Appendix G: Exercises - Codes: The Guide to Secrecy from Ancient to Modern Times - page 572

r . See pages 479 and 480 in Appendix A for a discussion of Euler's φ -
function. )

G.4 Chapter 4 Exercises

In Exercises 4.1-4.4, assume that the given ciphertext is formed via the
permutation given in Example 4.1 on page 163. Find the plaintext by
using the inverse permutation and converting to English text from Table
1.3 on page 11.

4.1 c =(8 , 6 , 18 , 19 , 7 , 5 , 11).

4.2 c =(0 , 13 , 12 , 20 , 8 , 2 , 17).

4.3 c =(0 , 18 , 2 , 8 , 18 , 2 , 11).

4.4. c = (24 , 11 , 7 , 18 , 8 , 18 , 19).

4.5. Prove that the DLP presented on page 164 is independent of the generator
m of F p .

( Hint: Pick two generators of

F p and show that the log of any element in
F p to one base can be written in terms of the log of that element to the
other base. This demonstrates that any procedure for calculating logs to
one base can be used to calculate logs to any other base that generates F p .

Hence, any such procedure is independent of the choice of base. )

The Generalized DLP (GDLP) is formulated as follows.

4.6. Given a finite

group G , and elements g,h G , find an integer e such that g e = h ,

assuming such an integer exists. Let e = L g ( h ). Prove that

∈

h )= L g ( h )+ L g ( h ) ,

L g ( h ∗

where h,h ∈ is the group operation, and g e g f = g e + f , for integers

G , ∗ ∗

e,f .

Exercises 4.7-4.10 pertain to the Pohlig-Hellman exponentiation cipher de-
scribed on page 165. In each case use the data to decrypt the ciphertext
and produce the plaintext via Table 1.3 on page 11.

4.7. Let p = 647, e = 67 and c = (119 , 346 , 32 , 499 , 115 , 63 , 346 , 617).

4.8. Let p = 919, e = 47 and c = (40 , 221 , 233 , 294 , 164 , 9 , 814).

4.9. Let p = 173, e = 99 and

c = (132 , 62 , 168 , 137 , 87 , 88 , 170 , 170 , 88 , 137 , 87 , 168 , 0 , 20) .