Tut 6 Qn 5, TB Pg' 192

1
Tut 6Qn 5, TB Pg. 192

• (a).
• Consider the signing equation s a-1(m-kr) (mod
  p-1). Show that the verification am (aa)srr
  (mod p) is a valid verification procedure.

2

• Substituting,
• r ak (mod p)
• s a-1(m-kr) (mod p-1)
• Fermats Little Theorem
• ak mod (p-1) (mod p) ak (mod p)
• RHS,
• (aa)srr (mod p)
• (aa) a-1(m-kr) (mod p-1) ak (mod p) r (mod p)
• ((aa) a-1(m-kr) (mod p-1) ) (mod p) (akr) (mod
  p)
• (a(m-kr) (mod p-1) ) (mod p) (akr) (mod p)
• a(m-kr) (akr) (mod p)
• am (mod p) LHS

3

• (b).
• Consider the signing equation s am kr (mod
  p-1). Show that the verification as (aa)mrr
  (mod p) is a valid verification procedure.

4

• Substituting,
• r ak (mod p)
• s am kr (mod p-1)
• Fermats Little Theorem
• ak mod (p-1) (mod p) ak (mod p)
• RHS,
• (aa)mrr (mod p)
• (aa)m ak (mod p) r (mod p)
• aam (mod p) (akr) (mod p)
• aam kr (mod p)
• as (mod p) LHS

5

• (c).
• Consider the signing equation s ar km (mod
  p-1). Show that the verification as (aa)rrm
  (mod p) is a valid verification procedure.

6

• Substituting,
• r ak (mod p)
• s ar km (mod p-1)
• Fermats Little Theorem
• ak mod (p-1) (mod p) ak (mod p)
• RHS,
• (aa)rrm (mod p)
• aar.ak (mod p) m (mod p)
• aar.akm (mod p)
• aar km (mod p)
• as (mod p) LHS