Decrypt the following and showing the mathematical steps for each: You must use
ID: 3713028 • Letter: D
Question
Decrypt the following and showing the mathematical steps for each:You must use RSA method- Public keys of n = 55 and e = 7, with private key 23
1502 2503 2415 2149 2003 Decrypt the following and showing the mathematical steps for each:
You must use RSA method- Public keys of n = 55 and e = 7, with private key 23
1502 2503 2415 2149 2003
You must use RSA method- Public keys of n = 55 and e = 7, with private key 23
1502 2503 2415 2149 2003 Public keys of n = 55 and e = 7, with private key 23
1502 2503 2415 2149 2003
Explanation / Answer
Given
n= 55
e (public key) = 7
d (private key) = 23
M(plain text) = Cd mod n
M = 1502 23 mod 55 {150223 is too big to be calculated so break it into parts so that it can be calculated)
= (150221 mod 55 * 1502 2 mod 55 ) mod 55
= (150221 mod 55 * 14 ) mod 55 {break 21 into 3 - 7 times}
= ((((( (15023 mod 55 * 15023 ?mod 55) mod 55 * 15023 ?mod 55) mod 55 *15023 ?mod 55)mod 55 * 15023 ?mod 55 )mod 55* 15023 ?mod 55)mod 55 * 15023 ?mod 55)mod 55 *14 ) mod 55
= ((((((((18 * 18) mod 55 * 18 )mod 55 * 18 )mod 55 * 18)mod 55) * 18 )mod55 *18)mod55 *14)mod 55)
=((((((49*18)mod 55 * 18 )mod 55 * 18)mod 55) * 18 )mod55 *18)mod55 *14)mod 55)
=(((((2 * 18 )mod 55 * 18)mod 55) * 18 )mod55 *18)mod55 *14)mod 55)
=((((36 * 18)mod 55) * 18 )mod55 *18)mod55 *14)mod 55)
=(((43 * 18 )mod55 *18)mod55 *14)mod 55)
=(((4 *18)mod55 *14)mod 55)
=((17 *14)mod 55)
=18
M= 250323 mod 55
M= 2503 mod 55 * 250322mod 55
M= 28 * 250322mod 55
where ,
250322mod 55 = (14 * 250320mod 55 )mod 55
2. 250320mod 55 = (250310 mod 55 * 250310? mod 55 )mod 55
3. 250310mod 55 = ((((25032mod 55 * 25032 mod 55)mod55 * 25032 mod 55)mod55 * 25032 mod 55)mod55 * 25032 mod 55)mod55 )mod 55
= (14 *14)mod55 * 14)mod55 *14 )mod55 *14 )mod 55
= (((31* 14)mod55 *14 )mod55 *14 )mod 55
= (((49 *14 )mod55 *14 )mod 55
=(((26*14)mod 55
=34
so , after backtracking---
250320mod 55 = (250310 mod 55 * 250310? mod 55 )mod 55 = 34*34 mod 55 = 1
250322mod 55 = (14 * 250320mod 55 )mod 55 = 14*1mod55 =14
M= 28 * 250322mod 55 = 28*14 mod 55 =7
M=7
M=241523mod 55
M=( 241522 mod 55 * 2415 mod 55)mod 55
M=( 241522 mod 55 * 50)mod 55
=(((( 25 * 25)mod55 *25)mod55 *25)mod55 *25)mod 55
=(((20 *25 )mod 25 *25)mod55 *25)mod 55
=(((5 *25)mod55 *25)mod 55= (15 *25 mod 55) = 45
SO after backtracking
241520 mod 55 = (241510 mod 55 * 241510 mod 55 )mod 55
= 45 * 45 mod 55 = 45
241522 mod 55 = ( 25 * 241520 mod 55 )mod 55 = 25 * 45 mod 55 = 25
M=( 241522 mod 55 * 50)mod 55 = 25 * 50 mod 55 = 40
M=40
214923 mod 55 = (2149 21mod 55 * 21492mod 55 )mod 55
=(2149 21mod 55 * 16 )mod 55
= ((((((9 *9) mod55 * 9 )mod55 *9)mod 55 *9)mod55*9)mod 55*9)mod 55
= ((((((49 * 9 )mod55 *9)mod 55 *9)mod55*9)mod 55*9)mod 55
= ((((1 *9)mod 55 *9)mod55*9)mod 55*9)mod 55
=((26*9)mod 55*9)mod 55
=14*9 mod 55= 16
M=214923 mod 55 = (2149 21mod 55 * 16 )mod 55 = 16 * 16 mod 55 = 36
M=36
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