one_way_functions_examples

Here are a few examples of mathematical operations that are easy to compute in one direction but extremely difficult to reverse:

Multiplication of large prime numbers:

It is easy to multiply two large prime numbers to obtain their product. However, given only the product, it is extremely difficult to factor it back into the original prime numbers. This concept is the basis of the RSA encryption algorithm.

Example:

Let's say we have two prime numbers: p = 17 and q = 23. Multiplying them is easy: n = p × q = 17 × 23 = 391. However, given only n = 391, it is computationally challenging to determine the original prime factors (p and q).

Modular exponentiation:

Computing the result of raising a number to a power modulo a large number is relatively easy. However, given the result and the modulus, finding the original exponent is extremely difficult (known as the Discrete Logarithm Problem). This concept is used in various cryptographic algorithms, such as the Diffie-Hellman key exchange.

Example:

Let's say we have a number a = 5, an exponent x = 12, and a modulus m = 23. Computing (5^12) mod 23 is easy: (5^12) mod 23 = 18. However, given only the result (18) and the modulus (23), finding the original exponent (12) is computationally infeasible.

Elliptic curve point multiplication:

Multiplying a point on an elliptic curve by a scalar is easy. However, given only the resulting point, finding the original scalar multiplier is extremely difficult (known as the Elliptic Curve Discrete Logarithm Problem). Elliptic curve cryptography relies on this property for secure key generation and digital signatures.

Example:

Let's say we have an elliptic curve point P and a scalar k. Computing the point multiplication Q = k × P is relatively straightforward. However, given only the resulting point Q, finding the original scalar k is computationally infeasible.

These mathematical operations form the foundation of various cryptographic algorithms and protocols. The difficulty of reversing these operations is what provides the security and integrity of the cryptographic systems. It's important to note that the difficulty of reversing these operations is based on the current state of mathematical knowledge and computational capabilities. As mathematical research advances and computational power increases, the security of these operations is continually evaluated and adjusted accordingly.