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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).
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
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.
New cube root algorithm based on third order linear recurrence relation in finite field, preprint, available from http://eprint.iacr
"... In this paper, we present a new cube root algorithm in finite field Fq with q a power of prime, which extends the CipollaLehmer type algorithms [4, 5]. Our cube root method is inspired by the work of Müller [8] on quadratic case. For given cubic residue c ∈ Fq with q ≡ 1 (mod 9), we show that there ..."
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In this paper, we present a new cube root algorithm in finite field Fq with q a power of prime, which extends the CipollaLehmer type algorithms [4, 5]. Our cube root method is inspired by the work of Müller [8] on quadratic case. For given cubic residue c ∈ Fq with q ≡ 1 (mod 9), we show that there is an irreducible polynomial f(x) = x 3 − ax 2 + bx − 1 with root α ∈ Fq3 efficient cube root algorithm based on third order linear recurrence sequence arising from f(x). Complexity estimation shows that our algorithm is better than previously proposed CipollaLehmer type algorithms. such that T r(α q2 +q−2 9) is a cube root of c. Consequently we find an
Square Root Algorithm in Fq for q ≡ 2 s + 1 (mod 2 s+1)
"... We present a square root algorithm in Fq which generalizes Atkins’s square root algorithm [6] for q ≡ 5 (mod 8) and Kong et al.’s algorithm [8] for q ≡ 9 (mod 16). Our algorithm precomputes a primitive 2 sth root of unity ξ where s is the largest positive integer satisfying 2 s q − 1, and is appli ..."
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We present a square root algorithm in Fq which generalizes Atkins’s square root algorithm [6] for q ≡ 5 (mod 8) and Kong et al.’s algorithm [8] for q ≡ 9 (mod 16). Our algorithm precomputes a primitive 2 sth root of unity ξ where s is the largest positive integer satisfying 2 s q − 1, and is applicable for the cases when s is small. The proposed algorithm requires one exponentiation for square root computation and is favorably compared with the algorithms of Atkin, Müller and Kong et al.
On rth Root Extraction Algorithm in Fq For q ≡ lr s + 1 (mod r s+1) with 0 < l < r and Small s
"... We present an rth root extraction algorithm over a finite field Fq. Our algorithm precomputes a primitive r sth root of unity ξ where s is the largest positive integer satisfying r s q − 1, and is applicable for the cases when s is small. The proposed algorithm requires one exponentiation for the ..."
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We present an rth root extraction algorithm over a finite field Fq. Our algorithm precomputes a primitive r sth root of unity ξ where s is the largest positive integer satisfying r s q − 1, and is applicable for the cases when s is small. The proposed algorithm requires one exponentiation for the rth root computation and is favorably compared to the existing algorithms.
Trace Expression of rth Root over Finite Field Email:
"... Efficient computation of rth root in Fq has many applications in computational number theory and many other related areas. We present a new rth root formula which generalizes Müller’s result on square root, and which provides a possible improvement of the CipollaLehmer type algorithms for general ..."
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Efficient computation of rth root in Fq has many applications in computational number theory and many other related areas. We present a new rth root formula which generalizes Müller’s result on square root, and which provides a possible improvement of the CipollaLehmer type algorithms for general case. More precisely, for given rth power c ∈ Fq, we show that there exists α ∈ Fqr ( such that T r α (∑r−1 i=0 qi)−r r2)r = c where T r(α) = α + αq + αq2 + · · · + αqr−1 and α is a root of certain irreducible polynomial of degree r over Fq.