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In an RSA-encryption scenario, Bob's public key pair $(n, e)$ is $(143, 43)$. An attacker Mallory tries brute-force and comes to $d = 7$ as the private key.

The value of $φ(143) = 120$ is not known to Mallory.

However from $43 \cdot d \equiv 1 \bmod 120$, one can calculate the first positive element $d = 67$ from congruence class $d = 67 + 120n$ and $n \in \mathbb{Z}$

$d = 7$ clearly doesn't fit in that congruence class, so how come it can successfully decrypt the encryption?

In an RSA-encryption scenario, Bob's public key pair $(n, e)$ is $(143, 43)$. An attacker Mallory tries brute-force and comes to $d = 7$ as the private key.

The value of $φ(143) = 120$ is not known to Mallory.

However from $43 \cdot d \equiv 1 \pmod{120}$, one can calculate the first positive element $d = 67$ from congruence class $d = 67 + 120n$ and $n \in \mathbb{Z}$

$d = 7$ clearly doesn't fit in that congruence class, so how come it can successfully decrypt the encryption?

In an RSA-encryption scenario, Bob's public key pair $(n, e)$ is $(143, 43)$. An attacker Mallory tries brute-force and comes to $d = 7$ as the private key.

The value of $φ(143) = 120$ is not known to Mallory.

However from $43 \cdot d \equiv 1 \bmod 120)$, one can calculate the first positive element $d = 67$ from multiplicative group $d = 120 + 67n$ and $n \in \mathbb{Z}$

$d = 7$ clearly doesn't fit in that multiplicative group, so how come it can successfully decrypt the encryption?

In an RSA-encryption scenario, Bob's public key pair $(n, e)$ is $(143, 43)$. An attacker Mallory tries brute-force and comes to $d = 7$ as the private key.

The value of $φ(143) = 120$ is not known to Mallory.

However from $43 \cdot d \equiv 1 \bmod 120$, one can calculate the first positive element $d = 67$ from congruence class $d = 67 + 120n$ and $n \in \mathbb{Z}$

$d = 7$ clearly doesn't fit in that congruence class, so how come it can successfully decrypt the encryption?

In an RSA-encryption scenario, Bob's public key pair $(n, e)$ is $(143, 43)$. An attacker Mallory tries brute-force and comes to $d = 7$ as the private key.

The value of $φ(143) = 120$ is not know to Mallory.

However from $43 ⊙ d ≡ 1$ $(mod$ $120)$, one can calculate the first positive element $d = 67$ from multiplicative group $d = 120 + 67n$ and $n ∈ ℤ$

$d = 7$ clearly doesn't fit in that multiplicative group, so how come it can successfully decrypt the encryption?

In an RSA-encryption scenario, Bob's public key pair $(n, e)$ is $(143, 43)$. An attacker Mallory tries brute-force and comes to $d = 7$ as the private key.

The value of $φ(143) = 120$ is not known to Mallory.

However from $43 \cdot d \equiv 1 \bmod 120)$, one can calculate the first positive element $d = 67$ from multiplicative group $d = 120 + 67n$ and $n \in \mathbb{Z}$

$d = 7$ clearly doesn't fit in that multiplicative group, so how come it can successfully decrypt the encryption?