Alice's RSA public key [N:e] is stored in binary as [1111010011000111:00001011]. Eve wants to find Alice's private key by factorising N.

(a) What are the factors of N and what is Alice's private key?

(b) What is the maximum number of ASCII letters that can be encrypted in one go and sent to Alice (assuming the message has no padding)? Explain your answer.

(c) Alice decides to change the e value in her public key. Which of the following choices are not allowed: 3,5,7,13,17,19,35,45? Explain your answer.
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N=1111010011000111 converts to octal: first, put the binary digits into 3s:

1 111 010 011 000 111 = 172307₈

Now convert to decimal:

172307

-2(using decimal arithmetic)

152

-30

1223

-244

  9790

-1958

 78327

-15664

62663=223×281.

So the decimal factors are 223 and 281.

Convert to octal:

223

22

+4(using octal arithmetic)

263

+54

337

 

281

28

+4

341

+70

431

These are 337₈ and 431₈ which convert to binary:

(a) 11011111 and 100011001 are factors in binary (223,281decimal).

 

e=11 (decimal), p=223, q=281, so N=pq=62663, (p-1)(q-1)=222×280=62160.

de=1 modulo (p-1)(q-1),

11d=1 mod 62160; 62161=5651×11+1, so 5651×11=1 mod 62160.

Therefore d=5651 and the private key is (62663:5651).

 

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