Question

Explain how Bob will decrypt the message. 
9) a) Solve the discrete logarithm problem equation using Baby-Step, Giant-Step algorithm. 

09 Jan 2026
Answer :
Word Count : 722
We are asked to solve (5^x \equiv 22 \pmod{47}) using the Baby-Step Giant-Step (BSGS) algorithm. Step 1: Determine (m = \lceil \sqrt{p-1} \rceil), where (p = 47). [ p-1 = 46, \quad \sqrt{46} \approx 6.78 \implies m = 7 ] Step 2: Compute the baby steps (5^j \mod 47) for (j = 0, 1, \dots, m-1). * (j=0: 5^0 \equiv 1 \pmod{47}) * (j=1: 5^1 \equiv 5 \pmod{47}) * (j=2: 5^2 \equiv 25 \pmod{47}) * (j=3: 5^3 \equiv 5^2 \cdot 5 \equiv 25 \cdot 5 = 125 \equiv 31 \pmod{47}) (since (125 - 2\cdot47 = 125-94=31)) * (j=4: 5^4 \equiv 31 \cdot 5 = 155 \equiv 155 - 3\cdot47 = 155-141=14 \pmod{47}) * (j=5: 5^5 \equiv 14 \cdot 5 = 70 \equiv 70-47=23 \pmod{47}) * (j=6: 5^6 \equiv 23 \cdot 5 = 115 \equiv 115 - 2\cdot47 = 115-94=21 \pmod{47}) Baby steps table: [ \begin{array}{c|c} j & 5^j \mod 47 \ \hline 0 & 1 \ 1 & 5 \ 2 & 25 \ 3 & 31 \ 4 & 14 \ 5 & 23 \ 6 & 21 \ \end{array} ] Step 3: Compute (c = 5^{-m} \mod 47) for the giant steps. [ m = 7 \implies c = 5^{-7} \mod 47 ] First, find (5^7 \mod 47): [ 5^7 = 5^6 \cdot 5 \equiv 21 \cdot 5 = 105 \equiv 105-2\cdot47 = 105-94=11 \pmod{47} ] Then find the inverse of 11 modulo 47. Solve (11y \equiv 1 \pmod{47}). Check multiples of 47 + 1 divisible by 11: [ 11 \cdot 43 = 473 \equiv 473-10\cdot47=473-470=3 \quad \text{too
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