db 2 ..."
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In this paper, a new specialpurpose factorization algorithm is presented, which finds a prime factor p of an integer n in polynomial time, if 4p − 1 has the form db 2
The Extended Riemann Hypothesis and its Application to Computation
, 2003
"... Many of Hilbert’s 23 famous problems are not of a prove or disprove nature; rather, they are openended, “of a purely investigative nature,” ..."
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Many of Hilbert’s 23 famous problems are not of a prove or disprove nature; rather, they are openended, “of a purely investigative nature,”
A DETERMINISTIC VERSION OF POLLARD’S p − 1 ALGORITHM
"... Abstract. In this article we present applications of smooth numbers to the unconditional derandomization of some wellknown integer factoring algorithms. We begin with Pollard’s p − 1 algorithm, which finds in random polynomial time the prime divisors p of an integer n such that p − 1issmooth.Weshow ..."
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Abstract. In this article we present applications of smooth numbers to the unconditional derandomization of some wellknown integer factoring algorithms. We begin with Pollard’s p − 1 algorithm, which finds in random polynomial time the prime divisors p of an integer n such that p − 1issmooth.Weshow that these prime factors can be recovered in deterministic polynomial time. We further generalize this result to give a partial derandomization of the kth cyclotomic method of factoring (k ≥ 2) devised by Bach and Shallit. We also investigate reductions of factoring to computing Euler’s totient function ϕ. We point out some explicit sets of integers n that are completely factorable in deterministic polynomial time given ϕ(n). These sets consist, roughly speaking, of products of primes p satisfying, with the exception of at most two, certain conditions somewhat weaker than the smoothness of p − 1. Finally, we prove that O(ln n) oracle queries for values of ϕ are sufficient to completely factor any integer n in less than exp (1 + o(1))(ln n) 1 3 (ln ln n) 2) 3 deterministic time. 1.
AMS Math Review Number 94d:11103.
"... A positive integer n is a perfect power if there exist integers x and k, both at least 2, such that n = x k. The usual algorithm to recognize perfect powers computes approximate kth roots for k ≤ log 2 n, and runs in time O(log 3 n log log log n). First, we improve this worstcase running time to O( ..."
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A positive integer n is a perfect power if there exist integers x and k, both at least 2, such that n = x k. The usual algorithm to recognize perfect powers computes approximate kth roots for k ≤ log 2 n, and runs in time O(log 3 n log log log n). First, we improve this worstcase running time to O(log 3 n) by using a modified Newton’s method to compute approximate kth roots. Parallelizing this gives an N C 2 algorithm. Second, we present a sieve algorithm that avoids kth root computations by seeing if the input n is a perfect kth power modulo small primes. If n is chosen uniformly from a large enough interval, the average running time is O(log 2 n). Third, we incorporate trial division to give a sieve algorithm with an average running time of O(log 2 n / log 2 log n) and a median running time of O(log n). The two sieve algorithms use a precomputed table of small primes. We give a heuristic argument and computational evidence that the largest prime needed in this table is (log n) 1+o(1) ; assuming the Extended Riemann Hypothesis, primes up to (log n) 2+o(1) suffice. The table can be computed in time roughly proportional to the largest prime it contains. We also present computational results indicating that our sieve algorithms perform extremely well in practice.
A New SpecialPurpose Factorization Algorithm
"... In this paper, a new factorization algorithm is presented, which finds a prime factor p of an integer n in time (D log n) , if 4p 1 = Db where D and b are integers. Hence this algorithm will factor a number efficiently, if it has a prime factor p such that 4p1 is a product of a small in ..."
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In this paper, a new factorization algorithm is presented, which finds a prime factor p of an integer n in time (D log n) , if 4p 1 = Db where D and b are integers. Hence this algorithm will factor a number efficiently, if it has a prime factor p such that 4p1 is a product of a small integer and a square. Such primes should be avoided when we select the RSA secret keys. Some generalizations of the algorithm are discussed in the paper as well.
Abelian Groups, Gauß Periods, and Normal Bases
"... . A result on finite abelian groups is first proved and then used to solve problems in finite fields. Particularly, all finite fields that have normal bases generated by general Gauss periods are characterized and it is shown how to find normal bases of low complexity. Dedicated to Professor Chao Ko ..."
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. A result on finite abelian groups is first proved and then used to solve problems in finite fields. Particularly, all finite fields that have normal bases generated by general Gauss periods are characterized and it is shown how to find normal bases of low complexity. Dedicated to Professor Chao Ko on his 90th birthday. 1. Introduction and main results We first prove a result on finite abelian groups. We use the standard notation < S, K > for the subgroup generated by the elements in S and K together, and G/K, or G K , for the quotient group of G by K. Theorem 1.1. Let G be any finite abelian group. Let S be a subset and K a subgroup of G such that G =< S, K >. Then, for any direct product G = G 1# G 2# # G t , there is a subgroup H of the form H = H 1# H 2# # H t , H i #G i , 1 # i # t, such that G =< S, H > and G H # = G K . Next we apply this theorem to some problems in finite fields that arise in the work of Feisel et al [7] on constructing normal bases ...
Draft. Aimed at Math. Comp. I’m rewriting [8] in light of this. HOW TO FIND SMOOTH PARTS OF INTEGERS
"... Abstract. Let P be a finite set of primes, and let S be a finite sequence of positive integers. This paper presents an algorithm to find the largest Psmooth divisor of each integer in S. The algorithm takes time b(lg b) 2+o(1), where b is the total number of bits in P and S. A previous algorithm by ..."
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Abstract. Let P be a finite set of primes, and let S be a finite sequence of positive integers. This paper presents an algorithm to find the largest Psmooth divisor of each integer in S. The algorithm takes time b(lg b) 2+o(1), where b is the total number of bits in P and S. A previous algorithm by the author takes time b(lg b) 3+o(1) to find all the factors from P of each integer in S; a variant by Franke, Kleinjung, Morain, and Wirth usually takes time b(lg b) 2+o(1) to find the largest Psmooth divisor of each integer in S; the algorithm in this paper always takes time b(lg b) 2+o(1) to find the largest Psmooth divisor of each integer in S. Positive integer x batch time b(lg b) 3+o(1) (Bernstein 2000)
DEDICATED TO CHAO KO FOR HIS 90TH BIRTHDAY
, 2000
"... We exhibit a deterministic algorithm for factoring polynomials in one variable over "nite "elds. It is e$cient only if a positive integer k is known for which Φ (p) is built up from small prime factors; here Φ denotes the kth cyclotomic polynomial, and p is the characteristic of the " ..."
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We exhibit a deterministic algorithm for factoring polynomials in one variable over "nite "elds. It is e$cient only if a positive integer k is known for which Φ (p) is built up from small prime factors; here Φ denotes the kth cyclotomic polynomial, and p is the characteristic of the "eld. In the case k"1, when Φ (p)"p!1, such an algorithm was known, and its analysis required the generalized Riemann hypothesis. Our algorithm depends on a similar, but weaker, assumption; speci"cally, the algorithm requires the availability of an irreducible polynomial of degree r over Z/pZ for each prime number r for which Φ (p) has a prime factor l with l,1 mod r. An auxiliary procedure is devoted to the construction of roots of unity by means of Gauss sums. We do not claim that our algorithm has any practical value. � 2000 Academic Press Key =ords: "nite "eld; algorithm; factoring polynomials; Gauss sum. 5