improve CIOS implementation

Author: clementjuventin

include/evmmax/evmmax.hpp

@@ -83,26 +83,48 @@ class ModArith
         // Based on 2.3.2 from
         // High-Speed Algorithms & Architectures For Number-Theoretic Cryptosystems
         // https://www.microsoft.com/en-us/research/wp-content/uploads/1998/06/97Acar.pdf
+        // and on 2.2 from
+        // EdMSM: Multi-Scalar-Multiplication for SNARKs and Faster Montgomery multiplication
+        // https://eprint.iacr.org/2022/1400.pdf
 
+        constexpr uint64_t most_significant_mod_word_limit {std::numeric_limits<uint64_t>::max() >> 1};
         constexpr auto S = UintT::num_words;  // TODO(C++23): Make it static
 
         intx::uint<UintT::num_bits + 64> t;
-        for (size_t i = 0; i != S; ++i)
+        if (mod[S - 1] < most_significant_mod_word_limit)
         {
-            uint64_t c = 0;
-            for (size_t j = 0; j != S; ++j)
-                std::tie(c, t[j]) = addmul(t[j], x[j], y[i], c);
-            auto tmp = intx::addc(t[S], c);
-            t[S] = tmp.value;
-            const auto d = tmp.carry;  // TODO: Carry is 0 for sparse modulus.
-
-            const auto m = t[0] * m_mod_inv;
-            std::tie(c, std::ignore) = addmul(t[0], m, mod[0], 0);
-            for (size_t j = 1; j != S; ++j)
-                std::tie(c, t[j - 1]) = addmul(t[j], m, mod[j], c);
-            tmp = intx::addc(t[S], c);
-            t[S - 1] = tmp.value;
-            t[S] = d + tmp.carry;  // TODO: Carry is 0 for sparse modulus.
+            for (size_t i = 0; i != S; ++i)
+            {
+                uint64_t c = 0;
+                for (size_t j = 0; j != S; ++j)
+                    std::tie(c, t[j]) = addmul(t[j], x[j], y[i], c);
+                auto const c_2 = c;
+                const auto m = t[0] * m_mod_inv;
+                std::tie(c, std::ignore) = addmul(t[0], m, mod[0], 0);
+                for (size_t j = 1; j != S; ++j)
+                    std::tie(c, t[j - 1]) = addmul(t[j], m, mod[j], c);
+                t[S - 1] = c_2 + c;
+            }
+        }
+        else
+        {
+            for (size_t i = 0; i != S; ++i)
+            {
+                uint64_t c = 0;
+                for (size_t j = 0; j != S; ++j)
+                    std::tie(c, t[j]) = addmul(t[j], x[j], y[i], c);
+                auto tmp = intx::addc(t[S], c);
+                t[S] = tmp.value;
+                const auto d = tmp.carry;  // TODO: Carry is 0 for sparse modulus.
+
+                const auto m = t[0] * m_mod_inv;
+                std::tie(c, std::ignore) = addmul(t[0], m, mod[0], 0);
+                for (size_t j = 1; j != S; ++j)
+                    std::tie(c, t[j - 1]) = addmul(t[j], m, mod[j], c);
+                tmp = intx::addc(t[S], c);
+                t[S - 1] = tmp.value;
+                t[S] = d + tmp.carry;  // TODO: Carry is 0 for sparse modulus.
+            }
         }
 
         if (t >= mod)