"""
    https://arxiv.org/abs/1508.06574
    "An encryption scheme is said to be homomorphic 
    if certain mathematical operations can be applied 
    directly to the cipher text in such a way that 
    decrypting the result renders the same answer as 
    applying the function to the original unencrypted 
    data."
    The function = GradientBoostingRegressor
    the cipher text = X_encrypted
    original data = X
    same answer = mean absolute error
    """
    import numpy as np
    from sklearn.metrics import mean_absolute_error
    from sklearn.ensemble import GradientBoostingRegressor

    # Replicability
    np.random.seed(0)

    # Create a data set with 1000 samples and 3 features
    X = np.random.randint(0, 60, (1000,3))

    # Create ground truth (the product of the three 
    # features - 100) / 11
    y = (np.prod(X, axis=1) - 100) / 11.
    
    # Encrypt y
    y_encrypted = y + 20

    # Encrypt X
    X_encrypted = X * -0.5

    # Init our model
    rgr = GradientBoostingRegressor(random_state=42)

    # Fit model on first 500 unencrypted features
    rgr.fit(X[:500], y[:500])

    # Predict the remaining 500 features
    preds = rgr.predict(X[500:])

    # Fit model on first 500 encrypted features
    rgr.fit(X_encrypted[:500], y[:500])

    # Predict the remaining encrypted features and decrypt
    preds_decrypted = rgr.predict(X_encrypted[500:]) - 20

    # Evaluate both functions
    print(mean_absolute_error(preds, y[500:]))
    print(mean_absolute_error(preds_decrypted, y[500:]))

    #>>> 323.09
    #>>> 323.72