Introduction to Modern Cryptography Sharif University Spring 2015 Data and Network Security Lab Sharif University of Technology Department of Computer Engineering Algebra & Cryptography Author & Instructor: Mohammad Sadeq Dousti 1 / 42

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Introduction to Modern Cryptography Sharif University Spring 2015  What is algebra?  Group-like structures o Groups  Ring-like structures o Rings o Fields - Finite Fields Outline 3 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 What is Algebra? 4 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 What is algebra? 5 / 42

Introduction to Modern Cryptography Sharif University Spring 2015  Methods for solving linear (ax + b = 0) and quadratic (ax 2 + bx + c = 0) equations were known for centuries.  General cubic and quartic equations were solved in the 16 th century CE. o The solutions were expressed in terms of basic arithmetic operations (+, , ,  ), as well as radicals.  No such method was known for general equations of degree 5 or higher. o Working independently, Abel and Galois proved that giving such method is impossible. o Along the way, they laid the foundation of abstract algebra. From solving equations to abstract algebra 6 / 42

Introduction to Modern Cryptography Sharif University Spring 2015  Niels Henrik Abel (1802 – 1829) o Norwegian mathematician o Lived in poverty o Contracted tuberculosis o Died at the age of 26 in Paris  Évariste Galois (1811 – 1832) o French mathematician o Lived a wealthy life o Got involved in army & politics o Died in a duel at the age of 20 in Paris Founders of abstract algebra 7 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 Group-like Structures 8 / 42

Introduction to Modern Cryptography Sharif University Spring 2015  A set S endowed with one or more finitary operations is called an algebraic structure.  Let S be a set, and  : S  S  S be a binary operation.  The pair (S,  ) is called a group-like structure.  Depending on the properties that  satisfies on S, the structure is called by various names (semicategory, category, groupoid, magma, quasigroup, loop, semigroup, monoid, group, Abelian group, …).  If  behaves like multiplication, it is denoted by , and the structure is called multiplicative.  If  behaves like addition, it is denoted by +, and the structure is called additive. Algebraic structures 9 / 42

Introduction to Modern Cryptography Sharif University Spring 2015  (S,  ) satisfies the closure (totality) property if for all x, y  S, we have x  y  S. Equivalently: o S is closed under . o  is closed over S.  Examples: o ( ℕ, +) o ( ℤ,  ) o ( ℚ  {0},  )  Non-examples: o ( ℕ,  ) o ( ℤ  {0},  ) o ( ℚ,  ) Closure (Totality) 10 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 Associativity 11 / 42

Introduction to Modern Cryptography Sharif University Spring 2015  (S,  ) has an identity element e  S if for all x  S, we have e  x = x  e = x. The identity element is often denoted by: o 1 in multiplicative structures. o 0 in additive structures.  Examples: o = = 25 o A   =   A = A  Non-examples: o ( ℤ,  ) o ( ℚ  {0},  ) Identity 12 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 Uniqueness of identity element 13 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 Invertibility (Divisibility) 14 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 Commutativity 15 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 Group-like structures at a glance No identity. How is divisibility possible?! 16 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 Groups 17 / 42

Introduction to Modern Cryptography Sharif University Spring 2015  Group: An algebraic structure G = (S,  ) satisfying four properties: 1. Closure (totality) 2. Associativity 3. Identity 4. Divisibility (invertibility)  Abelian group: A group satisfying commutativity.  Group membership: x  G if and only if x  S.  Group order: The number of elements in the group. o Denoted |G| = |S|.  Finite group: A group with finite order. Groups 18 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 Notational conventions 19 / 42

Introduction to Modern Cryptography Sharif University Spring 2015  The order of an element x of a group is the smallest positive integer m such that: o mx = e (additive groups) o x m = e (multiplicative groups)  If no such m exists, x is said to have infinite order.  Periodic group: A group in which every element has finite order.  Exponent of a periodic group: The LCM of all group elements, if it exists.  THEOREM: Any finite group has an exponent. It is a divisor of |G|. (See Lagrange’s theorem a few slides ahead) Order and exponent 20 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 Examples of finite groups 21 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 Cayley tables + (mod 2)  (mod 3) / 42

Introduction to Modern Cryptography Sharif University Spring 2015 Subgroups and cosets 23 / 42

Introduction to Modern Cryptography Sharif University Spring 2015  Let  denote addition modulo 18. o G = ℤ 18. o H = 3G = {0, 3, 6, 9, 12, 15}. o H is a group under . o [H : G] = |G| / |H| = 18 / 6 = 3. o 7  H = H  7 = {7, 10, 13, 16, 1, 4} is a coset of H. o K = 2H = {0, 6, 12}. o K is a group under . o [K : G] = |G| / |K| = 18 / 3 = 6. o 7  K = K  7 = {7, 13, 1} is a coset of K. Examples 24 / 42

Introduction to Modern Cryptography Sharif University Spring 2015  Let G = (S,  ) be a group, and T  S.  The generating set of T, denoted, is a subgroup of G whose members can be expressed as the combination (under  ) of finitely many elements of T and their inverses.  If T = {x}, we may write instead of. o is called a cyclic group.  If G =, then we say T generates G; and the elements in T are called generators or group generators. Generators and cyclic groups 25 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 Examples 26 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 Fermat–Euler theorem from Lagrange’s theorem 27 / 42

Introduction to Modern Cryptography Sharif University Spring 2015  Let G = (S,  ) be a group of prime order p.  THEOREM: Every non-identity element of G is a generator of G.  PROOF: Easy using Lagrange’s theorem. o The order of any cyclic subgroup of G is either 1 or p (since it must divide p). o The only cyclic subgroup of order 1 is. o For non-identity group element g, we have | | = p. o Therefore, = G. Groups of prime order 28 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 Groups of prime order (Cont’d) 29 / 42

Introduction to Modern Cryptography Sharif University Spring / 42

Introduction to Modern Cryptography Sharif University Spring 2015 Permutations and symmetric groups 31 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 (External) Direct product of groups 32 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 Group homomorphism 33 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 Group isomorphism 34 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 Ring-like Structures 35 / 42

Introduction to Modern Cryptography Sharif University Spring 2015  A set S endowed with two operations: o An “addition-like” operator + o A “multiplication-like” operator  is called a ring-like structure.  Depending on properties that + and  satisfy on S, various structures are defined: Rng, Semiring, Near- ring, Near-semiring, Ring, Commutative ring, Domain, Integral domain, Field, etc.  We only study rings and fields. Ring-like structures 36 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 Rings 37 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 Characteristic 38 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 Ring homomorphism 39 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 Fields 40 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 Finite fields 41 / 42

Introduction to Modern Cryptography Sharif University Spring 2015 Wikipedia. References 42 / 42