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18
An analysis of Shanks’s algorithm for computing square roots in finite fields
 in Proc. 5th Conf. Canadian Number Theory Assoc
, 1999
"... Abstract We rigorously analyze Shanks's algorithm for computing square roots modulo a prime number. The initialization always requires two exponentiations. Averaged over all primes and possible inputs, the body of the algorithm requires 8/3 additional multiplications. We obtain exact values for the ..."
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Abstract We rigorously analyze Shanks's algorithm for computing square roots modulo a prime number. The initialization always requires two exponentiations. Averaged over all primes and possible inputs, the body of the algorithm requires 8/3 additional multiplications. We obtain exact values for the mean and variance of the number of additional multiplications for a fixed prime, and finally show that the distribution is asymptotically normal.
On Solving Univariate Polynomial Equations over Finite Fields and Some Related Problems
, 2007
"... We show deterministic polynomial time algorithms over some family of finite fields for solving univariate polynomial equations and some related problems such as taking nth roots, constructing nth nonresidues, constructing primitive elements and computing elliptic curve “nth roots”. In additional, we ..."
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We show deterministic polynomial time algorithms over some family of finite fields for solving univariate polynomial equations and some related problems such as taking nth roots, constructing nth nonresidues, constructing primitive elements and computing elliptic curve “nth roots”. In additional, we present a deterministic polynomial time primality test for some family of integers. All algorithms can be proved by elementary means (without assuming any unproven hypothesis). The problem of solving polynomial equations over finite fields is a generalization of the following problems over finite fields • constructing primitive nth roots of unity, • taking nth roots, • constructing nth nonresidues, • constructing primitive elements (generators of the multiplicative group) for any positive n dividing the number of elements of the underlying field. By the TonelliShanks square root algorithm [21, 19] and its generalization for taking nth roots, constructing nth nonresidues
On taking square roots and constructing quadratic nonresidues over finite fields
, 2007
"... We present a novel idea to compute square roots over some families of finite fields. Our algorithms are deterministic polynomial time and can be proved by elementary means (without assuming any unproven hypothesis). In some particular finite fields Fq, there are algorithms for taking square roots wi ..."
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We present a novel idea to compute square roots over some families of finite fields. Our algorithms are deterministic polynomial time and can be proved by elementary means (without assuming any unproven hypothesis). In some particular finite fields Fq, there are algorithms for taking square roots with Õ(log2 q) bit operations. As an application of our square root algorithms, we show a deterministic primality testing algorithm for some form of numbers. For some positive integer N, this primality testing algorithm runs in Õ(log3 N).
Square Roots Modulo p
"... Abstract. The algorithm of Tonelli and Shanks for computing square roots modulo a prime number is the most used, and probably the fastest among the known algorithms when averaged over all prime numbers. However, for some particular prime numbers, there are other algorithms which are considerably fas ..."
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Abstract. The algorithm of Tonelli and Shanks for computing square roots modulo a prime number is the most used, and probably the fastest among the known algorithms when averaged over all prime numbers. However, for some particular prime numbers, there are other algorithms which are considerably faster. In this paper we compare the algorithm of Tonelli and Shanks with an algorithm based in quadratic field extensions due to Cipolla, and give an explicit condition on a prime number to decide which algorithm is faster. Finally, we show that there exists an infinite sequence of prime numbers for which the algorithm of Tonelli and Shanks is asymptotically worse. 1
On Taking Square Roots without Quadratic Nonresidues over Finite Fields
, 2009
"... We present a novel idea to compute square roots over finite fields, without being given any quadratic nonresidue, and without assuming any unproven hypothesis. The algorithm is deterministic and the proof is elementary. In some cases, the square root algorithm runs in Õ(log2 q) bit operations over f ..."
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We present a novel idea to compute square roots over finite fields, without being given any quadratic nonresidue, and without assuming any unproven hypothesis. The algorithm is deterministic and the proof is elementary. In some cases, the square root algorithm runs in Õ(log2 q) bit operations over finite fields with q elements. As an application, we construct a deterministic primality proving algorithm, which runs in Õ(log3 N) for some integers N. 1
NonParallelizable and NonInteractive Client Puzzles from Modular Square Roots
"... Abstract—Denial of Service (DoS) attacks aiming to exhaust the resources of a server by overwhelming it with bogus requests have become a serious threat. Especially protocols that rely on public key cryptography and perform expensive authentication handshakes may be an easy target. A wellknown coun ..."
