Public-Key Math Essentials Checkpoint

Worked scenarios: trace state, diagnose mistakes, and apply the module contract.

Get every question correct to pass this checkpoint. You can review feedback, revise your answers, and submit again.

1. Find the least nonnegative inverse of 7 modulo 20.

2. Why does 6 have no multiplicative inverse modulo 15?

Select one answer.

The extended Euclidean algorithm requires both inputs prime 6 is smaller than 15 gcd(6,15)=3, so 6x can never be congruent to 1 15 is odd

3. Solve 6x ≡ 8 (mod 14). Give both residues in ascending order, comma-separated.

4. Two residues satisfy x≡1 mod4 and x≡3 mod6. Which conclusion is correct?

Select one answer.

The unique modulus is always 24 No solution exists because the moduli are not coprime A solution exists modulo 12 because both residues agree modulo gcd(4,6)=2 Every integer is a solution

5. Find the least nonnegative x satisfying x≡2 mod5 and x≡3 mod7.

6. A programmer computes an RSA private exponent as e inverse modulo n. What is the conceptual error?

Select one answer.

RSA requires ordinary real-number division The inverse must be modulo phi(n) or lambda(n), governing exponent cycles The public exponent must equal n The inverse must always be modulo e

7. For toy RSA primes 5 and 11, what is lambda(n)=lcm(p-1,q-1)?

8. Square-and-multiply computes (a*b) % m using a fixed-width type. Both a,b are below m. Is overflow ruled out?

Select one answer.

No; their product can overflow before reduction Yes if m is prime Yes; modulo always happens before multiplication Only negative operands can overflow

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