DOI10.2307/1971363zbMath0629.10006OpenAlexW2005317356WikidataQ55899111 ScholiaQ55899111MaRDI QIDQ1093665
Hendrik W. jun. Lenstra
Publication date: 1987
Published in: Annals of Mathematics. Second Series (Search for Journal in Brave)
Full work available at URL: https://semanticscholar.org/paper/307ab08c3d4f551019297d2480597c614af8069c
On Group Structures Realized by Elliptic Curves over a Fixed Finite Field ⋮
Sieve algorithms for perfect power testing ⋮
Factoring multi-power RSA moduli with primes sharing least or most significant bits ⋮
Trapdoor DDH groups from pairings and isogenies ⋮
Optimal strong primes ⋮
Practical cryptanalysis of ISO 9796-2 and EMV signatures ⋮
Crypto Galore! ⋮
Fast generation of prime numbers and secure public-key cryptographic parameters. ⋮
Distribution of elliptic twin primes in isogeny and isomorphism classes ⋮
Factoring RSA moduli with primes sharing bits in the middle ⋮
Zeta functions, one-way functions, and pseudorandom number generators. ⋮
Elliptic curves in isogeny classes ⋮
Corrigendum: Elliptic curves with a given number of points over finite fields ⋮
Elliptic curves with large torsion and positive rank over number fields of small degree and ECM factorization ⋮
Sums of Euler products and statistics of elliptic curves ⋮
Variations in the distribution of principally polarized abelian varieties among isogeny classes ⋮
Sato-Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height ⋮
On efficient sparse integer matrix Smith normal form computations ⋮
On the oracle complexity of factoring integers ⋮
An integrated parallel GNFS algorithm for integer factorization based on linbox montgomery block Lanczos method over \(GF(2)\) ⋮
A verifiable secret shuffle of homomorphic encryptions ⋮
RSA: a number of formulas to improve the search for \(p+q\) ⋮
Small secret exponent attack on RSA variant with modulus \(N=p^rq\) ⋮
Computing the endomorphism ring of an ordinary abelian surface over a finite field ⋮
On the distribution of Atkin and Elkies primes ⋮
Analysis of the fault attack ECDLP over prime field ⋮
The distribution of the number of points modulo an integer on elliptic curves over finite fields ⋮
Spreading alerts quietly and the subgroup escape problem ⋮
Elliptic curve cryptography: the serpentine course of a paradigm shift ⋮
Computing the endomorphism ring of an ordinary elliptic curve over a finite field ⋮
Explicit methods in number theory. Abstracts from the workshop held July 18--24, 2021 (hybrid meeting) ⋮
Factoring multi power RSA moduli with a class of secret exponents ⋮
On oracle factoring of integers ⋮
Prime factorization using square root approximation ⋮
\(o\)-minimality and the André-Oort conjecture for \(\mathbb{C}^{n}\) ⋮
Some optimal codes from algebraic geometry and their covering radii ⋮
Points on curves in small boxes and applications ⋮
Elliptic curves, modular forms, and sums of Hurwitz class numbers ⋮
An elliptic curve analogue to the Fermat numbers ⋮
Elliptic curve implementation of zero-knowledge blobs ⋮
Computational complexity of sentences over fields ⋮
Use of SIMD-based data parallelism to speed up sieving in integer-factoring algorithms ⋮
Rigorous analysis of a randomised number field sieve ⋮
Generic hardness of inversion on ring and its relation to self-bilinear map ⋮
On powersmooth numbers ⋮
CRT-based fully homomorphic encryption over the integers ⋮
Metrics on the sets of nonsupersingular elliptic curves in simplified Weierstrass form over finite fields of characteristic two ⋮
Efficient randomized generation of optimal algorithms for multiplication in certain finite fields ⋮
Drinfeld modules with complex multiplication, Hasse invariants and factoring polynomials over finite fields ⋮
Polynomial time bounded distance decoding near Minkowski's bound in discrete logarithm lattices ⋮
Special prime numbers and discrete logs in finite prime fields ⋮
Cryptanalysis of ISO/IEC 9796-1 ⋮
Constructing elliptic curves over finite fields using double eta-quotients ⋮
Average Frobenius distributions for elliptic curves with 3-torsion ⋮
Numbers in a given set with (or without) a large prime factor ⋮
A new public key cryptosystem based on Edwards curves ⋮
Toric forms of elliptic curves and their arithmetic ⋮
On the ultimate complexity of factorials ⋮
