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Add utilities to make Strong Lucas tests easier

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The Strong Lucas test coming in the next PR will already be complicated
enough.  It'll be convenient, and less distracting, if we already have
functions for certain operations we'll need.

One thing we'll need to do is detect inputs that are perfect squares.
Fortunately, this is pretty easy to do robustly and quickly, with
Newton's method.  We _don't_ want to use `std::sqrt`, because that takes
us into the floating point domain for no good reason, which could give
us wrong answers for larger integers.

The other thing we need is Jacobi symbols.  These are a lot more
obscure, but thankfully, still resonably straightforward to compute.
The Wikipedia page (https://en.wikipedia.org/wiki/Jacobi_symbol) has a
good explanation, and in particular, good instructions for computing
values.

With these utilities in place, the Strong Lucas code should be easier to
review.
This commit is contained in:
Chip Hogg
2024-11-14 18:50:21 -05:00
parent 7e1baf65fa
commit d56ffc08b8
2 changed files with 120 additions and 0 deletions
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43
test/static/prime_test.cpp
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@ -122,4 +122,47 @@ static_assert(miller_rabin_probable_prime(2u, 9'007'199'254'740'881u), "Large kn
static_assert(miller_rabin_probable_prime(2u, 18'446'744'073'709'551'557u), "Largest 64-bit prime");
// Jacobi symbols --- a building block for the Strong Lucas probable prime test, needed for Baillie-PSW.
static_assert(jacobi_symbol(1, 1u) == 1, "Jacobi symbol always 1 when 'numerator' is 1");
static_assert(jacobi_symbol(1, 3u) == 1, "Jacobi symbol always 1 when 'numerator' is 1");
static_assert(jacobi_symbol(1, 5u) == 1, "Jacobi symbol always 1 when 'numerator' is 1");
static_assert(jacobi_symbol(1, 987654321u) == 1, "Jacobi symbol always 1 when 'numerator' is 1");
static_assert(jacobi_symbol(3, 1u) == 1, "Jacobi symbol always 1 when 'denominator' is 1");
static_assert(jacobi_symbol(5, 1u) == 1, "Jacobi symbol always 1 when 'denominator' is 1");
static_assert(jacobi_symbol(-1234567890, 1u) == 1, "Jacobi symbol always 1 when 'denominator' is 1");
static_assert(jacobi_symbol(10, 5u) == 0, "Jacobi symbol always 0 when there's a common factor");
static_assert(jacobi_symbol(25, 15u) == 0, "Jacobi symbol always 0 when there's a common factor");
static_assert(jacobi_symbol(-24, 9u) == 0, "Jacobi symbol always 0 when there's a common factor");
static_assert(jacobi_symbol(14, 9u) == +jacobi_symbol(7, 9u),
"Divide numerator by 2: positive when (denom % 8) in {1, 7}");
static_assert(jacobi_symbol(14, 15u) == +jacobi_symbol(7, 15u),
"Divide numerator by 2: positive when (denom % 8) in {1, 7}");
static_assert(jacobi_symbol(14, 11u) == -jacobi_symbol(7, 11u),
"Divide numerator by 2: negative when (denom % 8) in {3, 5}");
static_assert(jacobi_symbol(14, 13u) == -jacobi_symbol(7, 13u),
"Divide numerator by 2: negative when (denom % 8) in {3, 5}");
static_assert(jacobi_symbol(19, 9u) == +jacobi_symbol(9, 19u), "Flip is identity when (n % 4) = 1");
static_assert(jacobi_symbol(17, 7u) == +jacobi_symbol(7, 17u), "Flip is identity when (a % 4) = 1");
static_assert(jacobi_symbol(19, 7u) == -jacobi_symbol(9, 7u), "Flip changes sign when (n % 4) = 3 and (a % 4) = 3");
static_assert(jacobi_symbol(1001, 9907u) == -1, "Example from Wikipedia page");
static_assert(jacobi_symbol(19, 45u) == 1, "Example from Wikipedia page");
static_assert(jacobi_symbol(8, 21u) == -1, "Example from Wikipedia page");
static_assert(jacobi_symbol(5, 21u) == 1, "Example from Wikipedia page");
// Tests for perfect square finder
static_assert(is_perfect_square(0u));
static_assert(is_perfect_square(1u));
static_assert(!is_perfect_square(2u));
static_assert(is_perfect_square(4u));
constexpr uint64_t BIG_SQUARE = [](auto x) { return x * x; }((uint64_t{1u} << 32) - 1u);
static_assert(!is_perfect_square(BIG_SQUARE - 1u));
static_assert(is_perfect_square(BIG_SQUARE));
static_assert(!is_perfect_square(BIG_SQUARE + 1u));
} // namespace