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fix: resolve binary addition and substring issues in Karatsuba algorithm #2851

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fix: correct binary addition and substring handling in the Karatsuba algorithm

  • Fixed the add_strings function to handle binary addition correctly.
  • Improved the safe_substr function to manage leading zeros and extract substrings properly.
  • Ensured correct bit shifting by using 1LL << for 64-bit integers.
  • Added comprehensive self-tests to verify functionality.

This fixes incorrect results from the Karatsuba algorithm for binary multiplication.

Description of Change

Checklist

  • Added description of change
  • Added file name matches File name guidelines
  • Added tests and example, test must pass
  • Added documentation so that the program is self-explanatory and educational - Doxygen guidelines
  • Relevant documentation/comments is changed or added
  • PR title follows semantic commit guidelines
  • Search previous suggestions before making a new one, as yours may be a duplicate.
  • I acknowledge that all my contributions will be made under the project's license.

Notes:

fix: correct binary addition and substring handling in the Karatsuba algorithm

- Fixed the `add_strings` function to handle binary addition correctly.
- Improved the `safe_substr` function to manage leading zeros and extract substrings properly.
- Ensured correct bit shifting by using `1LL <<` for 64-bit integers.
- Added comprehensive self-tests to verify functionality.

This fixes incorrect results from the Karatsuba algorithm for binary multiplication.
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