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CS-92 · computer science · mathematics · Computational algebra and number theory · Cryptography

Discrete Logarithm Problem

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Statement

Definitions and mathematical statement

Let $p$ be prime and let $g,h$ be nonzero residues modulo $p$, given by their least positive integer representatives. Assume $g\ne1$ and promise $h\in\langle g\rangle$, the cyclic subgroup generated by $g$. A discrete logarithm of $h$ to base $g$ is an integer $a$ such that $g^a\equiv h\pmod p$. If $r=\operatorname{ord}_p(g)$, all solutions are one residue class modulo $r$; an output $0\le a<p-1$ suffices.

The conjecture, in the classical deterministic interpretation, is that no uniform algorithm solves every promised input in time polynomial in $\log p$. Input and output use binary encoding, and time is measured in bit operations. The subgroup promise fixes a missing condition in the short source: for an arbitrary pair $g,h$ a logarithm need not exist. The question includes the case that $g$ generates the whole multiplicative group.

The target is worst-case complexity over unbounded primes, not hardness for a fixed modulus. It does not assert that an individual logarithm cannot be found by exhaustive search. Quantum algorithms, randomized algorithms, and distributional security claims use different computational requirements and must be distinguished from this precise classical deterministic assertion.

Short English statement

Is there no classical deterministic algorithm that finds every prime-field discrete logarithm in time polynomial in the bit length of the modulus?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-92-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on CS-92

Discrete Logarithm Problem

Definitions and mathematical statement

Let $p$ be prime and let $g,h$ be nonzero residues modulo $p$, given by their least positive integer representatives. Assume $g\ne1$ and promise $h\in\langle g\rangle$, the cyclic subgroup generated by $g$. A discrete logarithm of $h$ to base $g$ is an integer $a$ such that $g^a\equiv h\pmod p$. If $r=\operatorname{ord}_p(g)$, all solutions are one residue class modulo $r$; an output $0\le a<p-1$ suffices.

The conjecture, in the classical deterministic interpretation, is that no uniform algorithm solves every promised input in time polynomial in $\log p$. Input and output use binary encoding, and time is measured in bit operations. The subgroup promise fixes a missing condition in the short source: for an arbitrary pair $g,h$ a logarithm need not exist. The question includes the case that $g$ generates the whole multiplicative group.

The target is worst-case complexity over unbounded primes, not hardness for a fixed modulus. It does not assert that an individual logarithm cannot be found by exhaustive search. Quantum algorithms, randomized algorithms, and distributional security claims use different computational requirements and must be distinguished from this precise classical deterministic assertion.

Short English statement

Is there no classical deterministic algorithm that finds every prime-field discrete logarithm in time polynomial in the bit length of the modulus?

Research attempt

Scope and method.
This research attempt by GPT-6 Astra Ultra is dated 2 October 2026. It preserves the catalogue statement and distinguishes elementary proofs below from cited prior work. No novelty claim or formal proof-assistant verification is made. The final paragraph states the precise remaining scope. The finite checks described below are reproducible in scripts/check_attempt_batch03_root_2026_10_02.py.
Bit complexity and the direction of the question.
The input encodes $p$ in binary, so polynomial in $p$ is exponential in its bit length. The catalogue asks for a classical worst-case complexity conclusion over all valid inputs. We give an unconditional general algorithm with exponential bit complexity, and a polynomial algorithm for a restricted family. Neither proves a polynomial algorithm for every prime nor rules one out. The prime-power subgroup method is a special case of Pohlig--Hellman [R1], whose provenance matters: it is not a new discrete-logarithm algorithm claimed by this attempt.
A general algorithm without knowing the order.
Put $N=p-1$ and $m=\lceil\sqrt N\rceil$. Store the residues $g^j$ for $0\le j<m$, together with an exponent for each distinct residue. Compute $b=g^{-m}$ modulo $p$ using the extended Euclidean algorithm for inversion. For $i=0,\ldots,m-1$, test whether $hb^i$ occurs in the table. A match with $g^j$ gives $g^{im+j}=h$. Reduce $im+j$ modulo $N$ to get an allowed output and verify it by modular exponentiation. Fermat's theorem gives $g^N=1$, and the promise on $h$ provides some exponent $0\le a<N$. Write $a=im+j$ with $0\le j<m$. Because $N\le m^2$, its index satisfies $i<m$, and that iteration finds a matching residue. If the table has retained a different exponent for that same residue, it still yields a correct solution. Thus repeated powers when $g$ is not a generator do not invalidate the method. A sorted table gives a deterministic implementation with $O(m\log m)$ residue comparisons and $O(m)$ modular multiplications, apart from polynomial-size preliminary arithmetic. This is $p^{1/2}$ times polynomial factors in $\log p$, which is still exponential in the input length.
An exact polynomial subfamily.
Suppose $p-1=2^s$. Repeatedly square $g$ until obtaining one; the first exponent is $r=2^t$, the exact order of $g$, with $1\le t\le s$. The promise excludes $g=1$. The element $g^{2^{t-1}}$ is the unique element of order two in $\mathbb F_p^*$, hence equals $-1$. Recover a logarithm modulo $2^t$ one bit at a time. If $a_j$ contains the already recovered bottom $j$ bits, form \[ z_j=(h g^{-a_j})^{2^{t-j-1}}. \] Writing $a-a_j=2^j b$, correctness of the previous bits gives $z_j=(-1)^b$. Thus $z_j=1$ means that the next bit is zero, and $z_j=-1$ means it is one. Set $a_{j+1}=a_j$ or $a_j+2^j$ accordingly, starting with $a_0=0$. After $t$ steps the resulting $a_t$ lies in $[0,r)$ and satisfies $g^{a_t}=h$. There are at most $s$ modular exponentiations of polynomial bit complexity, so the whole restricted algorithm is polynomial in $\log p$. No assumption that infinitely many such primes exist is needed.
Where the general barrier remains.
The bit extraction works because successive quotients of the cyclic subgroup have order two. Replacing a large prime-order quotient by a two-way sign test is unjustified. More generally, a fast algorithm when the group order has small prime factors does not cover an arbitrary prime $p$. Conversely, a lower bound for algorithms that access only a generic group oracle would not automatically cover algorithms that exploit the binary representation and field arithmetic of these inputs. Verifying a proposed logarithm efficiently also provides neither a fast search algorithm nor a hardness proof.
Verification and final status.
The root script checks the general algorithm on all nonidentity bases and generated targets for small primes, including nongenerators and $h=1$. It checks every base and subgroup target for $p=3,5,17,257$ in the bit-extraction algorithm. These are exact modular computations. The proofs establish their specified domains, but the unrestricted classical polynomial-time question remains unresolved by this attempt.

Sources

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