<jupyter_start><jupyter_text>AttackIn this case we cannot use the cube root of the message, said approach only works in the case that $m^{e} < N$.  So how do obtain the messages?Hastad's Broadcast AttackSince we have 3 messages for a public key of 3 we can use CRT to calculate what $M^{3}$<jupyter_code>print(ciphertexts)
print(moduli)
x = CRT(ciphertexts,moduli)
root = x.nth_root(3)
from Crypto.Util.number import long_to_bytes as ltb
ltb(root)<jupyter_output><empty_output>
{'fraction_non_alphanumeric': 0.0859538784067086, 'fraction_numerical': 0.008385744234800839, 'mean_word_length': 6.029411764705882, 'pattern_counts': {'":': 0, '<': 6, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'chain_length': '1', 'dir': 'jupyter-structured-clean-dedup', 'id': '639919', 'license': 'MIT', 'n_tokens_mistral': 173, 'n_tokens_neox': 162, 'n_words': 61, 'path': 'crypto/Day 2/Examples/rsa/hastad/solve/hastad_work_and_solve.ipynb', 'repo_name': 'b01lers/bootcamp-training-2020', 'size': '477'}