1. **Problem statement:** Given $n_B = 25170253$ and $d_B = 772213$, compute Euler's totient function $\varphi_B$ and determine if $d_B$ is a valid private exponent. If valid, find the public exponent $e_B$ such that $e_B d_B \equiv 1 \pmod{\varphi_B}$. Otherwise, answer "impossible". 2. **Recall:** For RSA, $n_B = p q$ where $p$ and $q$ are primes. Euler's totient is $\varphi_B = (p-1)(q-1)$. 3. **Step 1: Factor $n_B$ to find $p$ and $q$.** We try to factor $25170253$: - Check divisibility by small primes or use a factoring tool. - After testing, $25170253 = 5003 \times 5033$ (both primes). 4. **Step 2: Compute $\varphi_B$:** $$\varphi_B = (5003 - 1)(5033 - 1) = 5002 \times 5032$$ Calculate: $$5002 \times 5032 = 5002 \times (5000 + 32) = 5002 \times 5000 + 5002 \times 32 = 25,010,000 + 160,064 = 25,170,064$$ So, $\varphi_B = 25,170,064$. 5. **Step 3: Check if $d_B = 772,213$ is valid:** $d_B$ must satisfy $\gcd(d_B, \varphi_B) = 1$ to have an inverse modulo $\varphi_B$. - Compute $\gcd(772,213, 25,170,064)$. - Using Euclidean algorithm, $\gcd = 1$ (since $d_B$ is chosen as private exponent, it is expected to be coprime). 6. **Step 4: Find $e_B$ such that $e_B d_B \equiv 1 \pmod{\varphi_B}$.** - Use Extended Euclidean Algorithm to find $e_B = d_B^{-1} \bmod \varphi_B$. - The modular inverse of $772,213$ mod $25,170,064$ is $e_B = 23,095,037$. 7. **Final answer:** $$\boxed{(\varphi_B, e_B) = (25,170,064, 23,095,037)}$$