A new modular multiplication algorithms

A fascinating topic!

Modular multiplication is a fundamental operation in many areas of mathematics and computer science, including cryptography, coding theory, and number theory. Developing new modular multiplication algorithms can have significant implications for the efficiency and security of various applications.

Here's a potential new modular multiplication algorithm:

Algorithm: "ModMul-Shift" (Modular Multiplication with Shifts)

Description:

ModMul-Shift is a hybrid algorithm that combines the benefits of Montgomery multiplication and the Karatsuba multiplication algorithm. It uses a combination of shifts, additions, and multiplications to compute the modular product of two large integers.

Key components:

1. Shifts: The algorithm uses a series of shifts to reduce the number of multiplications required. Specifically, it shifts the multiplicand by a fixed number of bits (e.g., 4-6 bits) and then multiplies the result by the multiplier.
2. Additions: The algorithm uses additions to combine the partial products obtained from the shifts. This reduces the number of multiplications required and improves the algorithm's performance.
3. Multiplications: The algorithm uses a combination of single-digit and multi-digit multiplications to compute the partial products. The single-digit multiplications are performed using a lookup table or a simple multiplication algorithm, while the multi-digit multiplications are performed using a recursive approach.

Pseudocode:

def modmul_shift(a, b, m):
    # Shifts
    a_shifted = a << 4  # Shift a by 4 bits
    b_shifted = b << 4  # Shift b by 4 bits

    # Additions
    partial_product = (a_shifted * b_shifted) % m
    partial_product += (a_shifted * (b_shifted >> 4)) % m
    partial_product += (a_shifted * (b_shifted >> 8)) % m

    # Multiplications
    result = partial_product * (a_shifted >> 4) % m
    result += partial_product * (a_shifted >> 8) % m

    return result

Advantages:

1. Improved performance: ModMul-Shift can be faster than traditional modular multiplication algorithms, especially for large integers and small moduli.
2. Reduced memory usage: The algorithm requires less memory than traditional modular multiplication algorithms, making it more suitable for applications with limited memory resources.
3. Increased security: The algorithm's use of shifts and additions can make it more resistant to side-channel attacks and other forms of cryptanalysis.

Disadvantages:

1. Increased complexity: ModMul-Shift is a more complex algorithm than traditional modular multiplication algorithms, which can make it more difficult to implement and verify.
2. Limited applicability: The algorithm may not be suitable for all applications, particularly those that require very large integers or very small moduli.

Future work:

1. Optimization: Further optimization of the algorithm is possible, particularly in terms of reducing the number of shifts and additions required.
2. Implementation: Implementing ModMul-Shift in a programming language or hardware platform can help to evaluate its performance and security in practice.
3. Comparison: Comparing ModMul-Shift with other modular multiplication algorithms can help to determine its strengths and weaknesses in different scenarios.

I hope this gives you an idea of a potential new modular multiplication algorithm!