Fixed-Precision p-Adic Operations

Updated 3 December 2025

Fixed-precision p-adic operations are arithmetic procedures on truncated p-adic expansions, enabling computer algebra systems and cryptographic applications.
They integrate traditional digit-wise computation with advanced lattice-based precision tracking to maintain accuracy in complex operations.
Techniques like Newton iteration and FFT-based multiplication enhance performance and stability in evaluating p-adic series and algebraic algorithms.

A fixed-precision p-adic operation refers to arithmetic and algorithmic procedures performed in the ring of p-adic numbers truncated to a finite precision. Such operations arise in computer algebra systems, explicit number theory, and algorithmic applications ranging from cryptography to linear algebra. At the core of this paradigm are two conceptual axes: the arithmetic rules for p-adic expansions up to precision $N$ (modulo $p^N$ ), and the propagation and management of precision across complex algebraic computations, using both classical (interval or coordinate-wise) and lattice-based (differential precision) frameworks.

1. Fixed-Precision p-Adic Representation and Arithmetic

A fixed-precision p-adic number $x$ is realized by truncating its infinite base- $p$ expansion to $N$ digits: $x \equiv a_0 + a_1p + a_2p^2 + \cdots + a_{N-1}p^{N-1} \pmod{p^N}, \qquad a_i \in \{0,\dots,p-1\}$ Internally, implementations store $N$ digits as an array or vector ("limbs"), each a small machine integer. Variants allow packaging several base- $p$ digits into one word-sized limb for efficiency (Caruso, 2017, Cherkaoui et al., 25 Nov 2025). The fixed-precision model naturally extends to $p$ -adic coefficients in power series, matrices, and Laurent expansions. The truncation error on each coefficient is always $O(p^N)$ , and all computation is performed modulo $p^N$ .

Low-Level Algorithms

Addition/Subtraction: Digit-wise with carries/borrows; cost

O(N)

(XU et al., 2010, Caruso, 2017).

for i in 0..N-1:
    sum = a[i] + b[i] + carry
    c[i] = sum % p
    carry = sum // p

Multiplication: Convolution (schoolbook $O(N^2)$ $O (N^{2})$ ), Karatsuba ( $O(N^{\log_2 3})$ $O (N^{l o g_{2} 3})$ ), FFT ( $O(N\log N)$ $O (N lo g N)$ ) (Caruso, 2017, Caruso et al., 2021).
1 2 3
for i in 0..N-1: for j in 0..N-1: t[i+j] += a[i] * b[j]
Carries are propagated as in standard algorithms, respecting modulus $p^N$ $p^{N}$ .
Inversion: Via Newton iteration. For $B$ with $B[0] \not\equiv 0 \pmod{p}$ , set $X_0 = B[0]^{-1} \bmod p$ and iterate $X_{k+1} = X_k(2 - B X_k) \bmod p^{2^k}$ until desired digits are obtained. Total cost is $O(M(N))$ for multiplication algorithm cost $M(N)$ (Caruso, 2017, Cherkaoui et al., 25 Nov 2025, Caruso et al., 2021).
Division: Reduce to inversion and multiplication.

2. Precision Propagation: Lattice and Interval Methods

Propagation of error and control of digit-loss is central to fixed-precision p-adic computation. The naive (interval or coordinate-wise) approach tracks a separate absolute precision for each variable or coordinate, updating them individually. This is easily implemented but suffers from the loss of "diffused" digits in linear combinations and correlated calculations.

Lattice-Based Precision and Differential Propagation

The modern paradigm—pioneered by Caruso, Roe, and Vaccon (Caruso et al., 2018, Caruso et al., 2015, Caruso et al., 2014, Caruso et al., 2017, Lairez et al., 2016)—formalizes p-adic precision in terms of $\mathbb{Z}_p$ -lattices $H \subset E$ for a vector space $E$ . The inexact value is modeled as $v_0 + H$ , with $v_0$ an approximation and $H$ encoding precision. First-order precision propagation is controlled by the differential: $f(v_0 + H) = f(v_0) + df_{v_0}(H)$ That is, the output lattice is the image of the input lattice under the Jacobian. This is provably sharp for all $p$ -adic analytic maps with surjective differential (Caruso et al., 2018, Caruso et al., 2015).

Lattice tracking is realized in software (Sage ZpL, Magma+X, etc.) by associating with each live variable a lattice $H_t$ and updating an integral matrix $M_t$ whose columns encode the joint precision among all variables.

Basic operations (add, sub, mul, div): Update $M_t$ by the closed-form Jacobian columns—e.g., for $z = x + y$ , $C_z = C_x + C_y$ ; for $z = x y$ , $C_z = x C_y + y C_x$ (Caruso et al., 2018).
Matrix operations: $dZ = dM N + M dN$ for $Z = MN$ .
Polynomial division, ODEs, etc: Lattice update amounts to Newton polygon or differential-based rules (Caruso et al., 2016, Lairez et al., 2016).

The effectiveness of lattice tracking is seen in operations like determinant and characteristic polynomial, GCD of polynomials, and Newton iteration for ODEs: diffused digits can be preserved, and computations avoid unnecessary zero-detection failures (Caruso et al., 2018, Caruso et al., 2015).

