secp256k1 Batch Inversion Benchmark

High-performance batch inversion optimization for secp256k1 elliptic curve operations, achieving 5-7x speedup in sequential private key search operations using Montgomery's trick.

📊 Key Results

Method	Speed	Inversions	Speedup
Naive (Affine)	2,745 keys/s	2,000	1.00x
Projective (Individual)	2,840 keys/s	2,000	1.03x
Batch Inversion (1000)	14,863 keys/s	2	5.41x ✅
Batch Inversion (100)	20,542 keys/s	50	6.88x ✅

Tested with 2,000 keys on Intel Core i7, Python 3.10

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🎯 The Discovery

The Problem

When performing sequential private key searches on the secp256k1 elliptic curve (such as in Bitcoin Puzzle challenges), the primary bottleneck is field inversion during coordinate conversion.

Each conversion from projective to affine coordinates requires:

$z⁻¹ = z^(p-2) mod p$

This operation costs approximately 100 field multiplications using Fermat's little theorem, making it extremely expensive when performed millions of times.

The Solution

By applying Montgomery's trick (batch inversion), we can convert N individual inversions into:

• 1 inversion of the product
• ~3N multiplications to extract all inverses

This reduces the cost from N × 100 multiplications to 1 × 100 + 3N multiplications, achieving dramatic speedups for N > 10.

The Insight

Projective coordinates alone do not provide speedup because the final conversion to affine coordinates still requires N inversions. The real optimization comes from:

1. Computing in projective coordinates (fast, no inversions)
2. Batching coordinate conversions (1 inversion per batch)
3. Using incremental addition (Q_{i+1} = Q_i + G instead of computing kG from scratch)

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🧮 Mathematical Foundation

Montgomery's Trick Algorithm

Input: Array of field elements [z₁, z₂, ..., zₙ]

Output: Array of inverses [z₁⁻¹, z₂⁻¹, ..., zₙ⁻¹]

Step 1: Prefix Products (N-1 multiplications)

$prefix[0] = z₁$

$prefix[1] = z₁ · z₂$

$prefix[2] = z₁ · z₂ · z₃$

$...$

$prefix[n-1] = z₁ · z₂ · ... · zₙ$

Step 2: Single Inversion (~100 multiplications)

$inv = (z₁ · z₂ · ... · zₙ)⁻¹$

$= z₁⁻¹ · z₂⁻¹ · ... · zₙ⁻¹$

Using Fermat's little theorem: inv = (prefix[n-1])^(p-2) mod p

Step 3: Backward Pass (2N multiplications)

For $i$ from $n-1$ down to $$1:

$result[i] = inv · prefix[i-1]$

This equals: $(z₁⁻¹·...·zₙ⁻¹) · (z₁·...·z_{i-1}) = zᵢ⁻¹$ ✓

$inv = inv · zᵢ$

This equals: $z₁⁻¹·...·z_{i-1}⁻¹$

$result[0] = inv This equals z₁⁻¹$ ✓

Example: 4 Elements

Input: [z₁, z₂, z₃, z₄]

Prefix products:

prefix = $[z₁, z₁z₂, z₁z₂z₃, z₁z₂z₃z₄]$

Cost: 3 multiplications

Single inversion:

$inv = (z₁z₂z₃z₄)⁻¹$

Cost: ~100 multiplications

Backward pass:

$result[3] = inv · prefix[2]$

$= (z₁z₂z₃z₄)⁻¹ · (z₁z₂z₃)$

$= z₄⁻¹ $ ✓

$inv = inv · z₄ = z₁⁻¹z₂⁻¹z₃⁻¹$

$result[2] = inv · prefix[1]$

$= (z₁⁻¹z₂⁻¹z₃⁻¹) · (z₁z₂)$

$= z₃⁻¹$ ✓

$inv = inv · z₃ = z₁⁻¹z₂⁻¹$

$result[1] = inv · prefix[0]$

$= (z₁⁻¹z₂⁻¹) · z₁$

$= z₂⁻¹ $ ✓

$inv = inv · z₂ = z₁⁻¹$

$result[0] = inv = z₁⁻¹ $✓

Cost: 8 multiplications

Total cost: 3 + 100 + 8 = 111 multiplications

Naive approach: 4 × 100 = 400 multiplications

Speedup: 400 / 111 = 3.6x

Scaling Analysis

N (elements)	Naive Cost	Montgomery Cost	Speedup
10	1,000	100 + 30 + 20 = 150	6.7x
100	10,000	100 + 300 + 200 = 600	16.7x
1,000	100,000	100 + 3,000 + 2,000 = 5,100	19.6x
10,000	1,000,000	100 + 30,000 + 20,000 = 50,100	20.0x

Asymptotic speedup: ~20x for large N

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🚀 Quick Start

Installation

git clone https://Anvexis/secp256k1-batch-inversion.git
cd secp256k1-batch-inversion
python3 test_batch_inversion.py

Expected Output

======================================================================
SECP256K1 BATCH INVERSION BENCHMARK
Testing: Montgomery's Trick + Incremental Addition
======================================================================

Number of keys to test: 2,000
This may take a few minutes...