Index calculation attacks on RSA signature and encryption ⋮
On RSA moduli with prescribed bit patterns ⋮
On the representation of the number of integral points of an elliptic curve modulo a prime number ⋮
On computing ord\(_{N}(2)\) and its application ⋮
New number-theoretic cryptographic primitives ⋮
Realistic analysis of some randomized algorithms ⋮
One-way permutations on elliptic curves ⋮
A Simple and Improved Algorithm for Integer Factorization with Implicit Hints ⋮
A new generalization of the KMOV cryptosystem ⋮
Factoring with hints ⋮
Quadratic sieving ⋮
Recovering Fourier coefficients of some functions and factorization of integer numbers ⋮
Strongly nonzero points and elliptic pseudoprimes ⋮
Some remarks on primality proving and elliptic curves ⋮
Smoothness testing of polynomials over finite fields ⋮
Arithmetic of the level four theta model of elliptic curves ⋮
Polynomial factorization over finite fields by computing Euler-Poincaré characteristics of Drinfeld modules ⋮
An improved parallel block Lanczos algorithm over GF(2) for integer factorization ⋮
Statistics about elliptic curves over finite prime fields ⋮
Approximating rings of integers in number fields ⋮
Fast group operations on elliptic curves in Maple ⋮
Large-Order Multiple Recursive Generators with Modulus 231 − 1 ⋮
A New Attack on Three Variants of the RSA Cryptosystem ⋮
Refined analysis to the extended tower number field sieve ⋮
Faster cofactorization with ECM using mixed representations ⋮
Prime-number algorithm for public-key systems ⋮
Adversary-Dependent Lossy Trapdoor Function from Hardness of Factoring Semi-smooth RSA Subgroup Moduli ⋮
Factoring with the quadratic sieve on large vector computers ⋮
Extended Tower Number Field Sieve: A New Complexity for the Medium Prime Case ⋮
On the Sato–Tate conjecture on average for some families of elliptic curves ⋮
On counting and generating curves over small finite fields ⋮
Cyclicity statistics for elliptic curves over finite fields ⋮
Arithmetic statistics and noncommutative Iwasawa theory ⋮
Average Frobenius distribution for the degree two primes of a number field ⋮
The distribution of class groups of function fields ⋮
Strong pseudoprimes to base 2 ⋮
Some computational problems of cryptographic significance concerning elliptic curves over rings ⋮
An efficient deterministic test for Kloosterman sum zeros ⋮
Modular curves over number fields and ECM ⋮
Modifications to the number field sieve ⋮
Integers without large prime factors ⋮
Revisiting prime power RSA ⋮
Factoring integers and oracles for elliptic and hyperelliptic curves ⋮
Factorization, malleability and equivalent problems ⋮
On elliptic curve $L$-functions integrated encryption scheme ⋮
Smooth factors of integers and elliptic curve based factoring with an oracle ⋮
New amicable pairs of type $(2,2)$ and type $(3,2)$ ⋮
The Non-existence of Universal Carmichael Numbers ⋮
Unlikely Intersections with ExCM curves in A2 ⋮
New weak keys in RSA ⋮
Attacking RSA Using an Arbitrary Parameter ⋮
Diffie-Hellman Oracles ⋮
Algorithms for Black-Box Fields and their Application to Cryptography ⋮
Compositeness test with nodal curves ⋮
Attacking (EC)DSA Given Only an Implicit Hint ⋮
Finding elliptic curves with a subgroup of prescribed size ⋮
Evaluating Igusa functions ⋮
A classification of ECM-friendly families of elliptic curves using modular curves ⋮
Unnamed Item ⋮
SOME ARITHMETICAL PROPERTIES ON HYPERBOLA ⋮
New applications of the polynomial method: The cap set conjecture and beyond ⋮
Computational Number Theory and Cryptography ⋮
Elliptic Twin Prime Conjecture ⋮
Generalised Mersenne numbers revisited ⋮
A survey on the group of points arising from elliptic curves with a Weierstrass model over a ring ⋮
UNLIKELY INTERSECTIONS IN FINITE CHARACTERISTIC ⋮
Amicable pairs and aliquot cycles on average ⋮
Lang–Trotter and Sato–Tate distributions in single and double parametric families of elliptic curves ⋮
Subgroup Security in Pairing-Based Cryptography ⋮
New Attacks on RSA with Moduli N = p r q ⋮
Factoring RSA Moduli with Weak Prime Factors ⋮
Small discriminants of complex multiplication fields of elliptic curves over finite fields ⋮
Arithmetic statistics and Diophantine stability for elliptic curves ⋮
