Topic
Jacobi sum
About: Jacobi sum is a research topic. Over the lifetime, 74 publications have been published within this topic receiving 1749 citations.
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Book•
1 Jan 1985
TL;DR: In this paper, Mapes' algorithm is used to construct a compact prime table, which is then used to test compositeness of numbers of the form N = h * 2n +-k.
Abstract: 1. The Number of Primes Below a Given Limit.- What Is a Prime Number?.- The Fundamental Theorem of Arithmetic.- Which Numbers Are Primes? The Sieve of Eratosthenes.- General Remarks Concerning Computer Programs.- A Sieve Program.- Compact Prime Tables.- Hexadecimal Compact Prime Tables.- Difference Between Consecutive Primes.- The Number of Primes Below x.- Meissel's Formula.- Evaluation of Pk(x, a).- Lehmer's Formula.- Computations.- A Computation Using Meissel's Formula.- A Computation Using Lehmer's Formula.- A Computer Program Using Lehmer's Formula.- Mapes' Method.- Deduction of Formulas.- A Worked Example.- Mapes' Algorithm.- Programming Mapes' Algorithm.- Recent Developments.- Results.- Computational Complexity.- Comparison Between the Methods Discussed.- 2. The Primes Viewed at Large.- No Polynomial Can Produce Only Primes.- Formulas Yielding All Primes.- The Distribution of Primes Viewed at Large. Euclid's Theorem.- The Formulas of Gauss and Legendre for ?(x). The Prime Number Theorem.- The Chebyshev Function ?(x).- The Riemann Zeta-function.- The Zeros of the Zeta-function.- Conversion From f(x) Back to ?(x).- The Riemann Prime Number Formula.- The Sign of li x ? ?(x).- The Influence of the Complex Zeros of ?(s) on ?(x).- The Remainder Term in the Prime Number Theorem.- Effective Inequalities for ?(x), pn, and ?(x).- The Number of Primes in Arithmetic Progressions.- 3. Subtleties in the Distribution of Primes.- The Distribution of Primes in Short Intervals.- Twins and Some Other Constellations of Primes.- Admissible Constellations of Primes.- The Hardy-Littlewood Constants.- The Prime k-Tuples Conjecture.- Theoretical Evidence in Favour of the Prime k-Tuples Conjecture.- Numerical Evidence in Favour of the Prime k-Tuples Conjecture.- The Second Hardy-Littlewood Conjecture.- The Midpoint Sieve.- Modification of the Midpoint Sieve.- Construction of Superdense Admissible Constellations.- Some Dense Clusters of Primes.- The Distribution of Primes Between the Two Series 4n + 1 and 4n + 3.- Graph of the Function ?4,3(x) ? ?4,1(x).- The Negative Regions.- The Negative Blocks.- Large Gaps Between Consecutive Primes.- The Cramer Conjecture.- 4. The Recognition of Primes.- Tests of Primality and of Compositeness.- Factorization Methods as Tests of Compositeness.- Fermat's Theorem as Compositeness Test.- Fermat's Theorem as Primality Test.- Pseudoprimes and Probable Primes.- A Computer Program for Fermat's Test.- The Labor Involved in a Fermat Test.- Carmichael Numbers.- Euler Pseudoprimes.- Strong Pseudoprimes and a Primality Test.- A Computer Program for Strong Pseudoprime Tests.- Counts of Pseudoprimes and Carmichael Numbers.- Rigorous Primality Proofs.- Lehmer's Converse of Fermat's Theorem.- Formal Proof of Theorem 4.3.- Ad Hoc Search for a Primitive Root.- The Use of Several Bases.- Fermat Numbers and Pepin's Theorem.- Cofactors of Fermat Numbers.- Generalized Fermat Numbers.- A Relaxed Converse of Fermat's Theorem.- Proth's Theorem.- Tests of Compositeness for Numbers of the form N = h * 2n +- k.- An Alternative Approach.