Right-to-Left Parallel EC Scalar Multiplication

Updated 17 August 2025

Right-to-left parallel elliptic curve scalar point multiplication is a method that decomposes the scalar into digits and processes the least-significant digit first, enabling parallel scheduling of point doubling and addition.
It employs optimized representations, such as the nearly optimal non-adjacent form (NAF) and M-ary windowing, to reduce computation time and memory usage.
Architectural enhancements, including pipelined doubling chains and side-channel countermeasures, offer substantial speedups and improved security for cryptographic implementations.

Right-to-Left Parallel Elliptic Curve Scalar Point Multiplication is a class of algorithms and implementation techniques designed to efficiently compute scalar multiples $kP$ of elliptic curve points $P$ , by leveraging parallelism and right-to-left digit processing. In these approaches, the scalar $k$ is decomposed into digits or windows, and point operations such as doubling and addition are scheduled so as to enable high-throughput, low-latency calculations in both software and hardware, often crucial for cryptographic primitives. Modern research addresses both the mathematical foundations for optimal digit representations as well as architecture-aware hardware acceleration, security properties, and application-level integration.

1. Mathematical Foundations and Parallelism Model

The central computation in elliptic curve cryptography is scalar multiplication: $Q = kP$ , for a scalar $k$ and base point $P$ . In right-to-left (RTL) parallel algorithms, the scalar is processed least-significant digit first, decomposing $k$ (e.g., in binary or other bases) as $k = \sum_{i=0}^{l-1} k_i 2^i$ (binary). Parallelization is achieved by decoupling the pipeline of doubling (computing $2^i P$ ) and addition (accumulating $k_i 2^i P$ when $k_i\neq 0$ ).

In the mathematical model of (Phalakarn et al., 10 Aug 2025), the computation time of such a parallel algorithm is described recursively in terms of the cost of a point doubling $D$ and an addition $A$ : $T(\mathcal{N}_\mathcal{S}, i) = \begin{cases} 0 & \text{if } i=0, n_0=0 \ T(\mathcal{N}_\mathcal{S}, i-1) & \text{if } i>0, n_i=0 \ iD + (|n_i|-1)A & \text{if } i \geq 0, n_i\neq 0 \text{ least significant nonzero} \ \max(T(\mathcal{N}_\mathcal{S}, i-1), iD) + |n_i| A & \text{otherwise} \end{cases}$ where $\mathcal{N}_\mathcal{S}$ is a digit expansion of $k$ in some set $\mathcal{S}$ (e.g., $\{-1,0,1\}$ for NAF). This formalization allows for rigorous optimization of digit representations under parallel execution constraints.

2. Optimal Scalar Representations and Near-Optimality of NAF

A key question is to determine which digit expansion minimizes parallel computation time under the above model. (Phalakarn et al., 10 Aug 2025) introduces algorithms to generate representations that, for arbitrary $A$ and $D$ , minimize the modeled time. For $A\geq 2D$ , a modification of the conventional Non-Adjacent Form (NAF) is constructed, while for $1\leq A < 2D$ , a delay-minimizing scheme is employed via digit flipping.

A notable result is that, for arbitrary $A$ and $D$ , no representation can improve parallel computation time by more than 1% over conventional NAF:

For all practical purposes, NAF is nearly optimal in the right-to-left parallel setting.

This supports NAF-based RTL implementations, providing theoretical justification for their performance and obviating the need for complex alternative digit encodings.

3. Architectural and Algorithmic Strategies

3.1 Parallel Scheduling and Hardware Mapping

The RTL parallel approach naturally supports mapping point doublings and conditional additions onto separate compute units or threads. In hardware, especially FPGAs or VLIW/SIMD cores:

Doubling chains (computing $P, 2P, 4P, \ldots$ ) may be pipelined.
Adding units process the digit-encoded contributions as soon as input from doublings is available (Ohno et al., 17 Feb 2025).
Memory contention and synchronization are mitigated by task mapping strategies such as coarse, fine, and medium-grained partitioning to exploit device-specific bandwidth and vectorization capabilities.

For example, on the Versal ACAP with 400 AI Engines (Ohno et al., 17 Feb 2025), MSM point additions are distributed so that carry-propagation and accumulation are overlapped, achieving up to a 568 $\times$ speedup over the CPU baseline, and utilizing 50.2% of available DRAM bandwidth.

3.2 Precomputation and Windowing

Advanced RTL methods use M-ary windowing or mixed-base expansions to reduce multiplication costs. Precompute tables $M[i,j] = j\cdot (B^iP)$ for base $B$ and digit values $j$ enable replacing online multiplications with table lookups and a small number of additions, reducing online complexity from $\Theta(Q\log p)$ to $\Theta((Q\log p)/\log Q)$ and memory from $\Theta(Q\log p)$ to $\Theta((Q\log p)/\log^2 Q)$ (Wu et al., 3 May 2025). In practice, this yields 22–59% time reduction and 25–30% memory savings for batch cryptographic operations.

Mixed-base representations (Eid et al., 2019) (e.g., varying window size bases per window) further reduce the inversion count and support grouping doublings for parallel execution with a single inversion per group, critical in resource-constrained devices and protocols such as SIDH.

