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Index calculus algorithm

The index calculus algorithm is a probabilistic computational method for solving the discrete logarithm problem (DLP) in finite fields, particularly in the multiplicative group of a prime field \mathbb{Z}_p^*, where given a primitive root g and an element y = g^x \mod p, it efficiently computes the exponent x \mod (p-1).^[1] The algorithm operates in three main phases: collecting "relations" by identifying smooth values (factorable over a chosen factor base of small primes) in the group, solving a large sparse system of linear equations modulo p-1 to determine the discrete logarithms of the factor base elements, and finally computing the target logarithm by decomposing y into a product involving the base elements.^[1] It achieves subexponential time complexity, making it asymptotically faster than generic algorithms like baby-step giant-step or Pollard's rho for large prime fields.^[2] The origins of the index calculus method trace back to 1922, when Maurice Kraitchik described an early version in his book Théorie des Nombres, focusing on computing indices (discrete logarithms) using factorizations of small elements in the group. The technique was rediscovered and formalized in the context of cryptography in the 1970s, with Leonard Adleman's 1979 paper providing a rigorous probabilistic analysis and proving its subexponential runtime of L_p(1/2, 1 + o(1)), where L_p(a, c) = \exp(c (\log p)^a (\log \log p)^{1-a}).^[2] Subsequent improvements, such as those by Coppersmith, Odlyzko, and Schroeppel in 1986, refined the complexity to L_p(1/2, 1), while the number field sieve variant extended it to general finite fields with L_p(1/3, 1.923) time.^[2] In practice, the algorithm's efficiency relies on the density of smooth numbers in the field and optimized sieving techniques to gather relations, often requiring significant precomputation for the factor base.^[1] It has been adapted to other settings, including hyperelliptic curves and fields of small characteristic, though with varying success; for instance, recent quasi-polynomial time variants for small-characteristic fields were developed by Joux in 2013.^[2] Despite these advances, the method does not apply effectively to elliptic curve groups, where the DLP remains resistant to index calculus approaches.^[2] The index calculus remains a cornerstone for assessing the security of discrete logarithm-based cryptosystems like Diffie-Hellman in finite fields.^[2]

Introduction

Overview

The index calculus algorithm is a probabilistic method for solving the discrete logarithm problem in the multiplicative group of finite fields, particularly (\mathbb{Z}/q\mathbb{Z})^\times where q is a large prime. The discrete logarithm problem is defined as follows: given a generator g of the group and an element h = g^x \mod q with x \in \{0, 1, \dots, q-2\}, the task is to recover the secret exponent x.^[3] At a high level, the algorithm operates in three phases. First, in the relation collection phase, random powers of g are generated and tested for factorization into products of small primes from a predefined factor base, yielding equations that relate the discrete logarithms of these primes. Second, in the linear algebra phase, these relations form a system of linear equations over the exponents, which is solved to compute the discrete logarithms of all elements in the factor base. Finally, in the logarithm computation phase, the target h is incorporated into a similar relation, allowing the use of the precomputed logarithms to solve for x.^[1] The factor base is a carefully chosen set of small primes, typically of size r \approx L_q[1/2, 1], where L_q[u, c] = \exp\left( c (\log q)^u (\log \log q)^{1-u} \right), selected to balance the probability that random elements near q are smooth (i.e., factor completely over the base) with the cost of solving the resulting linear system. The algorithm's probabilistic nature stems from this reliance on random sieving and factorization attempts, with success depending on collecting enough independent relations. Its expected running time is subexponential in \log q, scaling as L_q[1/2, c] for a constant c > 0 that varies with implementation details and optimizations.^[4] In cryptographic applications, the index calculus algorithm undermines the security of schemes relying on the hardness of the discrete logarithm in prime fields, such as Diffie-Hellman key exchange and the Digital Signature Algorithm (DSA), by enabling efficient computation of logarithms for field sizes that were previously considered secure.^[5]

