Windowed Modular Exponentiation

• Windowed modular exponentiation is a technique that processes groups of exponent bits simultaneously, reducing the number of costly modular multiplications through precomputation.
• It includes fixed-window (m-ary) and sliding-window methods, where adaptive window sizes are selected based on exponent bit-length to optimize performance.
• Quantum adaptations use grouped control qubits and lookup-driven operations to achieve speed improvements, balancing precomputation costs with runtime gains.

Windowed modular exponentiation denotes a family of exponentiation algorithms for computing $g^e \bmod N$ by consuming multiple exponent bits at a time so as to reduce the number of expensive modular multiplications, at the price of precomputing powers of the base and storing them. In the classical literature represented here, the principal forms are fixed-window or $m$-ary exponentiation and sliding-window exponentiation; in the quantum literature, the analogous idea is to iterate control qubits in groups and replace many individually controlled operations by coherent table lookups (Ji et al., 2017, Gidney, 2019).

1. Formal setting and central tradeoff

The basic task is modular exponentiation,

$g^e \bmod N,$

with large integers $g,e,N$. The standard binary left-to-right method scans exponent bits from most significant to least significant, squares every step, and multiplies by $g$ when the current bit is $1$. Window methods generalize this by consuming more than one exponent bit per step, thereby reducing the number of modular multiplications while increasing precomputation and table storage (Ji et al., 2017).

In the classical setting, two families dominate the discussion in the supplied literature. Fixed-window exponentiation partitions the exponent into chunks of exactly $m$ bits. Sliding-window exponentiation instead consumes variable-length odd windows up to size $m$, so that runs of zeros are handled more efficiently. The adaptive formulations studied in the Python benchmark paper do not alter these algorithms internally; they choose the window parameter from the exponent bit-length (Ji et al., 2017).

In the quantum setting, the same tradeoff reappears in a different guise. The target map is

$$|e\rangle|x\rangle \mapsto |e\rangle|x\cdot k^e \bmod N\rangle,$$

and windowing means grouping exponent or control bits so that one multiplication by a window-dependent constant replaces several individually controlled multiplications. The price is a larger precomputed lookup table and additional coherent lookup logic (Gidney, 2019).

2. Fixed-window and $m$-ary exponentiation

The $m$0-ary method is the fixed-window baseline. The exponent is parsed as

$m$1

where each $m$2 is an $m$3-bit block. The precomputation phase sets

$m$4

The main loop initializes

$m$5

and then, for $m$6, squares $m$7, $m$8 times, followed by multiplication by the precomputed table entry $m$9 (Ji et al., 2017).

For $g^e \bmod N,$0-bit exponents satisfying $g^e \bmod N,$1, the paper models the average number of large-integer multiplications by

$g^e \bmod N,$2

The three terms are stated explicitly as precomputation cost $g^e \bmod N,$3, squaring cost $g^e \bmod N,$4, and expected table-multiplication cost $g^e \bmod N,$5 (Ji et al., 2017).

A central result is that $g^e \bmod N,$6 is treated as convex in $g^e \bmod N,$7, so one can choose an optimal integer $g^e \bmod N,$8 from the exponent bit-length. The paper tabulates the minimizing ranges: $g^e \bmod N,$9

$g,e,N$0

$g,e,N$1

This adaptive rule is the paper’s concrete refinement of fixed-window exponentiation: the algorithm remains standard, but the window width is selected from $g,e,N$2 instead of being hard-coded (Ji et al., 2017).

3. Sliding-window exponentiation and adaptive window selection

Sliding-window exponentiation replaces fixed-size chunks by variable-length odd windows. Its precomputation phase sets

$g,e,N$3

and then builds odd powers by

$g,e,N$4

During the left-to-right scan, if the current bit is $g,e,N$5, the accumulator is squared once. If the current bit is $g,e,N$6, the algorithm finds the longest bitstring

$g,e,N$7

such that $g,e,N$8 and $g,e,N$9, performs $g$0 squarings, multiplies by the corresponding odd precomputed power, and continues from $g$1 (Ji et al., 2017).

