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Algebra Review

Groups, Rings, Fields Elliptic Curves Algebra. Algebra Review. CSCI381 Fall 2004 GWU. Group theory. What is a group? A set of elements G with An additive operation  such that G is closed under the operation, i.e. if a, b G, so does a b

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Algebra Review

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  1. Groups, Rings, Fields • Elliptic Curves Algebra Algebra Review CSCI381 Fall 2004 GWU

  2. Group theory What is a group? • A set of elements G with • An additive operation  such that • G is closed under the operation, i.e. if a, b G, so does a b • The operation is associative, i.e. (a b) c = a (b c) • An identity exists and is in G, i.e. • e  G, s.t. e  g = g e = g • Every element has an inverse in G, i.e.  g  G  g-1  G s.t g  g-1 = e CS284/Spring04/GWU/Vora/RSA

  3. Multiplicative and additive groups • The group operation can be addition or multiplication • Consider Zn • Is it a multiplicative group? Additive? Fact: Zp* for prime p is cyclic, generated by a primitive element  {1, , 2, … p-1} Also Fp Examples of Zn - multiplicative and additive groups, prime and composite n, primitive elements CS284/Spring04/GWU/Vora/RSA

  4. A A B B C C D D Example of a non-abelian group: transformations of a square Dihedral group of order n: Dn Structure CS284/Spring04/GWU/Vora/RSA

  5. Lagrange’s theorem on the order of a group element Theorem: Suppose G is a group of order n and g G. Then the order of g divides n. Example: multiplicative group, additive group. CS284/Spring04/GWU/Vora/RSA

  6. Lagrange’s theorem on the order of a group element - II Proof: Consider the following relation: a  b iff axi = b for some i • is an equivalence relation because: • axo(x) = a • If a  bthen b = axi and a = bx-I and b  a • If a  b and b  c, then b = axi and c = bxj = axi+j and a  c Hence, the cosets of this relation partition the group and are of equal size. Example: the relation for some x and composite n CS284/Spring04/GWU/Vora/RSA

  7. Lagrange’s theorem on the order of a group element - III Hence, the size of any coset divides the size of the group if it is finite {e, x1, x2, …xo(x)} is a coset of size o(x) Because any coset that contains x = {a s.t axi = x  i} = {a = x1-i  i} = {xj  j } Hence o(x) | n Example, composite n CS284/Spring04/GWU/Vora/RSA

  8. Ring Theory • What is a ring? • What is a field? • Examples. • Apply Lagrange Theorem to a ring, field. CS284/Spring04/GWU/Vora/RSA

  9. Lagrange Thm. on order of a subgroup • Pf. As with order of element. CS284/Spring04/GWU/Vora/RSA

  10. Group using points on an elliptic curve For a, b Fp such that 4a3 + 27b2 0 (mod p) G = {(x, y) | y2 = x3 + ax + b (mod p); x, y, Fp}  {(x, ); x Fp} From handout CS284/Spring04/GWU/Vora/RSA

  11. The operation CS284/Spring04/GWU/Vora/RSA

  12. Problem For p > 3, show that the curve has no order 2 point if f(x) = x3 + ax + b is irreducible over Fp, and has 1 or 3 such points otherwise. CS284/Spring04/GWU/Vora/RSA

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