3. Precision Models: Flat, Jagged, Newton, Lattice

Various data representation strategies are employed, offering a trade-off between storage overhead and tightness of tracked precision:

Model	Data Tracked	Space
Flat	Single $N$ per object	$O(1)$
Jagged	One $N_i$ per coordinate	$O(d)$
Newton	Piecewise-linear convex $\varphi$	$O(d)$
Lattice	Full lattice matrix	$O(d^2)$

The lattice model is optimal but can be expensive. Newton/jagged models (using Newton polygons) capture most of the digit gains, with jagged being more practical for polynomials/power series and Newton for operations governed by Newton-polygon calculus (Caruso et al., 2016, Caruso et al., 2014).

4. Complexity Analysis and Engineering Aspects

Arithmetic cost is dominated by the following:

Addition/Subtraction: $O(N)$ word operations per variable (XU et al., 2010, Cherkaoui et al., 25 Nov 2025)
Multiplication: $O(N^2)$ schoolbook, $O(N\log N)$ FFT (Caruso et al., 2021, Caruso, 2017)
Inversion (Newton): $O(M(N) \log N)$ (Caruso, 2017, Cherkaoui et al., 25 Nov 2025)
Lattice tracking: Updates to the $M_t$ matrix in ZpL cost $O(m^2)$ per operation for $m$ variables (Caruso et al., 2018), with deletion potentially $O(m^2)$ in the worst case, though typically amortized by temporal locality.

Memory and Performance Optimizations

For microcontroller and cryptography applications (see (Cherkaoui et al., 25 Nov 2025)), arrays of $D$ base- $p$ digits ("limbs") are used per coefficient, with all arithmetic branch-regular and constant-time for side-channel resilience.
Intermediate precision (caps) and adaptive strategies are employed to minimize space use or transition between models as diffused digits or precision needs become large (Caruso et al., 2018, Caruso et al., 2014).

5. Algorithmic Applications and Representative Examples

Linear Algebra: Determinant, LU, Characteristic Polynomial

Lattice-based precision recovers several additional digits in the determinant/characteristic polynomial of $p$ -adic matrices compared to coordinate-wise tracking (Caruso et al., 2018, Caruso et al., 2015, Caruso et al., 2017). Precision gain can be predicted by analytic properties such as the precision polygon (lower convex hull of comatrix valuations) (Caruso et al., 2017).

Euclidean Algorithm, Polynomial GCD

Naive interval arithmetic may fail in GCD algorithms by declaring $B=0$ too early due to coarse overapproximation. Lattice tracking prevents premature zero and yields correct results (Caruso et al., 2018).

p-adic Differential Equations

Newton iteration for ODEs typically suffers a $O(\log_p N)$ digit loss per doubling step in naive flat/interval approaches. Differential-lattice tracking achieves exactly the sharp theoretical bound, propagating only the digits genuinely lost by division, and allowing precise estimation and automatic management of input/output precision needs (Lairez et al., 2016, Caruso et al., 2018).

Polynomial Factorization and Precision-Aware Euclidean Division

Newton-polygon-based models (with or without full lattices) optimize digit retention, ensuring factorization routines such as slope factorization operate at stability bounds, without excess precision loss (Caruso et al., 2016).

Gröbner Basis and FGLM

In algorithms like FGLM, precision loss is tightly bounded using Smith normal form analysis, with overall loss determined by a global condition number rather than local accumulation of worst-case digit loss (Renault et al., 2016).

6. Comparison of Interval vs Lattice Approaches and Trade-offs

Lattice-based methods consistently outperform coordinate-wise precision tracking, especially in computations with strong variable coupling or after many composition steps:

Interval (flat/jagged): Fastest; suitable where little digit-diffusion occurs, but can lead to catastrophic loss in dependent operations (e.g., inversion, linear algebra, resultant computations).
Lattice (differential): Maximally sharp; can capture "diffused digits" and propagate tight precision in linear and nonlinear (Jacobian-controlled) settings; comes with $O(m^2)$ or higher memory and time costs for $m$ variables; widely adopted for prototyping, debugging, and as a guide for hand-tuned production code (Caruso et al., 2018, Caruso et al., 2015, Caruso et al., 2014).

Typical practice is to prototype algorithms under full lattice tracking (ZpLC), identify critical "precision hotspots," and then optimize with floating-point models (ZpLF) or insert extra caps/adjustments only where necessary for production (Caruso et al., 2018).

7. Implementation Patterns and Practical Guidelines

Separation of Approximation and Precision: Storage of approximations (residue classes modulo $p^N$ ) is decoupled from the precision “lattice” (caps, jagged, Newton, or full matrix) (Caruso et al., 2014, Caruso et al., 2018).
Use of Automatic Differentiation: Operator overloading assures Jacobian updates are injected into each arithmetic primitive (addition, multiplication, division, etc.), guaranteeing correct propagation.
Adaptive Precision: Algorithms may split into segments, repeatedly recomputing or centering the lattice to minimize digit loss and avoid global over-rounding (Caruso, 2017).
Microcontroller/cryptographic coding: Fixed-point, limb-wise arithmetic with constant-time, branch-free implementation is essential for high-assurance cryptography; the Engel-Laurent approach exemplifies efficient fixed-precision expansion for $p$ -adic coefficients (Cherkaoui et al., 25 Nov 2025).

In summary, fixed-precision $p$ -adic operations subsume a comprehensive complex of algorithmic disciplines: classical schoolbook arithmetic, Newton/FFT acceleration, precision-lattice modeling, and application-specific strategies for stabilization and efficiency. The lattice/differential approach, as realized in packages such as ZpL, enables first-order optimality in precision tracking, critical in high-precision computations, algorithmic number theory, and cryptographic protocols (Caruso et al., 2018, Caruso, 2017, Caruso et al., 2014, Cherkaoui et al., 25 Nov 2025).