Tut 6 Qn 5, TB Pg. 192

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# Tut 6 Qn 5, TB Pg. 192 - PowerPoint PPT Presentation

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 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.

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

(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.

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

(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.

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