Mathematical Prerequisites

The index calculus algorithm is fundamentally based on computations in the multiplicative group of the finite field \mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}, where p is an odd prime. This group, denoted (\mathbb{Z}/p\mathbb{Z})^\times, consists of the integers from 1 to p-1 under multiplication modulo p and forms a cyclic group of order p-1.^[5] There exists a generator g \in (\mathbb{Z}/p\mathbb{Z})^\times, known as a primitive root modulo p, whose order is exactly p-1, meaning the powers g^0, g^1, \dots, g^{p-2} modulo p generate all elements of the group.^[5] The discrete logarithm problem (DLP) in this setting is to find an integer x such that g^x \equiv y \pmod{p} for given g and y \in (\mathbb{Z}/p\mathbb{Z})^\times, with $0 \leq x < p-1. The DLP is believed to be computationally hard in large prime-order subgroups of finite fields, forming the security basis for cryptographic primitives such as the Diffie-Hellman key exchange and ElGamal encryption.^[5] No polynomial-time algorithm is known for solving the general DLP, though subexponential methods like index calculus exist.^[5] A key concept is that of smooth numbers: a positive integer n is B-smooth if every prime factor of n is at most B. The density of B-smooth numbers up to x is approximated by the Dickman-de Bruijn function, with the proportion roughly u^{-u + o(u)} where u = \log x / \log B, enabling probabilistic estimates for finding such numbers efficiently.^[4] In index calculus, the factor base often includes the element -1 alongside small primes to facilitate complete factorizations of field elements lifted to integers in [1, p-1]. The discrete logarithm of -1 is \log_g(-1) = (p-1)/2, since g^{(p-1)/2} \equiv -1 \pmod{p} for any primitive root g, as this power is the unique element of order 2 in the group.^[6] The Legendre symbol (a/p), defined for odd prime p and integer a not divisible by p as

\left( \frac{a}{p} \right) = a^{(p-1)/2} \pmod{p},

equals 1 if a is a quadratic residue modulo p, -1 if a non-residue, and 0 if p divides a.^[7] For a = g^k, the symbol reveals the parity of k via \left( \frac{g^k}{p} \right) \equiv (-1)^k \pmod{p}, since g^{(p-1)/2} \equiv -1 \pmod{p}; quadratic residues thus correspond to even discrete logs, aiding in handling signs and base elements like -1 (where \left( \frac{-1}{p} \right) = (-1)^{(p-1)/2}).^[7] The algorithm collects relations that form a sparse linear system over \mathbb{Z}/(p-1)\mathbb{Z} for the unknown discrete logs of factor base elements, solved via Gaussian elimination or structured methods exploiting sparsity.^[4] Using the Legendre symbol, the system can be reduced modulo 2 to solve for the parities (least significant bits) of these logs over \mathbb{F}_2, simplifying the full modular solve when p-1 has known factorization.^[5] To identify B-smooth candidates efficiently during factorization, sieve methods can test for divisibility by factor base primes when evaluating powers g^k \mod p for random k, accumulating exponents to detect smoothness without complete trial division for each, achieving subexponential time in practice analogous to sieving in factoring algorithms.^[4]

Core Algorithm

Relation Collection Phase

The relation collection phase is the initial step of the index calculus algorithm, where a sufficient number of relations among the discrete logarithms of small primes are gathered to form the basis for solving the system. The factor base P = \{p_1, \dots, p_r\} is selected as the set of the smallest prime numbers, often augmented with -1 if the characteristic is odd to handle sign issues in factorizations. The size r of the factor base is chosen to balance the probability of smoothness and the cost of subsequent linear algebra, typically r \approx \exp\left( \left(\frac{1}{2} + o(1)\right) \sqrt{\ln p \ln \ln p} \right), where p is the prime.^[5] To generate relations, random integers k are selected, and the group element s_k = g^k \mod p is computed, where g is a generator of the group. If s_k is P-smooth—meaning all its prime factors lie in P—it is fully factored as s_k = \prod_{i=1}^r p_i^{e_i}, and the exponent vector (e_1, \dots, e_r) is recorded as a relation, corresponding to the equation k \equiv \sum_{i=1}^r e_i \log_g p_i \pmod{p-1}.^[8] The process continues until approximately m \approx r + 1 such relations are collected, ensuring the resulting matrix has full rank and linear independence over the integers modulo p-1.^[5] For efficiency, especially with large p, sieving techniques are employed to identify P-smooth candidates without exhaustive trial division for each s_k. Interval sieving processes g^k for consecutive k in a range, marking multiples of small primes in the factor base to detect potential smooth values quickly.^[9] More advanced lattice sieving, particularly in variants like the number field sieve for discrete logarithms, uses lattice structures to sieve higher-dimensional spaces of potential relations, reducing the time per candidate tested.^[9] Non-smooth s_k are typically discarded, requiring multiple trials until a smooth one is found, with the expected number of attempts governed by the smoothness probability u^{-u + o(u)} where u = \ln p / \ln B and B is the smoothness bound.^[5] In some variants, partial relations with large cofactors are stored and combined later to form complete relations. Precomputed tables, such as the Cunningham tables providing factorizations of algebraic numbers of special forms like b^{\pm k} + 1, are used in practice to supply additional smooth relations efficiently. This phase is highly parallelizable, as computations for different random k are independent and can be distributed across multiple processors without coordination.^[5]