Computational Number Theory, Past, Present, and Future ⋮
Arithmetic statistics for the fine Selmer group in Iwasawa theory ⋮
Another Generalization of Wiener’s Attack on RSA ⋮
Tables of Fibonacci and Lucas factorizations ⋮
Lattice enumeration for tower NFS: a 521-bit discrete logarithm computation ⋮
Cryptanalysis of the multi-power RSA cryptosystem variant ⋮
A Titchmarsh divisor problem for elliptic curves ⋮
Lattice enumeration and automorphisms for tower NFS: a 521-bit discrete logarithm computation ⋮
Revisiting cycles of pairing-friendly elliptic curves ⋮
Group structure of elliptic curves over \(\mathbb{Z}/N\mathbb{Z}\) ⋮
High-rank elliptic curves with given torsion group and some applications ⋮
JKL-ECM: an implementation of ECM using Hessian curves ⋮
An elliptic curve analogue of McCurley's key agreement scheme ⋮
Computing isomorphisms and embeddings of finite fields ⋮
An asymptotic for the average number of amicable pairs for elliptic curves ⋮
Aurifeuillian factorizations and the period of the Bell numbers modulo a prime ⋮
A Kilobit Special Number Field Sieve Factorization ⋮
A Subexponential Algorithm for Discrete Logarithms Over all Finite Fields ⋮
A Practical Analysis of the Elliptic Curve Factoring Algorithm ⋮
Average Frobenius distribution for elliptic curves defined over finite Galois extensions of the rationals ⋮
The Factorization of the Ninth Fermat Number ⋮
Elliptic Curves and Primality Proving ⋮
Application of ECM to a class of RSA keys ⋮
Finding the group structure of elliptic curves over finite fields ⋮
Compression on the Twisted Jacobi Intersection ⋮
An exploration of affine group laws for elliptic curves ⋮
ECM using Edwards curves ⋮
Finding Suitable Curves for the Elliptic Curve Method of Factorization ⋮
A deterministic version of Pollard’s $p-1$ algorithm ⋮
Factorization with genus 2 curves ⋮
Computing the order of points on an elliptic curve modulo \(N\) is as difficult as factoring \(N\) ⋮
Huff’s Model for Elliptic Curves ⋮
Fast construction of irreducible polynomials over finite fields ⋮
Using LLL-Reduction for Solving RSA and Factorization Problems ⋮
A $p$-adic algorithm to compute the Hilbert class polynomial ⋮
A generic approach to searching for Jacobians ⋮
A Rigorous Time Bound for Factoring Integers ⋮
On the asymptotic effectiveness of Weil descent attacks ⋮
Speeding up the computations on an elliptic curve using addition-subtraction chains ⋮
Another look at non-standard discrete log and Diffie-Hellman problems ⋮
Using number fields to compute logarithms in finite fields ⋮
Three new factors of Fermat numbers ⋮
Binary Huff Curves ⋮
Computing isogenies between elliptic curves over $F_{p^n}$ using Couveignes's algorithm ⋮
Information and computation: Classical and quantum aspects ⋮
Sufficient conditions for factoring a class of large integers ⋮
Complexity of inverting the Euler function ⋮
Computing Hilbert class polynomials with the Chinese remainder theorem ⋮
On the Security of Cryptosystems with Quadratic Decryption: The Nicest Cryptanalysis ⋮
ECM on Graphics Cards ⋮
Computing Prime Factorization And Discrete Logarithms: From Index Calculus To Xedni Calculus ⋮
THE ERROR TERM IN THE SATO–TATE THEOREM OF BIRCH ⋮
Cryptanalysis of RSA Using the Ratio of the Primes ⋮
Algorithm for factoring some RSA and Rabin moduli ⋮
Extension of elliptic curves on Krasner hyperfields ⋮
Construction of metrics on the set of elliptic curves over a finite field ⋮
Self-correctors for Cryptographic Modules ⋮
Computing prime divisors in an interval ⋮
Reliabilities of Double-Loop Networks ⋮
Factoring with Cyclotomic Polynomials ⋮
Unnamed Item ⋮
On the Number of Elliptic Pseudoprimes ⋮
A New Class of Weak Encryption Exponents in RSA ⋮
Factorization of the tenth Fermat number ⋮
Computing canonical heights with little (or no) factorization ⋮
Traces of Hecke operators and refined weight enumerators of Reed-Solomon codes ⋮
SMOOTH NUMBERS IN SHORT INTERVALS ⋮
Supersingular j-invariants and the class number of ℚ(−p) ⋮
Prime Factorization without Using Any Approximations ⋮
Ranks of elliptic curves ⋮
A new attack on RSA and Demytko’s elliptic curve cryptosystem ⋮
Forty years of attacks on the RSA cryptosystem: A brief survey
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