## MA-410 Homework 4

Due at the beginning of class,
Wednesday, April 23, 2008

Solutions may be submitted in person in class,
or you may email an ASCII text,
html, or postscipt/pdf-formatted document to me.

Note my office hours on my
schedule.

- Show that the El Gamal public key crpytosystem
is
**malleable** even if Alice encrypts with a randomly chosen *r* every time.
For the example *p = 13*, *g = 2*, *s = 6*, show how
Charlie can encrypt *M/2* with his own random *r'*
seeing Alice's ciphertext of *M* which is assumed to be an even residue.
- Let
*p* be an odd prime such that 3 does not
divide *p-1*. Prove that for any residue *a* modulo
*p* there exists a residue *b* modulo *p* such
that *b*^{3} is congruent to *a* modulo *p*.
- ENT, §9.3, Problem 1(d), page 190. Please use Eisenstein's method.
- ENT, §12.1, Problem 4, page 251.