Linear Algebra Phase

In the linear algebra phase of the index calculus algorithm, a sparse linear system is constructed and solved to compute the discrete logarithms of the factor base elements relative to the generator g. Suppose m relations have been collected, where m slightly exceeds r, the size of the factor base \{p_1, \dots, p_r\}. The matrix A is an m \times r matrix over \mathbb{Z}/(p-1)\mathbb{Z}, with entry A_{i,j} denoting the exponent of p_j in the factorization of the i-th smooth value y_i = g^{k_i} \mod p. The vector \mathbf{k} = (k_1, \dots, k_m)^T contains the known exponents k_i, and the unknown vector \mathbf{x} = (\log_g p_1, \dots, \log_g p_r)^T satisfies the overdetermined system

A \mathbf{x} \equiv \mathbf{k} \pmod{p-1}.

This setup arises because each relation equates k_i \equiv \sum_{j=1}^r A_{i,j} \log_g p_j \pmod{p-1}.^[10]^[11] Solving this system requires identifying linear dependencies among the relations to extract r independent equations for the r unknowns. For small factor bases, standard Gaussian elimination suffices, reducing the matrix to row echelon form and back-substituting to find \mathbf{x}. However, for large r, the sparsity of A (with typically few non-zero entries per row) is exploited using advanced techniques. Structured Gaussian elimination first prunes dependent or dense rows and columns to minimize computational fill-in and reduce the effective matrix size from roughly r \times r to a smaller dense core, often achieving this in O(r^2) time. Subsequent steps employ iterative methods like the Lanczos algorithm or Wiedemann's algorithm to compute a basis for the solution space or kernel of the reduced system, improving overall complexity beyond the O(r^3) of naive Gaussian elimination.^[9]^[5]^[12] The kernel computation yields basis vectors \mathbf{v} such that A \mathbf{v} \equiv \mathbf{0} \pmod{p-1}, corresponding to linear relations among the columns of A; combining these with the right-hand side via the augmented matrix [A \mid \mathbf{k}] provides differences of logarithms, such as \log_g p_i - \log_g p_l = \sum_j v_j k_j \pmod{p-1}. Fixing one logarithm (e.g., \log_g 2 via baby-step giant-step) determines the full set. For the 2-primary component of p-1, operations simplify over \mathbb{F}_2 using exponent parities, while the odd part uses the full modular arithmetic. All computations employ modular reduction modulo p-1 to manage large intermediate values and avoid overflow.^[5]^[12]^[9]

Logarithm Computation Phase

In the logarithm computation phase of the index calculus algorithm, the precomputed discrete logarithms of the factor base elements are applied to determine the discrete logarithm of the target element h with respect to the generator g in the multiplicative group \mathbb{F}_p^\times, where p is prime and the order is p-1. This phase begins by selecting a random integer s modulo p-1 and computing y = g^s \cdot h \pmod{p}. The value y is then tested for smoothness over the factor base P, a set of small primes. If y factors completely as y = \prod_{p_i \in P} p_i^{e_i} (where the e_i are non-negative integers), the discrete logarithm x = \log_g h satisfies \log_g y = s + x \pmod{p-1}, so x = \left( \sum_i e_i \log_g p_i - s \right) \pmod{p-1}, using the precomputed values \log_g p_i from the linear algebra phase.^[13]^[14] If y is not P-smooth, a new random s is chosen, and the process repeats. The expected number of trials required to find a smooth y is subexponential, on the order of L_p(1/2, 1). Factoring each candidate y (an integer between 1 and p-1) can be performed efficiently using trial division over P or more advanced methods like the elliptic curve method for larger factors.^[14]^[13] Once a candidate x is obtained, verification ensures correctness by checking whether g^x \equiv h \pmod{p}; if not, additional trials are conducted until a valid smooth y yields the correct logarithm. To accelerate this phase in practice, multiple independent searches for s can be run in parallel across processors, as each trial is independent and the first successful smooth y suffices. This parallelization is particularly effective given the subexponential expected trial count.^[13]^[14]