- Certificates of Primality.- Primality Tests of Lucasian Type.- Lucas Sequences.- The Fibonacci Numbers.- Large Subscripts.- An Alternative Deduction.- Divisibility Properties of the Numbers Un.- Primality Proofs by Aid of Lucas Sequences.- Lucas Tests for Mersenne Numbers.- A Relaxation of Theorem 4.8.- Pocklington's Theorem.- Lehmer-Pocklington's Theorem.- Pocklington-Type Theorems for Lucas Sequences.- Primality Tests for Integers of the form N = h * 2n ? 1, when 3?h.- Primality Tests for N = h * 2n ? 1, when 3?h.- The Combined N ? 1 and N + 1 Test.- Lucas Pseudoprimes.- Modern Primality Proofs.- The Jacobi Sum Primality Test.- Three Lemmas.- Lenstra's Theorem.- The Sets P and Q.- Running Time for the APRCL Test.- Elliptic Curve Primality Proving, ECPP.- The Goldwasser-Kilian Test.- Atkin's Test.- 5. Classical Methods of Factorization.- When Do We Attempt Factorization?.- Trial Division.- A Computer Implementation of Trial Division.- Euclid's Algorithm as an Aid to Factorization.- Fermat's Factoring Method.- Legendre's Congruence.- Euler's Factoring Method.- Gauss' Factoring Method.- Legendre's Factoring Method.- The Number of Prime Factors of Large Numbers.- How Does a Typical Factorization Look?.- The Erd?s-Kac Theorem.- The Distribution of Prime Factors of Various Sizes.- Dickman's Version of Theorem 5.4.- A More Detailed Theory.- The Size of the kth Largest Prime Factor of N.- Smooth Integers.- Searching for Factors of Certain Forms.- Legendre's Theorem for the Factors of N = an +- bn.- Adaptation to Search for Factors of the Form p = 2kn + 1.- Adaptation of Trial Division.- Adaptation of Fermat's Factoring Method.- Adaptation of Euclid's Algorithm as an Aid to Factorization.- 6. Modem Factorization Methods.- Choice of Method.- Pollard's (p ? 1)-Method.- Phase 2 of the (p ? 1)-Method.- The (p + 1)-Method.- Pollard's rho Method.- A Computer Program for Pollard's rho Method.- An Algebraic Description of Pollard's rho Method.- Brent's Modification of Pollard's rho Method.- The Pollard-Brent Method for p = 2kn + 1.- Shanks' Factoring Method SQUFOF.- A Computer Program for SQUFOF.- Comparison Between Pollard's rho Method and SQUFOF.- Morrison and Brillhart's Continued Fraction Method CFRAC.- The Factor Base.- An Example of a Factorization with CFRAC.- Further Details of CFRAC.- The Early Abort Strategy.- Results Achieved with CFRAC.- Running Time Analysis of CFRAC.- The Quadratic Sieve, QS.- Smallest Solutions to Q(x) ? 0 mod p.- Special Factors.- Results Achieved with QS.- The Multiple Polynomial Quadratic Sieve, MPQS.- Results Achieved with MPQS.- Running Time Analysis of QS and MPQS.- The Schnorr-Lenstra Method.- Two Categories of Factorization Methods.- Lenstra's Elliptic Curve Method, ECM.- Phase 2 of ECM.- The Choice of A, B, and P1.- Running Times of ECM.- Recent Results Achieved with ECM.- The Number Field Sieve, NFS.- Factoring Both in Z and in Z(z).- A Numerical Example.- The General Number Field Sieve, GNFS.- Running Times of NFS and GNFS.- Results Achieved with NFS. Factorization of F9.- Strategies in Factoring.- How Fast Can a Factorization Algorithm Be?.- 7. Prime Numbers and Cryptography.- Practical Secrecy.- Keys in Cryptography.- Arithmetical Formulation.- RSA Cryptosystems.- How to Find the Recovery Exponent.- A Worked Example.- Selecting Keys.- Finding Suitable Primes.- The Fixed Points of an RSA System.- How Safe is an RSA Cryptosystem?.- Superior Factorization.- Appendix 1. Basic Concepts in Higher Algebra.- Modules.