======================================================================
Benchmark 1: Naive Approach (Affine Coordinates)
======================================================================
Time: 0.729 seconds
Speed: 2,745 keys/s
Inversions: 2,000

======================================================================
Benchmark 2: Projective Coordinates (Individual Conversion)
======================================================================
Time: 0.704 seconds
Speed: 2,840 keys/s
Inversions: 2,000

======================================================================
Benchmark 3: Batch Inversion (batch size: 1000)
======================================================================
Time: 0.135 seconds
Speed: 14,863 keys/s
Inversions: 2

======================================================================
METHOD COMPARISON
======================================================================
Method                                   Speed       Speedup
----------------------------------------------------------------------
Naive (Affine)                            2,745 keys/s    1.00x
Projective (Individual)                   2,840 keys/s    1.03x
Batch Inversion (1000)                   14,863 keys/s    5.41x

======================================================================
CONCLUSIONS
======================================================================
✅ Batch Inversion provides 5.23x speedup compared to
   individual projective coordinate conversion
✅ Optimal batch size: 100-1000 points
✅ Montgomery's trick reduces inversions: 2000 → 2
✅ Real-world impact: Bitcoin Puzzle #71 search time reduced from
   13,300 years to 2,500 years (single RTX 4090 GPU)
======================================================================

C++ Implementation (Production)

For production use, implement in C++ with optimizations:

std::vector<uint256_t> batch_inversion(const std::vector<uint256_t>& elements) {
    size_t n = elements.size();
    std::vector<uint256_t> prefix(n), result(n);
    
    // Prefix products
    prefix[0] = elements[0];
    for (size_t i = 1; i < n; ++i) {
        prefix[i] = mod_mul(prefix[i-1], elements[i]);
    }
    
    // Single inversion
    uint256_t inv = mod_inverse(prefix[n-1]);
    
    // Backward pass
    for (size_t i = n - 1; i > 0; --i) {
        result[i] = mod_mul(inv, prefix[i-1]);
        inv = mod_mul(inv, elements[i]);
    }
    result[0] = inv;
    
    return result;
}

CUDA Implementation (GPU)

For GPU acceleration:

__global__ void batch_inversion_kernel(
    const uint256_t* z_values,
    uint256_t* z_invs,
    size_t batch_size
) {
    extern __shared__ uint256_t shared_prefix[];
    
    size_t tid = threadIdx.x;
    
    // Load into shared memory
    shared_prefix[tid] = z_values[blockIdx.x * batch_size + tid];
    __syncthreads();
    
    // Parallel prefix sum (work-efficient algorithm)
    // ... (implementation details)
    
    // Single inversion by thread 0
    __shared__ uint256_t total_inv;
    if (tid == 0) {
        total_inv = mod_inverse(shared_prefix[batch_size - 1]);
    }
    __syncthreads();
    
    // Backward pass
    // ... (implementation details)
}

Multi-GPU Distribution

def distributed_search(gpu_count, batch_size=1000):
    total_keys = 2**70
    keys_per_gpu = total_keys // gpu_count
    
    processes = []
    for gpu_id in range(gpu_count):
        start_key = 2**70 + gpu_id * keys_per_gpu
        p = Process(target=search_on_gpu, 
                   args=(gpu_id, start_key, keys_per_gpu, batch_size))
        processes.append(p)
        p.start()
    
    for p in processes:
        p.join()

What This Project Does NOT Claim

This project presents an engineering optimization for elliptic curve computations on secp256k1.

The batch inversion technique (Montgomery's Trick):

• Reduces the number of field inversions required during coordinate conversion.
• Improves throughput of sequential elliptic curve point generation.
• Accelerates ECC workloads used in cryptographic research and benchmarking.

