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Tut 6 Qn 5, TB Pg. 192

Tut 6 Qn 5, TB Pg. 192. (a). Consider the signing equation s = a -1 (m-kr) (mod p-1). Show that the verification α m ≡ ( α a ) s r r (mod p) is a valid verification procedure. Substituting, r = α k (mod p) s = a -1 (m-kr) (mod p-1) Fermat’s Little Theorem:

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Tut 6 Qn 5, TB Pg. 192

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  1. Tut 6Qn 5, TB Pg. 192 (a). Consider the signing equation s = a-1(m-kr) (mod p-1). Show that the verification αm≡ (αa)srr (mod p) is a valid verification procedure.

  2. Substituting, r = αk (mod p) s = a-1(m-kr) (mod p-1) Fermat’s Little Theorem: αk mod (p-1) (mod p) = αk (mod p) RHS, (αa)srr (mod p) ≡ (αa) a-1(m-kr) (mod p-1)αk (mod p) r (mod p) ≡ ((αa) a-1(m-kr) (mod p-1)) (mod p) (αkr) (mod p) ≡ (α(m-kr) (mod p-1)) (mod p) (αkr) (mod p) ≡ α(m-kr) (αkr) (mod p) ≡ αm (mod p) = LHS

  3. (b). Consider the signing equation s = am + kr (mod p-1). Show that the verification αs≡ (αa)mrr (mod p) is a valid verification procedure.

  4. Substituting, r = αk (mod p) s = am + kr (mod p-1) Fermat’s Little Theorem: αk mod (p-1) (mod p) = αk (mod p) RHS, (αa)mrr (mod p) ≡ (αa)mαk (mod p) r (mod p) ≡ αam (mod p) (αkr) (mod p) ≡ αam + kr (mod p) ≡ αs (mod p) = LHS

  5. (c). Consider the signing equation s = ar + km (mod p-1). Show that the verification αs≡ (αa)rrm (mod p) is a valid verification procedure.

  6. Substituting, r = αk (mod p) s = ar + km (mod p-1) Fermat’s Little Theorem: αk mod (p-1) (mod p) = αk (mod p) RHS, (αa)rrm (mod p) ≡ αar.αk (mod p) m (mod p) ≡ αar.αkm (mod p) ≡ αar + km (mod p) ≡ αs (mod p) = LHS

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