The RSA Cryptosystem - Understanding Cryptography - page 174

Alice Bob

message x = 4 1. choose p = 3and q = 11

2. n = p · q = 33
3. Φ( n )=(3 − 1)(11 − 1)=20
4. choose e = 3
5. d ≡ e − 1 ≡ 7 mod 20

k pub =(33 , 3)
←−−−−−−−−−−−−

y = x e 4 3 ≡ ≡ 31 mod 33

y =31
−−−−−−−−−−−−→ y d = 31 7

≡ 4 = x mod 33

Note that the private and public exponents fulfill the condition e · d = 3 · 7 ≡

1mod Φ ( n ).

Practical RSA parameters are much, much larger. As can be seen from Table 6.1,
the RSA modulus n should be at least 1024 bit long, which results in a bit length for
p and q of 512. Here is an example of RSA parameters for this bit length:

p = E 0 DF D 2 C 2 A 288 AC E BC 705 EFAB 30 E 4447541 A 8 C 5 A 47 A 37185 C 5 A 9
CB 98389 CE 4 DE 19199 AA 3069 B 404 FD 98 C 801568 CB 9170 EB 712 BF
10 B 4955 CE 9 C 9 DC 8 CE 6855 C 6123 h
q = EBE 0 FCF 21866 FD 9 A 9 F 0 D 72 F 7994875 A 8 D 92 E 67 AEE 4 B 515136 B 2
A 778 A 8048 B 149828 AEA 30 BD 0 BA 34 B 977982 A 3 D 42168 F 594 CA 99
F 3981 DDABFAB 2369 F 229640115 h
n = CF 33188211 FDF 6052 BDBB 1 A 37235 E 0 ABB 5978 A 45 C 71 FD 381 A 91
AD 12 FC 76 DA 0544 C 47568 AC 83 D 855 D 47 CA 8 D 8 A 779579 AB 72 E 635
D 0 B 0 AAAC 22 D 28341 E 998 E 90 F 82122 A 2 C 06090 F 43 A 37 E 0203 C 2 B
72 E 401 FD 06890 EC 8 EAD 4 F 07 E 686 E 906 F 01 B 2468 AE 7 B 30 CBD 670
255 C 1 FEDE 1 A 2762 CF 4392 C 0759499 CC 0 ABECF F 008728 D 9 A 11 ADF h
e = 40 B 028 E 1 E 4 CCF 07537643101 FF 72444 A 0 BE 1 D 7682 F 1 EDB 553 E 3
AB 4 F 6 DD 8293 CA 1945 DB 12 D 796 AE 9244 D 60565 C 2 EB 692 A 89 B 888
1 D 58 D 278562 ED 60066 DD 8211 E 67315 CF 89857167206120405 B 0
8 B 54 D 10 D 4 EC 4 ED 4253 C 75 FA 74098 FE 3 F 7 FB 751 FF 5121353 C 554
391 E 114 C 85 B 56 A 9725 E 9 BD 5685 D 6 C 9 C 7 EED 8 EE 442366353 DC 39 h
d = C 21 A 93 EE 751 A 8 D 4 FBFD 77285 D 79 D 6768 C 58 EBF 283743 D 2889 A 3
95 F 266 C 78 F 4 A 28 E 86 F 545960 C 2 CE 01 EB 8 AD 5246905163 B 28 D 0 B
8 BAABB 959 CC 03 F 4 EC 499186168 AE 9 ED 6 D 88058898907 E 61 C 7 CC
CC 584 D 65 D 801 CFE 32 DFC 983707 F 87 F 5 AA 6 AE 4 B 9 E 77 B 9 CE 630 E 2
C 0 DF 05841 B 5 E 4984 D 059 A 35 D 7270 D 500514891 F 7 B 77 B 804 BED 81 h