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asked 2022-11-11 01:09:16 +0100

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Anonymous

Problems with factoring exponents in a prime field

if a, b, m, n ϵ Fp, with p is prime, and if b = a^m and c = b^n, the following should be true: 

c == b^n == (a^m)^n == (a^n)^m == a^mn  == a^nm  
b * c == a^(m + mn) == a^(m * (1+n))

The following code exhibits error behavior for c == a^mn and c = a^nm and b*c == a^(m + mn) == a^(m * (1 + n)).

p = 3541

Fp = GF(p)

a = Fp(114)
m = Fp(526)
n = Fp(3350)


b = a^m
c = b^n

print("\nGiven b = a^m and c == b^n, in prime field")
print("  Test 1: c == b^n  == (a^m)^n == (a^n)^m == a^(m*n) == a^(n*m) ")
print("    where,a: %d, m: %d, n: %d, b: %d, c: %d , a_mn: %d " %(a, m, n, b, c, a^(m* n)))
c == b^n
c == (a^m)^n
c == (a^n)^m
c == a^(m*n)
c == a^(n*m)
a^(m*n) == a^(n*m)

print("  Test 2: b*c == a^m*b^n  == a^(m + mn)   == a^m(1 + n)")
print("    where, b*c: %d, a^m*b^n: %d, a^(m * ( 1 +n )): %d, a^(m * (1+n)): %d" %( b*c, a^m*b^n, a^(m * ( 1 +n )), a^(m * (1+n
))))
b*c == a^m*b^n
b*c == a^(m * ( 1 +n ))
b*c == a^(m * (1+n))
a^(m * (1+n)) == a^(m + m*n)

Given b = a^m and c == b^n, in prime field
  Test 1: c == b^n  == (a^m)^n == (a^n)^m == a^(m*n) == a^(n*m)
    where,a: 114, m: 526, n: 3350, b: 1494, c: 2584 , a_mn: 1310
True
True
True
False
False
True
  Test 2: b*c == a^m*b^n  == a^(m + mn)   == a^m(1 + n)
    where, b*c: 806, a^m*b^n: 806, a^(m * ( 1 +n )): 2508, a^(m * (1+n)): 2508
True
False
False
True

Can someone help with these problems in with exponents? Is there an alternative approach?

Problems with factoring exponents in a prime field

if a, b, m, n ϵ Fp, with p is prime, and if b = a^m and c = b^n, the following should be true: 

c == b^n == (a^m)^n == (a^n)^m == a^mn  == a^nm  
b * c == a^(m + mn) == a^(m * (1+n))

The following code exhibits error behavior for c == a^mn and c = a^nm and b*c == a^(m + mn) == a^(m * (1 + n)).

 p = 3541
13

Fp = GF(p)

a = Fp(114)
Fp(7)
m = Fp(526)
Fp(3)
n = Fp(3350)
Fp(6)


b = a^m
c = b^n

print("\nGiven b = a^m and c == b^n, in prime field")
print("  Test 1: c == b^n  == (a^m)^n == (a^n)^m == a^(m*n) == a^(n*m) ")
print("    where,a: %d, m: %d, n: %d, b: %d, c: %d , a_mn: %d " %(a, m, n, b, c, a^(m* n)))
c == b^n
c == (a^m)^n
c == (a^n)^m
c == a^(m*n)
c == a^(n*m)
a^(m*n) == a^(n*m)

print("  Test 2: b*c == a^m*b^n  == a^(m + mn)   == a^m(1 + n)")
print("    where, b*c: %d, a^m*b^n: %d, a^(m * ( 1 +n )): %d, a^(m * (1+n)): %d" %( b*c, a^m*b^n, a^(m * ( 1 +n )), a^(m * (1+n
))))
b*c == a^m*b^n
b*c == a^(m * + ( 1 +n m * n ))
b*c == a^(m * (1+n))
a^(m * (1+n)) == a^(m + m*n)

Given b = a^m and c == b^n, in prime field
  Test 1: c == b^n  == (a^m)^n == (a^n)^m == a^(m*n) == a^(n*m)
    where,a: 114, 7, m: 526, 3, n: 3350, 6, b: 1494, 5, c: 2584 12 , a_mn: 1310
11
True
True
True
False
False
True
  Test 2: b*c == a^m*b^n  == a^(m + mn)   == a^m(1 + n)
    where, b*c: 806, 8, a^m*b^n: 806, 8, a^(m * ( 1 +n )): 2508, 3, a^(m * (1+n)): 2508
3
True
False
False
True

Can someone help with these problems in with exponents? Is there an alternative approach?

Problems with factoring exponents in a prime field

if a, b, m, n ϵ Fp, with p is prime, and if b = a^m and c = b^n, the following should be true: 

c == b^n == (a^m)^n == (a^n)^m == a^mn  == a^nm  
b * c == a^(m + mn) == a^(m * (1+n))

The following code exhibits error behavior for has problems with c == a^mn and c = a^nm and b*c == a^(m + mn) == a^(m * (1 + n)).

