ec: Improve readability and clarity

- Move all of the ec functions into the Common::ec namespace.

- Give the public functions better names and some usage information.

- Move all of the "elt" related functions into an "elt" class including
  all of the arithmetic operations, so that the logic becomes clearer
  and feels less like assembly.

  This also makes it much more obvious what the parameters are, instead
  of only using unsigned char* (which doesn't tell anything about what
  the pointer is used for or the size).

- Similarly, add a new "Point" class and move point functions there.
  Overload the arithmetic operators to make calculations easier to read

Source/Core/Common/Crypto/ec.h

@@ -8,8 +8,14 @@
 
 #include "Common/CommonTypes.h"
 
-void generate_ecdsa(u8* R, u8* S, const u8* k, const u8* hash);
-
-void ec_priv_to_pub(const u8* k, u8* Q);
+namespace Common::ec
+{
+/// Generate a signature using ECDSA.
+std::array<u8, 60> Sign(const u8* key, const u8* hash);
 
+/// Compute a shared secret from a private key (30 bytes) and public key (60 bytes).
 std::array<u8, 60> ComputeSharedSecret(const u8* private_key, const u8* public_key);
+
+/// Convert a ECC private key (30 bytes) to a public key (60 bytes).
+std::array<u8, 60> PrivToPub(const u8* key);
+}  // namespace Common::ec

Source/Core/Core/IOS/IOSC.cpp

@@ -251,7 +251,7 @@ ReturnCode IOSC::ComputeSharedKey(Handle dest_handle, Handle private_handle, Han
 
   // Calculate the ECC shared secret.
   const std::array<u8, 0x3c> shared_secret =
-      ComputeSharedSecret(private_entry->data.data(), public_entry->data.data());
+      Common::ec::ComputeSharedSecret(private_entry->data.data(), public_entry->data.data());
 
   std::array<u8, 20> sha1;
   mbedtls_sha1(shared_secret.data(), shared_secret.size() / 2, sha1.data());
@@ -425,7 +425,8 @@ static Certificate MakeBlankSigECCert(const char* signer, const char* name, cons
   std::strncpy(reinterpret_cast<char*>(cert_out.data()) + 0xc4, name, 0x40);
   const u32 swapped_key_id = Common::swap32(key_id);
   std::memcpy(cert_out.data() + 0x104, &swapped_key_id, sizeof(swapped_key_id));
-  ec_priv_to_pub(private_key, cert_out.data() + 0x108);
+  const std::array<u8, 60> public_key = Common::ec::PrivToPub(private_key);
+  std::copy(public_key.cbegin(), public_key.cend(), cert_out.begin() + 0x108);
   return cert_out;
 }
 
@@ -454,11 +455,12 @@ void IOSC::Sign(u8* sig_out, u8* ap_cert_out, u64 title_id, const u8* data, u32
   std::copy(cert.begin(), cert.end(), ap_cert_out);
 
   mbedtls_sha1(ap_cert_out + 0x80, 0x100, hash.data());
-  generate_ecdsa(ap_cert_out + 4, ap_cert_out + 34, m_key_entries[HANDLE_CONSOLE_KEY].data.data(),
-                 hash.data());
+  auto signature = Common::ec::Sign(m_key_entries[HANDLE_CONSOLE_KEY].data.data(), hash.data());
+  std::copy(signature.cbegin(), signature.cend(), ap_cert_out + 4);
 
   mbedtls_sha1(data, data_size, hash.data());
-  generate_ecdsa(sig_out, sig_out + 30, ap_priv.data(), hash.data());
+  signature = Common::ec::Sign(ap_priv.data(), hash.data());
+  std::copy(signature.cbegin(), signature.cend(), sig_out);
 }
 
 constexpr std::array<u8, 512> ROOT_PUBLIC_KEY = {