- Euclid's Algorithm.- The Labor Involved in Euclid's Algorithm.- A Definition Taken from the Theory of Algorithms.- A Computer Program for Euclid's Algorithm.- Reducing the Labor.- Binary Form of Euclid's Algorithm.- Groups.- Lagrange's Theorem. Cosets.- Abstract Groups. Isomorphic Groups.- The Direct Product of Two Given Groups.- Cyclic Groups.- Rings.- Zero Divisors.- Fields.- Mappings. Isomorphisms and Homomorphisms.- Group Characters.- The Conjugate or Inverse Character.- Homomorphisms and Group Characters.- Appendix 2. Basic Concepts in Higher Arithmetic.- Divisors. Common Divisors.- The Fundamental Theorem of Arithmetic.- Congruences.- Linear Congruences.- Linear Congruences and Euclid's Algorithm.- Systems of Linear Congruences.- Carmichael's Function.- Carmichael's Theorem.- Appendix 3. Quadratic Residues.- Legendre's Symbol.- Arithmetic Rules for Residues and Non-Residues.- The Law of Quadratic Reciprocity.- Jacobi's Symbol.- Appendix 4. The Arithmetic of Quadratic Fields.- Appendix 5. Higher Algebraic Number Fields.- Algebraic Numbers.- Appendix 6. Algebraic Factors.- Factorization of Polynomials.- The Cyclotomic Polynomials.- Aurifeuillian Factorizations.- Factorization Formulas.- The Algebraic Structure of Aurifeuillian Numbers.- Appendix 7. Elliptic Curves.- Cubics.- Rational Points on Rational Cubics.- Homogeneous Coordinates.- Elliptic Curves.- Rational Points on Elliptic Curves.- Appendix 8. Continued Fractions.- What Is a Continued Fraction?.- Regular Continued Fractions. Expansions.- Evaluating a Continued Fraction.- Continued Fractions as Approximations.- Euclid's Algorithm and Continued Fractions.- Linear Diophantine Equations and Continued Fractions.- A Computer Program.- Continued Fraction Expansions of Square Roots.- Proof of Periodicity.- The Maximal Period-Length.- Short Periods.- Continued Fractions and Quadratic Residues.- Appendix 9. Multiple-Precision Arithmetic.- Various Objectives for a Multiple-Precision Package.- How to Store Multi-Precise Integers.- Addition and Subtraction of Multi-Precise Integers.- Reduction in Length of Multi-Precise Integers.- Multiplication of Multi-Precise Integers.- Division of Multi-Precise Integers.- Input and Output of Multi-Precise Integers.- A Complete Package for Multiple-Precision Arithmetic.- A Computer Program for Pollard's rho Method.- Appendix 10. Fast Multiplication of Large Integers.- The Ordinary Multiplication Algorithm.- Double Length Multiplication.- Recursive Use of Double Length Multiplication Formula.- A Recursive Procedure for Squaring Large Integers.- Fractal Structure of Recursive Squaring.- Large Mersenne Primes.- Appendix 11. The Stieltjes Integral.- Functions With Jump Discontinuities.- The Riemann Integral.- Definition of the Stieltjes Integral.- Rules of Integration for Stieltjes Integrals.- Integration by Parts of Stieltjes Integrals.- The Mean Value Theorem.- Applications.- Tables. For Contents.- List of Textbooks.
478 citations
TL;DR: Using the LLL-algorithm for finding short vectors in lattices, it is shown how to compute a Jacobi sum for the prime field Fp in Q(e2πi/n) in time O(log3p), useful in the construction of hyperelliptic cryptosystems.
Abstract: Using the LLL-algorithm for finding short vectors in lattices, we show how to compute a Jacobi sum for the prime field Fp in Q(e2πi/n) in time O(log3p), where n is small and fixed, p is large, and p = 1 (mod n). This result is useful in the construction of hyperelliptic cryptosystems.
