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Applied Symbolic Computation (CS 300) Modular Arithmetic

Applied Symbolic Computation (CS 300) Modular Arithmetic. Jeremy R. Johnson. Introduction. Objective: To become familiar with modular arithmetic and some key algorithmic constructions that are important for computer algebra algorithms. Modular Arithmetic

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Applied Symbolic Computation (CS 300) Modular Arithmetic

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  1. Applied Symbolic Computation (CS 300) Modular Arithmetic Jeremy R. Johnson

  2. Introduction • Objective: To become familiar with modular arithmetic and some key algorithmic constructions that are important for computer algebra algorithms. • Modular Arithmetic • Modular inverses and the extended Euclidean algorithm • Fermat’s theorem • Euler’s Identity • Chinese Remainder Theorem References: Rivest, Shamir, Adelman.

  3. Modular Arithmetic (Zn) Definition: a  b (mod n)  n | (b - a) Alternatively, a = qn + b Properties (equivalence relation) • a  a (mod n) [Reflexive] • a  b (mod n)  b  a (mod n) [Symmetric] • a  b (mod n) and b  c (mod n)  a  c (mod n) [Transitive] Definition: An equivalence class mod n [a] = { x: x  a (mod n)} = { a + qn | q  Z}

  4. Modular Arithmetic (Zn) It is possible to perform arithmetic with equivalence classes mod n. • [a] + [b] = [a+b] • [a] * [b] = [a*b] In order for this to make sense, you must get the same answer (equivalence) class independent of the choice of a and b. In other words, if you replace a and b by numbers equivalent to a or b mod n you end of with the sum/product being in the same equivalence class. a1 a2 (mod n) and b1 b2 (mod n)  a1+ b1 a2 + b2 (mod n) a1* b1 a2 * b2 (mod n) (a + q1n) + (b + q2n) = a + b + (q1 + q2)n (a + q1n) * (b + q2n) = a * b + (b*q1 + a*q2 + q1* q2)n

  5. Representation of Zn The equivalence classes [a] mod n, are typically represented by the representatives a. • Positive Representation: Choose the smallest positive integer in the class [a] then the representation is {0,1,…,n-1}. • Symmetric Representation: Choose the integer with the smallest absolute value in the class [a]. The representation is {-(n-1)/2 ,…, n/2 }. When n is even, choose the positive representative with absolute value n/2. • E.G. Z6 = {-2,-1,0,1,2,3}, Z5 = {-2,-1,0,1,2}

  6. Modular Inverses Definition: x is the inverse of a mod n, if ax  1 (mod n) The equation ax  1 (mod n) has a solution iff gcd(a,n) = 1. By the Extended Euclidean Algorithm, there exist x and y such that ax + ny = gcd(a,n). When gcd(a,n) = 1, we get ax + ny = 1. Taking this equation mod n, we see that ax  1 (mod n) By taking the equation mod n, we mean applying the mod n homomorphism: m Z  Zm, which maps the integer a to the equivalence class [a]. This mapping preserves sums and products. I.E. m(a+b) = m(a) + m(b), m(a*b) = m(a) * m(b)

  7. Fermat’s Theorem Theorem: If a  0 Zp, then ap-1 1 (mod p). More generally, if a  Zp, then ap a (mod p). Proof: Assume that a  0 Zp. Then a * 2a * … (p-1)a = (p-1)! * ap-1 Also, since a*i  a*j (mod p)  i  j (mod p), the numbers a, 2a, …, (p-1)a are distinct elements of Zp. Therefore they are equal to 1,2,…,(p-1) and their product is equal to (p-1)! mod p. This implies that (p-1)! * ap-1  (p-1)! (mod p)  ap-1 1 (mod p).

  8. Euler phi function • Definition: phi(n) = #{a: 0 < a < n and gcd(a,n) = 1} • Properties: • (p) = p-1, for prime p. • (p^e) = (p-1)*p^(e-1) •  (m*n) =  (m)* (n) for gcd(m,n) = 1. • (p*q) = (p-1)*(q-1) • Examples: • (15) = (3)* (5) = 2*4 = 8. = #{1,2,4,7,8,11,13,14} • (9) = (3-1)*3^(2-1) = 2*3 = 6 = #{1,2,4,5,7,8}

  9. Euler’s Identity • The number of elements in Zn that have multiplicative inverses is equal to phi(n). • Theorem: Let (Zn)* be the elements of Zn with inverses (called units). If a  (Zn)*, then a(n) 1 (mod n). Proof. The same proof presented for Fermat’s theorem can be used to prove this theorem.

  10. Chinese Remainder Theorem Theorem: If gcd(m,n) = 1, then given a and b there exist an integer solution to the system: x  a (mod m) and x = b (mod n). Proof: Consider the map x  (x mod m, x mod n). This map is a 1-1 map from Zmnto Zm  Zn, since if x and y map to the same pair, then x  y (mod m) and x  y (mod n). Since gcd(m,n) = 1, this implies that x  y (mod mn). Since there are mn elements in both Zmnand Zm  Zn, the map is also onto. This means that for every pair (a,b) we can find the desired x.

  11. Alternative Interpretation of CRT • Let Zm  Zn denote the set of pairs (a,b) where a  Zm and b  Zn. We can perform arithmetic on Zm  Zn by performing componentwise modular arithmetic. • (a,b) + (c,d) = (a+b,c+d) • (a,b)*(c,d) = (a*c,b*d) • Theorem: Zmn Zm  Zn. I.E. There is a 1-1 mapping from Zmn onto Zm  Zn that preserves arithmetic. • (a*c mod m, b*d mod n) = (a mod m, b mod n)*(c mod m, d mod n) • (a+c mod m, b+d mod n) = (a mod m, b mod n)+(c mod m, d mod n) • The CRT implies that the map is onto. I.E. for every pair (a,b) there is an integer x such that (x mod m, x mod n) = (a,b).

  12. Constructive Chinese Remainder Theorem Theorem: If gcd(m,n) = 1, then there exist em and en (orthogonal idempotents) • em 1 (mod m) • em 0 (mod n) • en 0 (mod m) • en 1 (mod n) It follows that a*em + b* en a (mod m) and  b (mod n). Proof. Since gcd(m,n) = 1, by the Extended Euclidean Algorithm, there exist x and y with m*x + n*y = 1. Set em = n*y and en = m*x

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