Complexity and Optimizations

Theoretical Complexity

The index calculus algorithm for computing discrete logarithms in the multiplicative group of a finite field \mathbb{F}_q achieves a subexponential expected running time of L_q[1/2, \sqrt{2} + o(1)], where L_q[a, c] = \exp\left( c (\ln q)^a (\ln \ln q)^{1-a} \right).^[4] This complexity arises from optimizing the trade-offs across its phases, significantly outperforming generic methods for large q. The algorithm's running time decomposes across its main phases. The relation collection phase requires finding approximately r smooth elements, where r is the size of the factor base plus a small excess for linear dependence; this phase has complexity O(r \cdot L_q[1/2, 1]).^[15] The subsequent linear algebra phase solves a sparse system of r equations in r variables, with complexity O(r^2).^[5] Finally, the individual logarithm computation phase, involving descent to the factor base, runs in O(L_q[1/2, 1]).^[4] With optimal choices, the dominant terms balance to yield the overall L_q[1/2, \sqrt{2} + o(1)] bound. Optimal performance is achieved by selecting the factor base size such that the smoothness bound B \approx L_q[1/2, 1], leading to r \approx B / \ln B \approx L_q[1/2, 1].^[15] The probability that a random element in \mathbb{F}_q^\times is B-smooth is approximately u^{-u}, where u = \ln q / \ln B \approx \sqrt{\ln q / \ln \ln q}; this probability governs the expected number of trials needed in the collection phase.^[4] The space complexity is O(r^2), primarily due to storing and processing the relation matrix in the linear algebra phase.^[5] This can be mitigated to O(r) space using structured techniques such as the block Wiedemann algorithm for solving sparse linear systems over finite fields.^[5] In comparison to the baby-step giant-step algorithm, which requires O(\sqrt{q}) time and space (exponential in \frac{1}{2} \ln q), the index calculus provides a subexponential advantage for sufficiently large q, making it the method of choice for prime fields of cryptographic size.^[4]

Practical Performance and Implementations

The practical performance of the index calculus algorithm has improved significantly over time, driven by advances in hardware and software optimizations. In the 1990s, computations for a 512-bit prime field could be completed in a few days to weeks on small clusters of workstations, marking an early demonstration of feasibility for cryptographic-sized instances. More recently, a record-breaking computation of a discrete logarithm in a 795-bit prime field was achieved in 2019, requiring approximately 3100 core-years of computation on distributed systems; this remains the largest such computation in general prime fields as of November 2025, highlighting the scalability challenges and the role of massive parallel resources in modern implementations.^[16]^[4] Key optimizations have enhanced efficiency, particularly in the relation collection phase. The large prime variation allows relations to include a small number of large prime factors beyond the factor base, reducing the required number of relations while increasing sieving complexity, which has been integrated into implementations for fields up to several hundred bits. In the number field sieve variant, the use of multiple polynomials enables better balancing of norms and smoother relations, improving overall runtime for medium- to high-characteristic finite fields. Hardware accelerations, such as GPU implementations for sieving and FPGA designs for modular arithmetic, have been explored to speed up specific subroutines, though their adoption remains limited compared to CPU clusters for full-scale runs.^[17]^[18] Prominent implementations include the open-source CADO-NFS suite, which supports the full index calculus pipeline for discrete logarithms in prime fields and has been used in records since 2017, incorporating advanced filtering and linear algebra solvers. The GMP-ECM library is commonly employed for elliptic curve-based smoothness testing during relation collection, complementing sieving methods in hybrid approaches.^[19] A primary bottleneck in large-scale computations arises during the linear algebra phase, where solving sparse systems over matrices with more than 10^6 rows dominates runtime due to high memory demands and communication overhead in parallel settings. Parallelization strategies, such as block Wiedemann algorithms and distributed matrix partitioning, mitigate this by enabling scalability across thousands of cores, though I/O and synchronization remain limiting factors.^[20] These advancements have significant cryptographic implications, rendering discrete logarithms feasible in fields with q < 2^{1024}, which influenced NIST's post-2020 recommendations to deprecate 1024-bit keys for Diffie-Hellman and related protocols in favor of at least 2048 bits to ensure long-term security.