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Abstract—Denial of Service (DoS) attacks aiming to exhaust the resources of a server by overwhelming it with bogus requests have become a serious threat. Especially protocols that rely on public key cryptography and perform expensive authentication handshakes may be an easy target. A wellknown countermeasure against DoS attacks are client puzzles. The victimized server demands from the clients to commit computing resources before it processes their requests. To get service, a client must solve a cryptographic puzzle and submit the right solution. Existing client puzzle schemes have some drawbacks. They are either parallelizable, coarsegrained or can be used only interactively. In case of interactive client puzzles where the server poses the challenge an attacker might mount a counterattack on the clients by injecting fake packets containing bogus puzzle parameters. In this paper we introduce a novel scheme for client puzzles which relies on the computation of square roots modulo a prime. Modular square root puzzles are nonparallelizable, i. e., the solution cannot be obtained faster than scheduled by distributing the puzzle to multiple machines or CPU cores, and they can be employed both interactively and noninteractively. Our puzzles provide polynomial granularity and compact solution and verification functions. Benchmark results demonstrate the feasibility of our approach to mitigate DoS attacks on hosts in 1 or even 10 GBit networks. In addition, we show how to raise the efficiency of our puzzle scheme by introducing a bandwidthbased cost factor for the client. Keywords—client puzzles, Denial of Service (DoS), network protocols, authentication, computational puzzles
A HighSpeed Square Root Algorithm in Extension Fields
"... Abstract. A square root (SQRT) algorithm in GF (p m)(m=r0r1 ···rn−12 d, ri: odd prime, d>0: integer) is proposed in this paper. First, the TonelliShanks algorithm is modified to compute the inverse SQRT in GF (p 2d where most of the computations are performed in the corresponding subfields GF (p 2 ..."
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Abstract. A square root (SQRT) algorithm in GF (p m)(m=r0r1 ···rn−12 d, ri: odd prime, d>0: integer) is proposed in this paper. First, the TonelliShanks algorithm is modified to compute the inverse SQRT in GF (p 2d where most of the computations are performed in the corresponding subfields GF (p 2i) for 0 i d − 1. Then the Frobenius mappings with an addition chain are adopted for the proposed SQRT algorithm, in which a lot of computations in a given extension field GF (p m) are also reduce to those in a proper subfield by the norm computations. Those reductions of the field degree increase efficiency in the SQRT implementation. More specifically the TonelliShanks algorithm and the proposed algorithm in GF (p 22), GF (p 44) and GF (p 88) were implemented on a Pentium4 (2.6 GHz) computer using the C++ programming language. The computer simulations showed that, on average, the proposed algorithm accelerates the SQRT computation by 25 times in GF (p 22), by 45 times in GF (p 44), and by 70 times in GF (p 88), compared to the TonelliShanks algorithm, which is supported by the evaluation of the number of computations. 1
STRUCTURE COMPUTATION AND DISCRETE LOGARITHMS IN FINITE ABELIAN pGROUPS
"... Abstract. We present a generic algorithm for computing discrete logarithms in a finite abelian pgroup H, improving the Pohlig–Hellman algorithm and its generalization to noncyclic groups by Teske. We then give a direct method to compute a basis for H without using a relation matrix. The problem of ..."
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Abstract. We present a generic algorithm for computing discrete logarithms in a finite abelian pgroup H, improving the Pohlig–Hellman algorithm and its generalization to noncyclic groups by Teske. We then give a direct method to compute a basis for H without using a relation matrix. The problem of computing a basis for some or all of the Sylow psubgroups of an arbitrary finite abelian group G is addressed, yielding a Monte Carlo algorithm to compute the structure of G using O(G  1/2) group operations. These results also improve generic algorithms for extracting pth roots in G. 1.
Taking Roots over High Extensions of Finite Fields
"... We present a new algorithm for computing mth roots over the finite field Fq, where q = p n, with p a prime, and m any positive integer. In the particular case m = 2, the cost of the new algorithm is an expected O(M(n) log(p) + C(n) log(n)) operations in Fp, where M(n) and C(n) are bounds for the co ..."
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We present a new algorithm for computing mth roots over the finite field Fq, where q = p n, with p a prime, and m any positive integer. In the particular case m = 2, the cost of the new algorithm is an expected O(M(n) log(p) + C(n) log(n)) operations in Fp, where M(n) and C(n) are bounds for the cost of polynomial multiplication and modular polynomial composition. Known results give M(n) = O(n log(n) log log(n)) and C(n) = O(n 1.67), so our algorithm is subquadratic in n.