Here's a simplified explanation of how the Kangaroo algorithm works in this context:

Tame Kangaroos: These kangaroos make small jumps and keep track of their positions as they move along the curve. They're responsible for exploring the region near the expected position of
𝑘
k and
𝑄
Q.

Wild Kangaroos: These kangaroos make large jumps and are used to cover a wider range of possible positions of
𝑘
k and
𝑄
Q.

Collision Detection: The algorithm looks for collisions, which occur when a tame kangaroo and a wild kangaroo land on the same point on the curve. This indicates a potential match for
𝑘
k and
𝑄
Q.

Private Key Extraction: Once a collision is detected, you can extract the private key
𝑘
k from the positions of the kangaroos at the collision point.

The key concept here is that you're not searching through a numerical range of
𝑘
k values linearly.

Instead, you're using kangaroos to explore the curve in a smart way, leveraging collisions to pinpoint the private key without exhaustively trying every possible
𝑘
k value.