Applications

Discrete Logarithm in Finite Fields

The index calculus algorithm finds primary application in solving the discrete logarithm problem (DLP) within the multiplicative group of finite prime fields, denoted (\mathbb{Z}/p\mathbb{Z})^*, where p is a large prime and the group order is p-1.^[5] This setting is particularly amenable to the algorithm because elements can be represented as integers modulo p, allowing efficient sieving for smooth relations during the relation collection phase.^[21] In cryptographic contexts, computations are often confined to large prime-order subgroups of (\mathbb{Z}/p\mathbb{Z})^* to mitigate attacks like the small subgroup confinement vulnerability, requiring the factorization of p-1 to identify such a subgroup of order q.^[22] If p-1 has a smooth factorization—meaning it factors into small primes—this confinement becomes computationally straightforward, enabling the algorithm to target the harder DLP in the subgroup rather than the full group.^[5] Several foundational cryptographic protocols rely on the presumed hardness of the DLP in these finite field groups, rendering them vulnerable to successful index calculus attacks. The Diffie-Hellman key exchange protocol, introduced in 1976, uses the DLP in (\mathbb{Z}/p\mathbb{Z})^* to enable secure key agreement between parties, but an adversary solving the DLP could compute shared secrets directly.^[5] Similarly, the ElGamal encryption scheme bases its semantic security on the same problem, where breaking the DLP allows decryption of ciphertexts without the private key.^[5] The Digital Signature Algorithm (DSA), standardized by NIST, employs the DLP in a subgroup of prime order q dividing p-1 for signature generation and verification, making it susceptible if the index calculus recovers the discrete log efficiently.^[22] To enhance security against certain attacks, such as small subgroup confinement, cryptographic standards recommend using safe primes p = 2q + 1, where both p and q are primes; this structure ensures the large subgroup of order q is the primary target for the DLP, simplifying implementation while maintaining hardness assumptions.^[22] A landmark demonstration of the algorithm's power occurred in 2016, when researchers computed a 768-bit discrete logarithm in a prime field using a variant of the number field sieve integrated with index calculus techniques, requiring approximately 4,000 core-years of computation on distributed systems.^[23] This record highlighted vulnerabilities in legacy systems still employing 768-bit or smaller parameters for protocols like Diffie-Hellman and DSA, prompting recommendations to migrate to at least 2048-bit primes.^[23] Although index calculus remains a classical algorithm unaffected by quantum threats like Shor's, it retains relevance in hybrid cryptographic schemes that combine finite field DLP-based systems with post-quantum primitives, as standardized by bodies like NIST through 2025.^[24] These hybrids leverage the efficiency of finite field operations for key exchange or signatures alongside quantum-resistant elements, ensuring backward compatibility while addressing long-term security needs.^[24]

Extensions to Other Algebraic Structures

The index calculus algorithm, originally developed for discrete logarithms in finite fields, has been adapted to various algebraic structures beyond prime fields, though with varying degrees of success due to differences in group structure and factorization properties. In elliptic curves over finite fields, the method proves ineffective in general because the group lacks an analogous notion of "primes" for smooth decomposition, preventing efficient relation collection similar to the integer case.^[25] Extensions to hyperelliptic curves in the 1990s focused on the Jacobian variety, where the algorithm exploits the Mumford representation for divisor decomposition. These adaptations, pioneered through work on small-genus curves, treat ideals in the function field as analogs to smooth integers, but practical implementations remain slower than the Pollard rho method for cryptographic sizes, primarily due to the higher-dimensional nature of the Jacobian.^[26] In function fields over \mathbb{F}_q(t), the index calculus finds a natural analogy, where smoothness is defined via factorization into irreducible polynomials of bounded degree, mirroring prime factorization in number fields. This setup allows subexponential-time algorithms for computing discrete logarithms, with relation collection involving random elements factored against a base of low-degree irreducibles.^[27] For pairing-friendly elliptic curves, index calculus adaptations yield limited success, achieving subexponential complexity only for specific curve families where the embedding degree permits efficient reduction to the extension field discrete logarithm problem. In these cases, the method exploits the field's structure but requires careful parameter selection to avoid vulnerability in cryptographic applications.^[28] Recent theoretical advancements from 2022 to 2025 have addressed gaps in genus 2 hyperelliptic curves by providing improved complexity bounds for index calculus attacks, incorporating efficient endomorphism-based decomposition to reduce the factorization base size; however, these remain impractical for large-scale computations due to high constant factors.^[29]

Historical Development

Early Origins

The conceptual foundations of the index calculus algorithm emerged from early investigations in analytic number theory, building on ideas about the factorization of powers and the structure of multiplicative groups. A significant precedent was provided by Alfred Zsigmondy's 1892 theorem on primitive prime factors, which established that for relatively prime integers a > b > 0 and n > 1, a^n - b^n has a prime divisor that does not divide a^k - b^k for any k < n, except in specific small cases. This result influenced subsequent approaches to generating smooth factorizations by highlighting how successive powers introduce new small prime factors, laying groundwork for relation collection in index-based methods.^[30] The first explicit formulation of an index calculus-like method appeared in Maurice Kraitchik's 1922 work on number theory, where he developed a technique to compute indices (discrete logarithms) in quadratic fields specifically for determining class numbers.^[31] Kraitchik's approach involved identifying smooth ideals in the ring of integers of the quadratic field and using relations among them to solve for the logarithmic structure, enabling practical computations of class groups without exhaustive enumeration.^[31] This method marked the initial application of systematic relation gathering and linear solving to logarithmic problems in algebraic number fields, though it remained computationally intensive for larger discriminants.^[2] In 1968, A. E. Western and J. C. P. Miller adapted these ideas to the discrete logarithm problem in the multiplicative group (\mathbb{Z}/q\mathbb{Z})^* for prime q, introducing a probabilistic strategy for collecting relations by randomly selecting elements likely to factor smoothly over a fixed base of small primes.^[31] Their work, detailed in tables of precomputed indices up to moderate primes, formalized the algorithm's core phases—relation collection via smoothness testing and resolution via linear algebra over the exponents—achieving subexponential efficiency for the era's computational resources.^[2] Throughout these early developments, the primary motivation was theoretical advancement in analytic number theory, such as tabulating indices for studying primitive roots and class groups, rather than practical applications like cryptography.^[31]