The operation-count model given for sliding windows is

$g$2

Here the precomputation cost is $g$3, the squaring cost is $g$4, and the expected number of table multiplications is $g$5. The paper gives the second derivative

$g$6

and uses its positivity to justify convexity-based selection of $g$7 (Ji et al., 2017).

The resulting adaptive thresholds are: $g$8

$g$9

$1$0

This is a direct adaptive policy for sliding-window exponentiation, not a new recurrence (Ji et al., 2017).

In benchmark results against CPython and PyPy, the practical gains are modest but systematic for very large exponents. The paper states that compared to the industry-standard efficient implementations of the modular power function in CPython and PyPy, the adaptive methods can reduce computing time by about $1$1–$1$2 for exponents with more than $1$3 bits. For $1$4-bit exponents, adaptive sliding-window exponentiation improves over CPython pow by $1$5 and over PyPy pow by $1$6, while also outperforming adaptive $1$7-ary at the same sizes (Ji et al., 2017).

4. Constant-time and SIMD software realizations

A more implementation-driven line of work treats windowed modular exponentiation as a constant-time systems problem. The AVX512 paper implements a constant-time left-to-right fixed-window algorithm for $1$8 simultaneous exponentiations using word-slicing and Intel’s VPMADD52 instructions. For each batch element, the computation is

$1$9

with the pipeline: expand operands to a 52-bit word-sliced batch format, precompute powers

$m$0

scan the exponent left-to-right by windows of width $m$1, select the corresponding table entry in constant time, perform $m$2 modular squarings, perform one modular multiplication, and finally contract the result back to standard form (Didier et al., 2024).

The constant-time property is enforced by a fixed operation schedule and constant-time table selection. Every window performs exactly $m$3 squarings and one multiplication, independent of the window value. The paper explicitly identifies the $m$4 case with square-and-multiply-always. This removes leakage through the presence or absence of a multiplication and through exponent-dependent memory behavior (Didier et al., 2024).

The arithmetic backend is Montgomery-based. Standard Montgomery reduction is written as

$m$5

with $m$6. The paper’s main low-level optimization is truncated Montgomery reduction: instead of computing all of $m$7, it computes only

$m$8

and a carry

$m$9

then forms

$m$0

This truncated variant yields speed gains of almost $m$1 over the conventional non-truncated versions in the Montgomery multiplication layer (Didier et al., 2024).

Window-size tuning remains decisive. The paper tests $m$2 through $m$3 and reports the best choices as $m$4 for $m$5-bit moduli and $m$6 for $m$7-bit and $m$8-bit moduli. For $m$9 fixed-window exponentiations, the best timings are truncated Schoolbook at $$|e\rangle|x\rangle \mapsto |e\rangle|x\cdot k^e \bmod N\rangle,$$0 and $$|e\rangle|x\rangle \mapsto |e\rangle|x\cdot k^e \bmod N\rangle,$$1 bits and truncated Karatsuba at $$|e\rangle|x\rangle \mapsto |e\rangle|x\cdot k^e \bmod N\rangle,$$2 bits, with reported speedups over OpenSSL BN_mod_exp_mont_consttime of $$|e\rangle|x\rangle \mapsto |e\rangle|x\cdot k^e \bmod N\rangle,$$3, $$|e\rangle|x\rangle \mapsto |e\rangle|x\cdot k^e \bmod N\rangle,$$4, and $$|e\rangle|x\rangle \mapsto |e\rangle|x\cdot k^e \bmod N\rangle,$$5, respectively. Against OpenSSL BN_mod_exp_mont_consttimex2, the reported speedups are $$|e\rangle|x\rangle \mapsto |e\rangle|x\cdot k^e \bmod N\rangle,$$6 at $$|e\rangle|x\rangle \mapsto |e\rangle|x\cdot k^e \bmod N\rangle,$$7 bits and $$|e\rangle|x\rangle \mapsto |e\rangle|x\cdot k^e \bmod N\rangle,$$8 at $$|e\rangle|x\rangle \mapsto |e\rangle|x\cdot k^e \bmod N\rangle,$$9 bits (Didier et al., 2024).