63 citations
TL;DR: All strong pseudoprimes (spsp's) n < 10 24 to the first ten prime bases 2,3, ..., 29, which have the form n = pq with p,q odd primes and q- 1 = k(p- 1), k = 2, 3, 4 are tabulated.
Abstract: Define ψ m to be the smallest strong pseudoprime to all the first m prime bases. If we know the exact value of ψ m , we will have, for integers n < ψ m , a deterministic primality testing algorithm which is not only easier to implement but also faster than either the Jacobi sum test or the elliptic curve test. Thanks to Pornerance et al. and Jaeschke, ψ m are known for 1 ≤ m ≤ 8. Upper bounds for ψ 9 , ψ 10 and ψ 1 were given by Jaeschke. In this paper we tabulate all strong pseudoprimes (spsp's) n < 10 24 to the first ten prime bases 2,3, ..., 29, which have the form n = pq with p,q odd primes and q- 1 = k(p- 1), k = 2, 3, 4. There are in total 44 such numbers, six of which are also spsp(31), and three numbers are spsp's to both bases 31 and 37. As a result the upper bounds for ψ 10 and ψ 11 are lowered from 28- and 29-decimal-digit numbers to 22-decimal-digit numbers, and a 24-decimal-digit upper bound for ψ 12 is obtained. The main tools used in our methods are the biquadratic residue characters and cubic residue characters. We propose necessary conditions for n to be a strong pseudoprime to one or to several prime bases. Comparisons of effectiveness with both Jaeschke's and Arnault's methods are given.
32 citations
1 Mar 1988
TL;DR: The connection between cubic Fermat curves and cubic Jacobi sums was first observed by Gauss [G], who used it to study such sums and showed that Frobenius is a Gauss or Jacobi sum.
Abstract: The following article offers an explanation of the relationship of Jacobi and Gauss sums to Fermat and Artin-Schreier curves which is an analogue of the proof of Stickleberger's theorem. 1. Correspondences. The connection between cubic Fermat curves and cubic Jacobi sums was first observed by Gauss [G], who used it to study such sums. That one can compute the number of points on a Fermat curve over a finite field using Jacobi sums has long been known. The same is true for Artin-Schreier curves and Gauss sums and were applied by Davenport and Hasse [D-H] to compute Artin L-functions associated with such curves. This computation, in turn, yielded the Hasse-Davenport identity. Ultimately, Weil [W] interpreted these computations via a Lifshetz fixed point formula by showing that the eigenvalues of Frobenius acting on what is now known as the etale cohomology groups of Fermat curves are Jacobi sums and on Artin-Schreier curves are Gauss sums. This can be interpreted as an identity which says, in a suitable sense, that Frobenius is a Gauss or Jacobi sum (see below). One can now use this to produce annihilators of the divisor class groups of these curves which are also given by the Brumer-Stark conjecture for function fields proven in [T]. Below, we will give an elementary proof of the aforementioned identity which is analogous to Stickelberger's proof of Stickelberger's theorem. We just write down especially simple functions with appropriate divisors. Let p be a rational prime and m an integer prime to p. Let K be a field of characteristic p. Let Am and Fm denote the complete nonsingular curves over K with affine equations: A: yP-y=xm, Fm: um+vm=1. Suppose q = pf =_ 1 mod m and suppose K D Fq. Define the homomorphisms V): Fq -Aut(A,,), X: Fq -Aut(A,,), X0, Xl: Fq Aut(Fm) by 4)(a)(x,y) = (x,y+TFq/Fp(a)), X(b)(x,y) =(b(q-l)/mX y) Xo(b)(u,v) = (b(q-l)/mu,v), Xi(b)(u,v) =(u,b(q )/ V) for a E F+ and b E F. Now recall that a correspondence between a curve X to a curve Y is a divisor Z on X x Y with no vertical or horizontal components. Among other things, Z gives Received by the editors April 15, 1986 and, in revised form, December 9, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 11G20. ?1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page
20 citations
TL;DR: In this paper, an asymptotic formula for the number of representations of an element of a finite field as a weighted sum of two prescribed powers of primitive elements is given.
15 citations
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