Key Advancements and Milestones

The introduction of the Diffie-Hellman key exchange protocol in 1976 underscored the cryptographic significance of the discrete logarithm problem, thereby increasing the demand for efficient solution methods such as index calculus. In 1979, Leonard Adleman refined the index calculus algorithm by optimizing the selection of the factor base, achieving subexponential complexity and establishing it as a practical tool for discrete logarithms in finite fields. During the 1980s, initial practical implementations of index calculus enabled the computation of discrete logarithms in finite fields of 100 to 200 bits, marking the transition from theoretical analysis to applied computation on early hardware.^[16] In the 1990s, the emergence of the number field sieve variant facilitated a crossover from integer factorization techniques, enhancing index calculus for larger instances; notably, in 1998, a 426-bit discrete logarithm in a prime field was solved as part of McCurley's challenge using the special number field sieve.^[32] From the 2000s to 2010s, distributed computing efforts set successive records, including a 431-bit prime field discrete logarithm in 2005 using optimized sieving on clusters, and a 530-bit instance in 2007 that demonstrated scalability improvements in linear algebra phases. By 2016, a 768-bit prime field discrete logarithm was computed via the number field sieve variant, requiring over two years of effort on hundreds of cores and underscoring the role of massive parallelism.^[23] In the 2020s, post-2017 advancements have pushed records further, such as the 795-bit discrete logarithm solved in late 2019 through refined sieving and GPU-accelerated linear algebra using the CADO-NFS software, which has accelerated the shift toward quantum-resistant cryptographic standards by demonstrating the ongoing viability of classical attacks on legacy systems.^[33]

