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Extended Euclidean Algorithm in CS 282 Lecture by Richard Fateman

Learn how the Extended Euclidean Algorithm (EEA) solves equations involving gcd, integers, polynomials, and inverses. Discover unique solutions with respect to variables and normal forms, including examples with polynomials like Knuth’s. Follow the EEA algorithm steps using Macsyma. Uncover insights on reducing the GCD result for accurate computations. Delve into the process to determine A, B, and G, while understanding the importance of corrections.

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Extended Euclidean Algorithm in CS 282 Lecture by Richard Fateman

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  1. Extended Euclidean Algorithm Lecture eea Richard Fateman CS 282 Lecture eea

  2. The EEA solves A¢ P + B¢ Q = G Given integers P and Q. Determine A, B, G such that G=gcd(A,B) A¢ P + B¢ Q = G Where uniqueness is asserted by deciding on |A|<|Q|, |B|<|P|, and G is the (positive) GCD.. 1*39-1*26= 13 because gcd(39,26)=13. 13*25-9*36=1 because gcd(25,36)=1. Richard Fateman CS 282 Lecture eea

  3. The EEA finds inverses mod Q from A¢ P + B¢ Q = G Assume Q is a prime integer, and P ≠ Q. Determine A, B such that A¢ P + B¢ Q = 1. (gcd(P,Q)=1 ) Where uniqueness requires 0<A<Q-1, or |A|<(Q-1)/2 If we do all our arithmetic modulo Q, then Q  0 And so A*P=1 mod Q. Thus A is the inverse of P mod Q. Example: (-5)*5 +2*13=1, so -5 is the inverse of 5 mod 13. -5 8 mod 13… Richard Fateman CS 282 Lecture eea

  4. The EEA solves A¢ P + B¢ Q = G, polynomials Given polynomials (coefficient field??) P and Q. Determine A, B, G such that G=gcd(A,B) A¢ P + B¢ Q = G Where uniqueness is asserted by deciding on a main variable x, with respect to which deg(A)<deg(Q), deg(B)<deg(P), and G is in some normal form. For example, over rationals, we would insist that G be unit normal. E.g. if it were an integer, G would be 1. Richard Fateman CS 282 Lecture eea

  5. The EEA solves A¢ P + B¢ Q = G For Knuth’s polynomials we would like … P=x8+x6-3*x4-3*x3+8*x2+2*x-5, Q=3*x6+5*x4-4*x2-9*x+21,  A=(13989*x5+18450*x4+40562*x3+67125*x2+5149*x-9737)/130354 B=-(4663*x7+6150*x6+10412*x5+18275*x4-9888*x3-21579*x2-3820*x-3889)/130354 G=1 Richard Fateman CS 282 Lecture eea

  6. The EEA algorithm (in Macsyma) extended_gcd(u,v,x):= block([u1,u2,u3,v1,v2,v3,t1,t2,t3], u: rat(u,x), v: rat(v,x), [u1,u2,u3]:[rat(1),rat(0),u], [v1,v2,v3]:[rat(0),rat(1),v], while v3#0 do (q: quotient(u3,v3,x), [t1,t2,t3]:[u1,u2,u3]-q*[v1,v2,v3], [u1,u2,u3]:[v1,v2,v3],[v1,v2,v3]:[t1,t2,t3]), [u1,u2,u3]) Richard Fateman CS 282 Lecture eea

  7. The EEA algorithm (in Macsyma) Actually, we lied, and the GCD instead of being 1 comes out as -1288744821/543589225. We have to make a correction here.. Richard Fateman CS 282 Lecture eea

  8. The EEA algorithm, reducing the result eea(u,v,x):= /* smallest gcd */ block([u1,u2,u3,v1,v2,v3,t1,t2,t3, realgcd:gcd(u,v),correction:1], u: rat(u,x), v: rat(v,x), [u1,u2,u3]:[rat(1),rat(0),u], [v1,v2,v3]:[rat(0),rat(1),v], while v3#0 do ( print([v1,v2,v3]), q: quotient(u3,v3,x), /* here is where we might like to patch*/ [t1,t2,t3]: [u1,u2,u3]-q*[v1,v2,v3], [u1,u2,u3]:[v1,v2,v3],[v1,v2,v3]:[t1,t2,t3]), correction:realgcd/u3, /* the patch we used */ [u1*correction,u2*correction,realgcd]), Richard Fateman CS 282 Lecture eea

  9. The EEA algorithm, reducing the result Note that in particular, the terms A, B are not directly derived from G=gcd(P,Q), but part of the process. Richard Fateman CS 282 Lecture eea

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