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You want to share a secret within a group in a secure way? You want to make sure, that more than one person is needed to recover the secret so no one can do thing single-handed? You need e.g. four-eyes principle?
Cryptography knows a shared secret as a piece of data, which is only known to parties involved. The data might a text, any number, a passphrase or just bits and bytes.
The slides explain the requirements on how to share a secret, which approaches their are and is showing one solution using an algorithm called "Shamir's Secret Sharing".
This cryptographic algorithm was created by Adi Shamir. The algorithm will divide the secret into parts, where each participant receives its own unique part. Most important is, that some of the parts or all of them are needed in order to reconstruct the secret.
In cryptography Shamir's Secret Sharing is a so called threshold scheme. It means, that not all of the parts are needed, but a number of x parts are sufficient to recover the original secret.
The PHP implementation of the algorithm can be found at https://github.com/teqneers/shamir.
The slides are based on a blog called Data Genetics (http://datagenetics.com/blog/november22012/).