Variants

Number Field Sieve Integration

The Number Field Sieve for discrete logarithms (NFS-DL) represents a sophisticated integration of index calculus principles into the framework of algebraic number fields, enabling subexponential-time solutions for the discrete logarithm problem in large prime fields \mathbb{F}_p. Developed as an adaptation of the factoring-oriented Number Field Sieve, NFS-DL operates over two interconnected rings: the rational integers \mathbb{Z} (the rational side) and the ring of integers \mathcal{O}_K of an algebraic number field K = \mathbb{Q}/(f(x)), where f(x) is a monic irreducible polynomial of degree d chosen such that f(m) \equiv 0 \pmod{p} for some integer m, identifying a root \alpha \equiv m \pmod{p} in \mathbb{F}_p. This setup leverages homomorphisms from \mathcal{O}_K to both \mathbb{Z} (via the norm) and \mathbb{F}_p (via reduction modulo p), allowing index calculus to exploit smoothness in both domains simultaneously for enhanced efficiency.^[34] Central to NFS-DL is the collection of relations through sieving for elements \pi = a - b\alpha \in K, where a, b \in \mathbb{Z} are coprime integers in a suitable range, such that the norm N_{K/\mathbb{Q}}(\pi) factors smoothly over a factor base of small primes in \mathbb{Z} and the projection \pi \pmod{p} = a - bm \in \mathbb{F}_p factors smoothly over a corresponding factor base in \mathbb{F}_p. These smoothness conditions yield logarithmic relations via the index calculus paradigm: the discrete logarithm of N_{K/\mathbb{Q}}(\pi) in \mathbb{Z}^\times equals the sum of logarithms of its prime factors, while the discrete logarithm of \pi \pmod{p} in \mathbb{F}_p^\times follows similarly, connected by the field embedding that equates \log_g N_{K/\mathbb{Q}}(\pi) \equiv \log_g (\pi \pmod{p}) \pmod{q-1} for generator g and prime q = p. Sufficient relations form a sparse linear system over the factor base logarithms, solved via Gaussian elimination to compute individual logs and extend to the target logarithm, mirroring the core phases of basic index calculus but with algebraic norms boosting relation yield. The factor bases consist of small rational primes and principal ideals in \mathcal{O}_K, with sieving optimized over polynomial pairs to detect smooth candidates efficiently.^[34] The heuristic complexity of NFS-DL is L_p\left[\frac{1}{3}, \left(\frac{64}{9}\right)^{1/3} + o(1)\right], where L_p[u, c] = \exp\left(c (\log p)^u (\log \log p)^{1-u} + o(1)\right), marking a significant improvement over the classical index calculus's L_p\left[\frac{1}{2}, c\right] for some constant c > 0, due to the balanced optimization of sieving and linear algebra steps across the dual smoothness probabilities. This complexity arises from choosing the polynomial degree d \approx 1.44 (\log p / \log \log p)^{1/3} to minimize the overall runtime, with relation collection dominating for large p. NFS-DL has been instrumental in breaking large prime-field instances, exemplified by the 2019 computation of a discrete logarithm in a 795-bit safe prime field \mathbb{F}_p (approximately 240 decimal digits), requiring about 3,100 core-years on distributed hardware and demonstrating practical scalability for cryptographic sizes.^[34] The Function Field Sieve (FFS) extends the index calculus framework to the discrete logarithm problem over function fields \mathbb{F}_q(t), particularly effective for finite fields of small characteristic where the field can be represented as quotient rings of polynomials. This adaptation leverages analogies to the number field sieve but operates in characteristic p using polynomial factor bases and sieving techniques for relation collection. In formulations involving Drinfeld modules, which generalize elliptic curves in positive characteristic, FFS facilitates efficient arithmetic and descent steps tailored to function field geometry. The heuristic complexity of FFS is L_q(1/3, (32/9)^{1/3}), where L_q(\alpha, c) = \exp((c + o(1)) (\log q)^\alpha (\log \log q)^{1-\alpha}), making it subexponential and superior to generic methods for small q. A significant advancement came with Joux's 2013 algorithm, which refines index calculus for small characteristic fields by generating relations through high-degree polynomials that perturb the target element, yielding a quasi-polynomial number of smooth relations without exhaustive sieving. This approach shifts the bottleneck from relation collection to linear algebra over a larger factor base, achieving a heuristic complexity of L_q(1/4 + [\epsilon](/page/Epsilon), c) for arbitrary small \epsilon > 0 and some constant c > 0. The method has been implemented for fields with up to $2^{6120} elements (as of 2013), demonstrating practical feasibility for large instances in small characteristic.^[35] In the 2020s, hybrid approaches combining index calculus with quantum methods have emerged for pre-quantum security analysis, particularly to evaluate classical-quantum transitions in discrete logarithm settings. These hybrids transform the relation-finding step of index calculus into quadratic unconstrained binary optimization problems solvable via quantum annealing, offering potential speedups in the near-term quantum regime while retaining classical sieving for large-scale relation validation. Such methods provide insights into post-quantum resilience without full fault-tolerant quantum hardware. Progress on elliptic curve variants from 2016 to 2025 has explored index calculus approaches for special families, such as subfield curves, achieving improved but still exponential complexities in targeted scenarios, without posing a general threat to elliptic curve cryptography. Kummer lines, as quotient models of elliptic curves by the

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-torsion action, enable lifted representations that facilitate computations in characteristic not dividing 2, but do not yield subexponential attacks for practical curves.

