Additive secret sharing and share automation - using Python

Additive secret sharing is a method of sharing a secret to a group of participants, such that the secret can only be reconstructed when a certain number of participants assemble and offer their shares. This technology is widely used in cryptography and secure multi-party computation. In this article, we will discuss the concepts of additive secret sharing and share preactivation and how to implement them using Python.

Introduction to Additive Secret Sharing

Additive secret sharing is a technique that allows a group of participants to share a secret without revealing the secret to any individual participant. The secret is divided into shares, with each participant receiving one. The secret can only be reconstructed when a certain number of participants come together and offer their shares. This technique is also known as threshold secret sharing.

The basic idea of ​​additive secret sharing is to express the secret as a polynomial of degree (t-1), where t is the threshold. Choose the coefficients of the polynomial to be random integers. Each participant is then assigned a point on the polynomial that represents their share. By interpolating using points provided by participants, the secret can be reconstructed.

For example, suppose we have a secret S that we want to share among three participants A, B, and C. We choose a 2nd degree polynomial with random coefficients. This polynomial is expressed as S(x) = a0 a1x a2x^2. Each participant is then assigned a point (x, S(x)) on the polynomial, where x is a unique value chosen for each participant.

Participant A receives the point (1, S(1))
Participant B receives the point (2, S(2))
Participant C receives the point (3, S(3))

Now, if any two players come together and provide their points, they can reconstruct the polynomial and thus the secret. However, if a single participant only provides their points, they cannot reconstruct the secret.

Implementing additive secret sharing in Python

To implement additive secret sharing in Python, we can use a Python library called "secretsharing", which provides an easy-to-use API to Implement threshold secret sharing. The library provides a function called "split_secret" which accepts a secret, number of shares, and threshold as input and returns a list of shares.

This is an example of sharing a secret between three participants using the "split_secret" function -

from secretsharing import SecretSharer

# Secret to be shared
secret = "mysecret"

# Number of shares to be generated
n = 3

# Threshold value
t = 2

# Generate shares
shares = SecretSharer.split_secret(secret, n, t)

# Print the shares
for share in shares:
   print(share)

In the above example, we There is a secret "mysecret" that we want to share among three participants. We set the threshold value to 2, which means any two participants can reconstruct the secret. The "split_secret" function generates three shares and prints them.

Reconstruct the secret

To reconstruct the secret, we can use the "recover_secret" function provided by the "secretsharing" library. This function accepts a list of shares as input and returns the secret. Here is an example of reconstructing a secret using the "recover_secret" function:

from secretsharing import SecretSharer

# List of shares
shares = ["1-mysecret", "2-mysecret", "3-mysecret"]

# Reconstruct the secret
secret = SecretSharer.recover_secret(shares)

print(secret)

In the above example, we have a list of stocks that we want to use to reconstruct the secret. The "recover_secret" function takes as input a list of stocks and returns the secret. In this example, the secret is "mysecret".

Introducing Share Proactivization

Share Proactivization is a technology that allows a group of participants to proactively refresh their shares without revealing secrets. This technique is used in situations where secrets may need to be changed frequently, such as in the case of shared passwords.

The basic idea of ​​Share Proactivation is that each participant generates a new share for a new secret and sends it to others. New shares are then combined to reconstruct new secrets. Old shares are discarded and new shares are used to reconstruct new secrets.

Implementing Share Activation in Python

To implement Share Proactivization in Python, we can use the same "secretsharing" library as the additive secret sharing. The library provides a function called "create_share" which accepts a secret and a unique share ID as input and returns a new secret share.

This is an example of actively refreshing a share using the "create_share" function -

from secretsharing import SecretSharer

# New secret
new_secret = "mynewsecret"

# Share ID
share_id = "1"

# Create a new share
new_share = SecretSharer.create_share(new_secret, share_id)

print(new_share)

In the above example, we have a new secret "mynewsecret" that we want to be shared among three participants shared between. We also have a share ID "1" representing the participant who will generate the new share. The "create_share" function accepts a new secret and share ID as input and returns the new share of the secret.

To reconstruct a new secret, we can use the same "recover_secret" function as before. However, this time we will use the new shares generated by each participant.

from secretsharing import SecretSharer

# List of new shares
new_shares = ["1-mynewsecret", "2-mysecret", "3-mysecret"]

# Reconstruct the new secret
new_secret = SecretSharer.recover_secret(new_shares)

print(new_secret)

In the example above, we have a new stock list that we want to use to reconstruct new secrets. The "recover_secret" function takes as input a list of new stocks and returns a new secret, in this case "mynewsecret".

Conclusion

Additive secret sharing and sharing automation are powerful techniques that can be used to securely share sensitive information among a group of participants. The "secretsharing" library in Python provides an easy-to-use API for implementing these techniques. By using the "split_secret" and "create_share" functions, we can easily implement additive secret sharing and sharing automation respectively. Moreover, by using the "recover_secret" function, we can easily reconstruct the secret or a new secret.

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