 p = 13

Fp = GF(p)

a = Fp(7)
m = Fp(3)
n = Fp(6)


b = a^m
c = b^n

print("\nGiven b = a^m and c == b^n, in prime field")
print("  Test 1: c == b^n  == (a^m)^n == (a^n)^m == a^(m*n) == a^(n*m) ")
print("    where,a: %d, m: %d, n: %d, b: %d, c: %d , a_mn: %d " %(a, m, n, b, c, a^(m* n)))
c == b^n
c == (a^m)^n
c == (a^n)^m
c == a^(m*n)
c == a^(n*m)
a^(m*n) == a^(n*m)

print("  Test 2: b*c == a^m*b^n  == a^(m + mn)   == a^m(1 + n)")
print("    where, b*c: %d, a^m*b^n: %d, a^(m * ( 1 +n )): %d, a^(m * (1+n)): %d" %( b*c, a^m*b^n, a^(m * ( 1 +n )), a^(m * (1+n
))))
b*c == a^m*b^n
b*c == a^(m + ( m * n ))
b*c == a^(m * (1+n))
a^(m * (1+n)) == a^(m + m*n)

Given b = a^m and c == b^n, in prime field
  Test 1: c == b^n  == (a^m)^n == (a^n)^m == a^(m*n) == a^(n*m)
    where,a: 7, m: 3, n: 6, b: 5, c: 12 , a_mn: 11
True
True
True
False
False
True
  Test 2: b*c == a^m*b^n  == a^(m + mn)   == a^m(1 + n)
    where, b*c: 8, a^m*b^n: 8, a^(m * ( 1 +n )): 3, a^(m * (1+n)): 3
True
False
False
True

Can someone help with these problems in with exponents? Is there an alternative approach?

Problems with factoring exponents in a prime field

if a, b, m, n ϵ Fp, with p is prime, and if b = a^m and c = b^n, the following should be true: 

c == b^n == (a^m)^n == (a^n)^m == a^mn  == a^nm  
b * c == a^(m + mn) == a^(m * (1+n))

The following code has problems with c == a^mn and , c = a^nm and b*c == a^(m + mn) , b*c == a^(m + mn),andb*c == a^(m * (1 + n)).

 p = 13

Fp = GF(p)

a = Fp(7)
m = Fp(3)
n = Fp(6)


b = a^m
c = b^n

print("\nGiven b = a^m and c == b^n, in prime field")
print("  Test 1: c == b^n  == (a^m)^n == (a^n)^m == a^(m*n) == a^(n*m) ")
print("    where,a: %d, m: %d, n: %d, b: %d, c: %d , a_mn: %d " %(a, m, n, b, c, a^(m* n)))
c == b^n
c == (a^m)^n
c == (a^n)^m
c == a^(m*n)
c == a^(n*m)
a^(m*n) == a^(n*m)

print("  Test 2: b*c == a^m*b^n  == a^(m + mn)   == a^m(1 + n)")
print("    where, b*c: %d, a^m*b^n: %d, a^(m * ( 1 +n )): %d, a^(m * (1+n)): %d" %( b*c, a^m*b^n, a^(m * ( 1 +n )), a^(m * (1+n
))))
b*c == a^m*b^n
b*c == a^(m + ( m * n ))
b*c == a^(m * (1+n))
a^(m * (1+n)) == a^(m + m*n)

Given b = a^m and c == b^n, in prime field
  Test 1: c == b^n  == (a^m)^n == (a^n)^m == a^(m*n) == a^(n*m)
    where,a: 7, m: 3, n: 6, b: 5, c: 12 , a_mn: 11
True
True
True
False
False
True
  Test 2: b*c == a^m*b^n  == a^(m + mn)   == a^m(1 + n)
    where, b*c: 8, a^m*b^n: 8, a^(m * ( 1 +n )): 3, a^(m * (1+n)): 3
True
False
False
True

Can someone help with these problems in with exponents? Have i missed something? Is there an alternative a fix/alternative approach?

Problems with factoring exponents in a prime field

if a, b, m, n ϵ Fp, with p is prime, and if b = a^m and c = b^n, the following should be true: 

c == b^n == (a^m)^n == (a^n)^m == a^mn  == a^nm  
b * c == a^(m + mn) == a^(m * (1+n))

The following code has problems with c == a^mn , c = a^nm , b*c == a^(m + mn),andb*c == a^(m * (1 + n)).

 p = 13

Fp = GF(p)

a = Fp(7)
m = Fp(3)
n = Fp(6)


b = a^m
c = b^n

print("\nGiven b = a^m and c == b^n, in prime field")
print("  Test 1: c == b^n  == (a^m)^n == (a^n)^m == a^(m*n) == a^(n*m) ")
print("    where,a: %d, m: %d, n: %d, b: %d, c: %d , a_mn: %d " %(a, m, n, b, c, a^(m* n)))
c == b^n
c == (a^m)^n
c == (a^n)^m
c == a^(m*n)
c == a^(n*m)
a^(m*n) == a^(n*m)

print("  Test 2: b*c == a^m*b^n  == a^(m + mn)   == a^m(1 + n)")
print("    where, b*c: %d, a^m*b^n: %d, a^(m * ( 1 +n )): %d, a^(m * (1+n)): %d" %( b*c, a^m*b^n, a^(m * ( 1 +n )), a^(m * (1+n
))))
b*c == a^m*b^n
b*c == a^(m + ( m * n ))
b*c == a^(m * (1+n))
a^(m * (1+n)) == a^(m + m*n)

Given b = a^m and c == b^n, in prime field
  Test 1: c == b^n  == (a^m)^n == (a^n)^m == a^(m*n) == a^(n*m)
    where,a: 7, m: 3, n: 6, b: 5, c: 12 , a_mn: 11
True
True
True
False
False
True
  Test 2: b*c == a^m*b^n  == a^(m + mn)   == a^m(1 + n)
    where, b*c: 8, a^m*b^n: 8, a^(m * ( 1 +n )): 3, a^(m * (1+n)): 3
True
False
False
True

Have i missed something? Is there a fix/alternative approach?work-around?

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