THE GARDEN

Matrix Discrete Log via Characteristic Polynomial Factorization (Extension Fields + CRT)

Updated 13 Sep 2026

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  What it is

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    Factor a square-free reduction polynomial into irreducible factors. Project the power relation into their extension fields, solve each DLP modulo the actual multiplicative order of its root, and combine compatible residues with generalized CRT. Smaller factors may reduce cost, but roots need not generate the whole field’s multiplicative group and the remaining DLPs can still be difficult.

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  When to apply

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    A matrix-based discrete log needs solving, the reduction polynomial has no repeated roots, but its characteristic polynomial factors into several irreducible pieces of much smaller degree than the full matrix dimension — check this before assuming the discrete log is intractable, and be deliberate about which discrete-log algorithm you invoke: generic algorithms (plain Baby-Step Giant-Step) can be dramatically slower than a specialized finite-field log routine for large binary extension fields, even for the "same size" problem.

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  Symbols and assumptions

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    For a square-free reduction polynomial $f=\prod_i f_i$ over $GF(q)$, let $F_i=GF(q)[X]/(f_i)$ and $\alpha_i=X\bmod f_i$. Assume the base matrix is invertible so these roots are nonzero.

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  Math — step by step

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    1. Reduce the known remainder a modulo each irreducible factor. This gives $a(\alpha_i)=\alpha_i^s$ in a smaller extension field of size $q^{d_i}$, where $d_i=\deg f_i$.
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    2. Solve each DLP modulo $n_i=\operatorname{ord}(\alpha_i)$. This order divides $q^{d_i}-1$, but need not equal it: an extension-field generator as a field is not necessarily a primitive multiplicative generator.
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    3. Combine $s\equiv s_i\pmod{n_i}$ with generalized CRT, checking compatibility when moduli share factors. The result is determined modulo $\operatorname{lcm}(n_i)$. Add a bound to recover a unique integer exponent.
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    4. Small factor degrees help, but do not guarantee practical DLPs. Use the actual subgroup orders, appropriate finite-field algorithms and full matrix/vector verification.
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  Worked example

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    Take factors $X^2+X+1$ and $X^3+X+1$ over $GF(2)$. Their roots $\alpha,\beta$ have orders 3 and 7. Let $a(X)=X^3+X^2+1$.

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    In the first field $\alpha^2=\alpha+1$ and $\alpha^3=1$, so $a(\alpha)=1+(\alpha+1)+1=\alpha+1=\alpha^2$. Thus $s\equiv2\pmod3$.

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    In the second field $\beta^3=\beta+1$, so $a(\beta)=\beta^2+\beta$. Since $\beta^4=\beta^2+\beta$, $s\equiv4\pmod7$. Write $s=4+7k$: modulo 3, $1+k=2$, so k=1 and $s=11\pmod{21}$. Check: 11 mod3=2 and 11 mod7=4.

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  Python

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    # In each extension field Fi, let alpha = X mod factor.
    order_i = alpha.multiplicative_order()
    target_i = a_poly(alpha)
    # Solve target_i = alpha**s modulo order_i.
    # Combine compatible (s_i, order_i) with generalized CRT.
    # Verify the final exponent against the original observation.
    

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  SageMath

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    For large binary extension fields, compare available finite-field-specific algorithms with generic discrete-log methods. A full-order generic square-root attack on an 89-bit group is impractical. The dump reports that PARI fflog worked for its instance; this is not a runtime guarantee for every extension field.

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    F = GF(2**d, 'x', modulus=factor)
    x = F.gen()
    target = a_poly(x)
    dlog_i = int(target.__pari__().fflog(x.__pari__()))   # NOT the generic discrete_log()
    

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  Related

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    Cayley-Hamilton Reduction of Matrix Exponentiation to Polynomial Coefficients

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    Pohlig-Hellman Algorithm

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    Finite Fields (Galois Fields: GF(p) and GF(2^n))

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    Chinese Remainder Theorem

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  Cards

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    Why does factoring the characteristic polynomial into irreducible pieces help solve the discrete log?

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      Factoring exposes power relations in smaller extension fields. Each gives a residue modulo the actual order of its root; generalized CRT combines compatible residues. Feasibility still depends on those orders and available DLP algorithms.

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    Why consider a finite-field-specific discrete-log routine?

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      It may exploit field structure that generic group algorithms ignore. Check actual subgroup orders and benchmark the instance; no single runtime claim applies to every binary extension field.

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