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In this section, though, we will show that the linear equations produced by the index-calculus algorithm can be solved in time essentially S 2, where S is ...
[6]
[PDF] Discrete Logarithm Factory - Cryptology ePrint Archive
Discrete logarithms in cryptography are not restricted to prime fields. Several cryptographic proto- cols rely on the hardness of the discrete logarithm problem ...
[7]
Legendre Symbol -- from Wolfram MathWorld
The Legendre symbol is a number theoretic function (a/p) which is defined to be equal to +/-1 depending on whether a is a quadratic residue modulo p.
[8]
None
### Summary of Relation Collection Phase in Index Calculus Algorithm
[9]
[PDF] improvements to the general number field sieve for discrete ...
Nov 4, 2002 · McCurley, Lattice sieving and trial division, Proceedings of the ANTS-I conference, Lecture Notes in Computer Science, vol. 877, Springer ...
[10]
[PDF] The Discrete Logarithm Problem in Cryptography - CS - Huji
Jan 9, 2007 · First step of index calculus: pick a number r at random, and ... This is a matrix equation, and we can try to use linear algebra to ...
[11]
[PDF] Discrete Log
Index Calculus. ❑ Given p, g, x = ga (mod p), determine a. ❑ Analogous to Dixon's algorithm o Except linear algebra phase comes first. ❑ Choose bound B and ...
[12]
https://inria.hal.science/hal-01420485
[13]
Discrete logarithms: The effectiveness of the index calculus method
The index calculus method, using smoothness in rings, is used for discrete logarithms in finite fields, but its computation lags behind factoring of integers.
[14]
[PDF] 11 Index calculus, smooth numbers, and factoring integers
Mar 15, 2017 · Algorithm 11.1 (Index calculus in a prime field Fp). 1 ... By contrast the largest discrete logarithm computation over a safe prime field ...
[15]
[PDF] Factoring and Discrete Logarithms in Subexponential Time
15.5.5 The Joux-Lercier Algorithm. The function field sieve of Adleman is a general algorithm for discrete logarithms in Fpn where p is relatively small ...
[16]
[PDF] discrete.logs.future.pdf
Jul 19, 1999 · ... calculus algorithms have been developed for a variety of discrete log ... methods, the basic number field sieve, and lattice sieving) have come ...
[17]
[PDF] The Multiple Number Field Sieve for Medium and High ...
The main idea in this case is to rely on the polynomial selection of NFS-HD that permits to balance both the degrees of the two polynomials and the sizes of ...
[18]
[PDF] First step toward an implementation of the Tower Number Field Sieve
Jun 21, 2019 · Index calculus algorithm [Western–Miller 68, Adleman 79], prequel of the Number Field Sieve algorithm (NFS). ▷ p prime, (p − 1)/2 prime, G = (Z/ ...
[19]
[PDF] Improving NFS for the Discrete Logarithm Problem in Non ... - Hal-Inria
Jun 3, 2016 · Abstract. The aim of this work is to investigate the hardness of the discrete logarithm problem in fields GF(pn) where n is a small integer ...
[20]
[PDF] Diffie–Hellman, discrete logarithm computation - Inria
Record computations with the CADO-NFS software. • Important software development effort since 2007. • 250k lines of C/C++ code, 60k for relation collection only ...
[21]
[PDF] 11.6 Discrete logarithms over finite fields
11.6.42 Remark Index calculus algorithms for discrete logs require the solution of linear equations modulo q − 1, where q is the size of the field.
[22]
Recent Advances in the Index Calculus Method for Solving the ECDLP
... isogenies between supersingular elliptic curves. The main technical idea in our scheme is that we transmit the images of torsion bases under the isogeny in ...
[23]
Index calculus attack for Jacobian of hyperelliptic curves of small ...
This paper introduces a fast algorithm for solving the DLP of Jacobian of hyperelliptic curve of small genus. To solve the DLP, Gaudry first shows that the ...
[24]
[PDF] A new index calculus algorithm with complexity L(1/4 + o(1)) in small ...
The new index calculus algorithms proposed in this paper hinges on a few basic ideas, which can be arranged into a functional discrete logarithm algorithm.
[25]
[PDF] Pairing-Friendly Elliptic Curves: Revisited Taxonomy, Attacks ... - arXiv
attack on FFDLP is the index calculus method solved in sub-exponential time: exp¥8c +. 𝑜(1)9(𝑙𝑜𝑔 𝑞")8/%(𝑙𝑜𝑔 𝑙𝑜𝑔 𝑞")8/%¨. There have been seen ...
[26]
Index calculus attacks on hyperelliptic Jacobians with efficient ...
Apr 12, 2022 · We exploit the endomorphism of the Jacobian to reduce the size of the factorization base and improve the complexity of the index calculus attack.Missing: 1990s | Show results with:1990s
[27]
[PDF] Primitive Indexes, Zsigmondy Numbers, and Primoverization - arXiv
Oct 25, 2018 · Abstract. We define a primitive index of an integer in a sequence to be the index of the term with the integer as a primitive divisor.Missing: 1892 smooth
[28]
[PDF] Computing Discrete Logarithms - Cryptology ePrint Archive
In §2 we introduce elliptic curves and pairings over finite fields and consider various discrete logarithm algorithms. Then in §3 we consider some groups in ...
[29]
[PDF] Computation of a 768-bit prime field discrete logarithm
On average each additional discrete logarithm requires two core days. This result is a record for computing prime field discrete logarithms. It closes the gap ...
[30]
Discrete Logarithms in G ⁡ F ⁡ ( P ) Using the Number Field Sieve
Discrete logarithms: The effectiveness of the index calculus method. Algorithmic Number Theory | 2 June 2005. Computing discrete logarithms with the general ...
[31]
A new index calculus algorithm with complexity $L(1/4+o(1))$ in ...
Feb 21, 2013 · Abstract. In this paper, we describe a new algorithm for discrete logarithms in small characteristic. This algorithm is based on index calculus ...Missing: optimized sieving 2020-2025