Question

Problem 6: There are three users with pairwise relatively prime moduli n, n and n3. Suppose that their encryption exponents are all e3. The same message m is sent to each of them and you intercept the three ciphertexts ci mrs (mod n.), for i-1, 2, 3. (a) Show that 0 m3< nin2n (b) Show how to use the CRT to find m3 (as an exact integer, not only as m3 (mod ninns)) and, therefore also m c) Suppose that and the corresponding ciphertexts are 1985835273318, 778146848031, 38058141961777 These were all encrypted with e 3. Find the message

Answer 1

b the hin Re maind ? 一,jeor( m, the soluㅓ,on fo (,- ms nod ne ond

Using mod n, n AK have ni : 246 924 7531 693 151 ond 29 99 43 have exd-1(mod g(n)) also we ol as:

d: 164 616 2148 67 Encrpt a mesiog m with the pohtic ky (ne) tl: ciphi txt mod n ond the maagA "As 21121,8,'211,2 M2 211211212 p 3333383 Q 3333 401 Φ(n) : P-1)x(q-): 1: 111 49653 8800 4m·Hu chosen values ob P,q, and e, AI dl: 4o7 6b3 70 3367

21, 2112112112 P: 666661 66666 82 r 444222 2 111 44419 2088 88 11 d 246 29 292 0oTs