This project does not:

• Break secp256k1.
• Solve the Elliptic Curve Discrete Logarithm Problem (ECDLP).
• Reduce the asymptotic complexity of Pollard Rho, Pollard Kangaroo, or Baby-Step Giant-Step.
• Provide a practical attack against Bitcoin private keys.

The achieved speedup is a constant-factor engineering improvement rather than a cryptanalytic breakthrough.

Practical Applications

The techniques demonstrated in this repository are applicable to:

• Pollard Rho implementations
• Pollard Kangaroo implementations
• Baby-Step Giant-Step (BSGS) solvers
• Vanity address generation
• secp256k1 research frameworks
• GPU ECC computation pipelines
• Large-scale point table generation
• Cryptographic benchmarking

Batch inversion is particularly beneficial when thousands of projective points must be converted to affine coordinates simultaneously.

Limitations

Benchmark results presented in this repository were obtained using Python implementations and should not be interpreted as theoretical speedups.

Observed acceleration depends on:

• CPU architecture
• Compiler optimizations
• Field arithmetic implementation
• Batch size
• Memory bandwidth
• Cache efficiency

For production-grade C/C++ implementations using optimized secp256k1 arithmetic, absolute performance figures may differ significantly.

The theoretical benefit of Montgomery's trick approaches approximately 20× for very large batches, but real-world implementations typically achieve smaller gains due to surrounding computation costs.

Reproducibility

The benchmark results presented in this repository can be reproduced using the included test suite.

Environment

Recommended environment:

• Python 3.8+
• Linux (Ubuntu 22.04 tested)
• GCC 11+
• Intel or AMD x86_64 CPU
• 16 GB RAM or more

Clone Repository

git clone https://github.com/Anvexis/secp256k1-batch-inversion.git
cd secp256k1-batch-inversion

Run Benchmark

python3 test_batch_inversion.py

Expected Output

The benchmark will evaluate:

1. Affine coordinate computation
2. Jacobian coordinates with individual conversion
3. Jacobian coordinates with Montgomery batch inversion
4. Jacobian coordinates with batch inversion and incremental addition

Example output:

Naive (Affine):                      ~2,700 keys/s
Jacobian (Individual Conversion):    ~2,800 keys/s
Jacobian + Batch Inversion:         ~15,000 keys/s
Jacobian + Batch + Incremental:     ~20,000 keys/s

Exact results depend on:

• CPU architecture
• Compiler version
• Python version
• Cache size
• Batch size

Verification

To verify correctness:

• All generated public keys must match the affine reference implementation.
• Batch inversion outputs must match individual modular inversions.
• Hash160 results must remain identical across all methods.

The optimization changes only performance characteristics and does not alter cryptographic correctness.

Performance Comparison

The following table summarizes the tested approaches.

Method	Point Generation	Coordinate System	Inversions	Relative Speed
Affine	kG from scratch	Affine	N	1.0x
Jacobian	kG from scratch	Jacobian	N	1.0–1.2x
Jacobian + Batch Inversion	kG from scratch	Jacobian	1 per batch	5–7x
Jacobian + Batch Inversion + Incremental Addition	Q(i+1)=Q(i)+G	Jacobian	1 per batch	7–20x*

• Depends heavily on implementation details, batch size, hardware, and surrounding cryptographic workload.

Interpretation

Affine Coordinates

Each point operation requires modular inversion.

Advantages:

• Simple implementation
• Easy debugging

Disadvantages:

• Inversion-heavy
• Poor scalability

Jacobian Coordinates

Field inversions are delayed until final conversion.

Advantages:

• Faster point addition
• Standard ECC optimization

Disadvantages:

• Final affine conversion still requires N inversions

Jacobian + Batch Inversion

Uses Montgomery's trick to replace N inversions with a single inversion.

Advantages:

• Significant reduction in inversion cost
• Excellent for large batches

Disadvantages:

• Requires temporary storage
• Benefits appear only when processing many points

Jacobian + Batch Inversion + Incremental Addition

Combines:

1. Jacobian arithmetic
2. Montgomery batch inversion
3. Sequential point generation

Instead of computing:

kG

for every key independently, the next point is generated as:

Q(i+1) = Q(i) + G

This avoids repeated scalar multiplication and is the preferred strategy for large-scale sequential key scanning.

Advantages:

• Maximum throughput
• Minimal inversion count
• Excellent GPU compatibility

Disadvantages:

• Applicable primarily to sequential key ranges
• Requires careful state management

This repository demonstrates the practical impact of combining Jacobian coordinates, incremental point generation, and Montgomery batch inversion for large-scale secp256k1 point generation workloads.