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52 Bit Limbs

A well-optimized elliptic curve library, such as libsecp256k1, uses a specialized format to optimize arithmetic operations. Here, I will explain how this works and the reasoning behind it.

Why uint64_t[5] and 32 bytes for secp256k1?

To understand the design of the secp256k1 field implementation, we need to break down its field element representation and base system.

Field Element Representation:

The secp256k1 field is a prime field defined by the modulus \(p=2^{256}-2^{32}-977\). Any field element can be represented by a number in the range \(0\) to \(p-1\).

Base \(2^{52}\) Representation:

The field element is represented using five 64-bit unsigned integers (uint64_t). Each integer represents a limb in a base-\(2^{52}\) system. Specifically, the field element \(fe\) is represented as:

\[ fe=\sum_{i=0}^4f.n[i]\cdot 2^{52\cdot i}\ mod\ p \] This means the overall bit-length of the representation is \(5\times 52 = 260\) bits. The reason for choosing \(2^{52}\) as the base is that it allows efficient use of 64-bit operations while providing some "headroom" to handle carries in arithmetic operations.

Why 64-bit Limbs:

Each limb is stored in a 64-bit integer, which fits naturally in 64-bit processors, allowing for efficient computation. The extra bits beyond 52 in each 64-bit limb can be used to handle carries without immediate overflow during intermediate calculations.

32-Byte Total Size:

Although the logical bit-length of the representation is 260 bits, the physical storage uses 320 bits ( \(5 \times 64\) bits). However, practical implementations and the standard often consider the byte size as 32 bytes because the actual field modulus \(p\) fits within 32 bytes, and arithmetic operations typically normalize results to fit within this range.

Understanding the Structure

The structure of a field element in the secp256k1 library is as follows: 

typedef struct {
    uint64_t n[5];
    SECP256k1_FE_VERIFY_FIELDS
} secp251k1_fe;

The uint64_t n[5] array contains five 64-bit "limbs" that represent the field element in base \(2^{52}\). The Magnitude Constraints ensure that each limb (or chunk) doesn't become too large during calculations, keeping the arithmetic operations manageable and efficient.

For a field element to be considered normalized:

  • Each limb \(n[i]\) for \(i = 0\) to \(3\) must be within the range \(0\) to \(2^{53} - 1\).
  • The last limb \(n[4]\) should be within \(0\) to \(2^{48} - 1\).

After any computation, results are reduced modulo \(p\) (the field prime) to ensure they remain within the valid field range.

Reasons for choosing 52 Bits

Choosing 52 bits as the limb size in the representation of field elements is a carefully considered decision based on a balance of several factors:

  1. Efficient Use of 64-bit Registers: Modern processors are optimized for 64-bit operations, making uint64t a natural choice for limbs. A 52-bit limb fits well within a 64-bit register, leaving 12 bits for carry handling during arithmetic operations. This reduces the need for frequent normalization and allows efficient multi-precision arithmetic.
  2. Carry Propagation: Using 52 bits leaves 12 bits for carries, which is significant enough to handle intermediate results during addition and multiplication without immediate overflow. This headroom is essential for reducing the overhead of carry propagation, improving performance.
  3. Alignment and Memory Efficiency: Memory access is more efficient when data is aligned to natural word boundaries (e.g., 64 bits). Using 52 bits per limb ensures that limbs are stored in 64-bit aligned memory, making memory access faster and more efficient.
  4. Simplified Arithmetic Operations: Arithmetic operations (addition, subtraction, multiplication) are simpler and faster when the limbs are within the processor’s natural word size. Operations on 52-bit limbs can utilize the full width of 64-bit registers, optimizing the use of CPU resources.
  5. Balancing Limb Count and Complexity: Choosing 52 bits results in 5 limbs for a 256-bit field element (since ⌈256/52⌉=5⌈256/52⌉=5). This is a manageable number of limbs that balances complexity and performance. Fewer limbs would increase complexity and slow down operations, while more limbs would increase memory usage and possibly decrease efficiency.

Example of an Operation

While it seems like there are wasted bits in each limb, these bits are not truly wasted because they contribute to the overall efficiency and performance of the implementation. The trade-off between bit usage and computational efficiency is common in cryptographic implementations.

Consider the following example of an addition operation: 

void field_add(secp256k1_fe *r, const secp256k1_fe *a, const secp256k1_fe *b) {
    uint64_t carry = 0;
    for (int i = 0; i < 5; i++) {
        uint64_t temp = a->n[i] + b->n[i] + carry;
        r->n[i] = temp & ((1ULL << 52) - 1);  // Mask to 52 bits
        carry = temp >> 52;  // Extract the carry
    }
    // Additional reduction modulo p might be needed
}

The field_add function adds two field elements a and b, storing the result in r. It iterates over the limbs, adding corresponding limbs from a and b, along with any carry from the previous limb. r->n[i] = temp & ((1ULL << 52) - 1); ensures that only the lower 52 bits of the sum are stored in the current limb. carry = temp >> 52; extracts the higher bits of the sum that did not fit into the current limb and propagates the carry to the next limb.

Summary

The choice of 52-bit limbs in the representation of secp256k1 field elements is a deliberate design decision that optimizes performance and efficiency. By balancing the use of 64-bit registers, carry handling, memory alignment, and computational efficiency, this approach ensures that cryptographic operations are both fast and reliable. Understanding these design choices is crucial for implementing and optimizing cryptographic algorithms.

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