5. Quantum windowing: lookup-driven modular arithmetic

In quantum arithmetic, windowing is not a metaphor but a direct circuit transformation. The target operation is

$m$0

A non-windowed implementation applies one controlled modular multiplication by $m$1 for each exponent bit $m$2. The windowed alternative partitions the exponent into windows of size $m$3, with

$m$4

and replaces $m$5 separate multiplications in each block by one multiplication by the window-dependent constant

$m$6

The same paper simultaneously windows the multiplication itself, partitioning the multiplicand into windows of size $m$7 and using small lookup tables to realize each modular product addition (Gidney, 2019).

The resulting Toffoli complexity is

$m$8

where $m$9 is the exponent width and $m$00 is the modular register size. With balanced windows

$m$01

the lookup size satisfies $m$02, yielding

$m$03

The paper interprets this as saving two logarithmic factors relative to the naive bitwise-controlled approach (Gidney, 2019).

Later work sharpens the cost of the lookup-addition primitive itself. The optimization paper on windowed modular arithmetic starts from the observation that modular exponentiation can be assembled from repeated lookup-add-unlookup blocks. It then introduces four refinements: reducing the cost of unlookups by $m$04 asymptotically in the number of bits, bypassing certain trivial addresses, merging multiple lookup-addition operations into a single larger initial lookup, and reducing the depth of unary conversion for unlookups. At the logical level, these changes produce about a $m$05 improvement in Toffoli count and Toffoli depth for modular exponentiation circuits relevant to cryptographic applications, and for a given number of physical qubits they reduce the expected runtime for factoring $m$06-$m$07 integers by $m$08 to $m$09 (Luongo et al., 24 Feb 2025).

Resource-estimation work on Shor-style implementations reaches a closely related conclusion from a different direction. In that study, windowed modular exponentiation is parameterized by a window size $m$10 with fitted runtime model

$m$11

which leads to the heuristic

$m$12

The paper reports that the non-windowed LYY modular exponentiation is best in physical qubit count, whereas the optimized windowed variant LYY-W-Opt is best in runtime, making the space–time tradeoff explicit (Fedoriaka et al., 6 Sep 2025).

6. Boundaries of the term and recurrent misconceptions

A recurring source of confusion is that many papers on modular exponentiation, especially in the Shor literature, are not about windowing even when they use precomputed powers or binary decomposition. Pedagogical and architectural treatments based on the modular multiplication operator $m$13 and its controlled powers $m$14 are examples: they implement standard repeated-squaring over exponent bits, sometimes with direct construction of $m$15, but they do not introduce fixed-window, sliding-window, or $m$16-ary grouping of exponent bits (Jr, 2023, Pavlidis et al., 2012).

The same distinction applies to custom-synthesis work on modular multipliers. Reversible-circuit synthesis for fixed constants $m$17 can reduce the cost of each multiplication stage by large constant factors, yet the exponentiation schedule may still use one controlled modular multiplication per exponent bit. This is complementary to windowing, not a replacement for it (Markov et al., 2013, Markov et al., 2012).

Other neighboring directions are conceptually adjacent but algorithmically different. Secure outsourcing rewrites the exponent as

$m$18

and verifies results through affinely related second exponentiations; this is exponent masking and verification, not fixed-window or sliding-window exponentiation (Zhou et al., 2016). Transformer studies that learn the map

$m$19

address mechanistic interpretability of the input–output function, not classical window schedules or precomputed odd-power tables (Africa et al., 30 Jun 2025). Likewise, proposals for truncated or orbit-restricted modular exponentiation operators in Shor’s algorithm exploit the initial state $m$20 and the periodic orbit of modular multiplication, not windowing in the standard arithmetic sense (Jr, 2024).

The most stable meaning of “windowed modular exponentiation” across the supplied literature is therefore precise. In classical computation it means fixed-window or sliding-window processing of exponent digits with precomputed base powers. In quantum arithmetic it means grouping control or exponent bits and replacing many small controlled updates by lookup-driven operations on precomputed tables. Techniques based only on powers of two, constant-specific multiplier synthesis, exponent masking, truncated unitaries, or learned function approximation are related to modular exponentiation, but they are not windowed modular exponentiation in the usual sense (Ji et al., 2017, Gidney, 2019).