Ĭan you sign the message Trust me, I have root access and send it to the server to receive the FLAG? # Importsįrom import bytes_to_long, long_to_bytes, getPrime Using this the totient(N), we choose the public key e such that 1. We note that this totient function requires knowledge of p and q and can therefore not be computed unless we know the values p and q. The fact that it is computationally infeasible to obtain p and q if we are only given N is the prime (pun intended) observation behind the RSA cryptosystem, because knowledge of p and q is required to create our private key d.Īfter we generated N, we compute Charmichael's totient function totient(N), which is defined as totient(N) = (p-1) * (q-1). However, if we are only given a number N, it is computationally infeasible to obtain the primes p and q used to generate N if they are sufficiently large. It is straightforward to compute N as it is a simple multiplication. Here, we will simply explain how to choose such keys.Īs a first step, we construct N by multiplying two randomly chosen prime numbers p and q, such that N = p * q.īecause of the Fundamental theorem of arithmetic, we know that the only way to write N as the product of primes is using the unique combination of p and q. Instead, we must specifically choose these values such that applying encryption and decryption actually yield the original message.Ĭhoosing these values is based on Fermat's little theorem. However, RSA does not work with any value for N, e and d.
M = pow(c, d, N) # Decryption, d is a private value Setup
RSA is based on modular exponentiation in a group N such that if we have a message m, a public key e and a private key d we can compute the ciphertext c as:Ĭ = pow(m, e, N) # Encryption, e and N are public valuesĪnd compute the original message from the ciphertext as: This is different from symmetric-key cryptosystems such as AES which use the same key for encryption and decryption. This means that encryption and decryption use different components.įor encryption, we use a public component that everybody can use to encrypt messages.įor decryption, we use a private component that only the person who wants to decrypt messages may know. RSA is a widely used public-key (also known as asymmetric-key) cryptosystem.
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