3.3 Efficient Coordinate Systems and Field Arithmetic

Use of projective or alternative coordinate systems (Jacobian, López-Dahab, $\boldsymbol{\mu}_4$ -normal, twisted Edwards) to minimize inversions (Kohel, 2020, Kohel, 2016).
Efficient adder, squarer, and especially hybrid Karatsuba multipliers enable hardware acceleration with significant reductions in area-delay product (e.g., for GF(2 $^{163}$ ), 13.31 ns delay at 6,812 LUTs in the hybrid design, a 39.8% reduction vs. bit-parallel) (Kumari et al., 11 Jun 2025, Kumari et al., 14 Jun 2025).
Dedicated squarers and optimized inversion circuits (e.g., Extended Euclidean in $\leq 326$ cycles) further lower RTL point multiplication time (Kumari et al., 14 Jun 2025).

4. Security and Side-Channel Analysis

Implementations based on side-channel countermeasures, particularly those using atomic patterns (e.g., the MNAMNAA sequence for field operations) (Li, 10 Sep 2024), structurally enforce a constant, fixed-order of field operations for doubling/addition regardless of the scalar digit. Despite this, inherent vulnerabilities can arise:

Multiplication vs. squaring leakage: As revealed in (Sigourou et al., 4 Dec 2024), hardware register addressing can make squarings distinguishable from multiplications based on differential power consumption, as the hardware multiplexer activity differs for $M(a,b)$ vs. $S(a)=M(a,a)$ .
Key-dependent operation sequences can thus be revealed to side-channel attackers in both left-to-right and right-to-left, and in single-threaded as well as parallel RTL EC scalar multiplication.
Mitigations include dummy memory access, randomization of register addressing, and masking, but care must be taken at hardware design, microarchitecture, and algorithmic levels.

5. Genus 2, Endomorphism, and Multi-dimensional Parallel Methods

High-performance EC scalar multiplication can be further improved using endomorphism-based decomposition:

In the four-dimensional GLV method (Birkner et al., 2011), $kP$ is decomposed as $kP=k_1P + k_2\Phi(P) + k_3\Psi(P) + k_4\Psi\Phi(P)$ , with $|k_i|\leq C_2 n^{1/4}$ . All four scalar multiplications are parallelizable, offering near fourfold speedup for curves admitting suitable endomorphisms.
Scalar decomposition techniques (Smith, 2013) (especially with ready-made short lattice bases) transform scalar multiplication into multiple shorter, independent scalar multiplications, all suitable for concurrent execution.
Multi-scalar multiplication (MSM), essential in zero-knowledge proofs and signature aggregation, leverages right-to-left bucket-based parallel accumulation schedules tightly coupled to parallel EC point addition hardware (Ohno et al., 17 Feb 2025).

6. Applications and Practical Implications

RTL parallel EC scalar multiplication frameworks are deployed across:

Hardware cryptographic engines (FPGAs, ASICs, SoCs with SIMD/VLIW cores), especially in applications requiring ultra-low-latency signature verification, ECDSA, SIDH, and privacy protocols (Ohno et al., 17 Feb 2025, Kumari et al., 14 Jun 2025).
Resource-constrained IoT devices benefiting from low area-delay product field arithmetic and flexible coordinate systems (Kumari et al., 11 Jun 2025, Kohel, 2020).
Quantum circuit designs employing parallel RTL scheduling for low T-gate depth (translated into quantum circuit resource reductions in Shor’s algorithm for ECDLP) (Häner et al., 2020).

7. Summary Table: Models and Tradeoffs

Representation/Approach	Online Time Complexity	Memory Complexity	Security/Notes
Standard NAF (RTL-Parallel)	$\Theta(Q\log p)$	Small (window dep.)	Nearly optimal (Phalakarn et al., 10 Aug 2025)
M-ary Precompute (Wu et al., 3 May 2025)	$\Theta(\frac{Q \log p}{\log Q})$	$\Theta(\frac{Q\log p}{\log^2 Q})$	Excellent for batched ops
4-GLV Decomposition (Birkner et al., 2011)	$\approx$ quarter the sequential cost	Lattice reduction offline	Endomorphism needed; parallelism
Atomic Patterns (Li, 10 Sep 2024)	As underlying method	Small (code/data)	SPA vulnerabilities possible

References to Key Results

Mathematical parallel runtime model, optimality margin of NAF (Phalakarn et al., 10 Aug 2025)
Fourfold parallel GLV decomposition (Birkner et al., 2011), efficient decomposition (Smith, 2013)
Precompute-based M-ary windowing acceleration (Wu et al., 3 May 2025)
Hardware acceleration and hybrid multipliers (Kumari et al., 11 Jun 2025, Kumari et al., 14 Jun 2025)
Security of atomic patterns, inherent SPA leakage (Li, 10 Sep 2024, Sigourou et al., 4 Dec 2024)
Pippenger MSM parallel implementations (Ohno et al., 17 Feb 2025)
Mixed-base representations reducing inversion count (Eid et al., 2019)

Conclusion

Right-to-left parallel elliptic curve scalar point multiplication emerges as a culmination of algorithmic scheduling, representation theory, device-aware architectural optimization, and cryptographic security analysis. Theoretical results confirm the near-optimality of NAF-based right-to-left processing. Implementation strategies, including hybrid multipliers, task mapping for parallel architectures, and higher-dimensional decompositions, deliver significant speed and area savings in hardware and batch-optimized software contexts. However, attention to side-channel leakage at the implementation level remains crucial to preserve cryptographic security in practical deployments.