Example #1
0
        /// <summary>
        /// An inner function to calculate scalar * point
        /// </summary>
        /// <param name="input"></param>
        /// <param name="baseIn"></param>
        /// <returns></returns>
        private static byte[] ScalarMult(byte[] input, byte[] baseIn)
        {
            var e = new byte[32];

            Array.Copy(input, e, 32); //copy(e[:], input[:])
            e[0]  &= 248;
            e[31] &= 127;
            e[31] |= 64;

            FieldElement x1, x2, z2, x3, z3, tmp0, tmp1;

            z2 = new FieldElement();
            // feFromBytes(&x1, base)
            x1 = new FieldElement(baseIn); //SECOND NUMBER
            //feOne(&x2)
            x2 = new FieldElement();
            x2.One();
            //feCopy(&x3, &x1)
            x3 = new FieldElement();
            FieldElement.Copy(ref x3, x1);
            //feOne(&z3)
            z3 = new FieldElement();
            z3.One();

            int swap = 0;

            for (int pos = 254; pos >= 0; pos--)
            {
                byte b = Convert.ToByte(e[pos / 8] >> (pos & 7));
                b    &= 1;
                swap ^= (int)(b);
                FieldElement.CSwap(ref x2, ref x3, swap);
                FieldElement.CSwap(ref z2, ref z3, swap);
                swap = (int)(b);

                tmp0  = x3 - z3; //feSub(&tmp0, &x3, &z3)
                tmp1  = x2 - z2; //feSub(&tmp1, &x2, &z2)
                x2   += z2;      //feAdd(&x2, &x2, &z2)
                z2    = x3 + z3; //feAdd(&z2, &x3, &z3)
                z3    = tmp0.Multiply(x2);
                z2    = z2.Multiply(tmp1);
                tmp0  = tmp1.Square();
                tmp1  = x2.Square();
                x3    = z3 + z2; //feAdd(&x3, &z3, &z2)
                z2    = z3 - z2; //feSub(&z2, &z3, &z2)
                x2    = tmp1.Multiply(tmp0);
                tmp1 -= tmp0;    //feSub(&tmp1, &tmp1, &tmp0)
                z2    = z2.Square();
                z3    = tmp1.Mul121666();
                x3    = x3.Square();
                tmp0 += z3; //feAdd(&tmp0, &tmp0, &z3)
                z3    = x1.Multiply(z2);
                z2    = tmp1.Multiply(tmp0);
            }

            FieldElement.CSwap(ref x2, ref x3, swap);
            FieldElement.CSwap(ref z2, ref z3, swap);

            z2 = z2.Invert();
            x2 = x2.Multiply(z2);
            return(x2.ToBytes());
        }
Example #2
0
        /// <summary>
        /// Calculates f*f. Can overlap h with f.
        /// </summary>
        /// <returns></returns>
        public FieldElement Square()
        {
            int  f0      = _elements[0];
            int  f1      = _elements[1];
            int  f2      = _elements[2];
            int  f3      = _elements[3];
            int  f4      = _elements[4];
            int  f5      = _elements[5];
            int  f6      = _elements[6];
            int  f7      = _elements[7];
            int  f8      = _elements[8];
            int  f9      = _elements[9];
            int  f0_2    = 2 * f0;
            int  f1_2    = 2 * f1;
            int  f2_2    = 2 * f2;
            int  f3_2    = 2 * f3;
            int  f4_2    = 2 * f4;
            int  f5_2    = 2 * f5;
            int  f6_2    = 2 * f6;
            int  f7_2    = 2 * f7;
            int  f5_38   = 38 * f5; // 1.31*2^30
            int  f6_19   = 19 * f6; // 1.31*2^30
            int  f7_38   = 38 * f7; // 1.31*2^30
            int  f8_19   = 19 * f8; // 1.31*2^30
            int  f9_38   = 38 * f9; // 1.31*2^30
            long f0f0    = (long)(f0) * (long)(f0);
            long f0f1_2  = (long)(f0_2) * (long)(f1);
            long f0f2_2  = (long)(f0_2) * (long)(f2);
            long f0f3_2  = (long)(f0_2) * (long)(f3);
            long f0f4_2  = (long)(f0_2) * (long)(f4);
            long f0f5_2  = (long)(f0_2) * (long)(f5);
            long f0f6_2  = (long)(f0_2) * (long)(f6);
            long f0f7_2  = (long)(f0_2) * (long)(f7);
            long f0f8_2  = (long)(f0_2) * (long)(f8);
            long f0f9_2  = (long)(f0_2) * (long)(f9);
            long f1f1_2  = (long)(f1_2) * (long)(f1);
            long f1f2_2  = (long)(f1_2) * (long)(f2);
            long f1f3_4  = (long)(f1_2) * (long)(f3_2);
            long f1f4_2  = (long)(f1_2) * (long)(f4);
            long f1f5_4  = (long)(f1_2) * (long)(f5_2);
            long f1f6_2  = (long)(f1_2) * (long)(f6);
            long f1f7_4  = (long)(f1_2) * (long)(f7_2);
            long f1f8_2  = (long)(f1_2) * (long)(f8);
            long f1f9_76 = (long)(f1_2) * (long)(f9_38);
            long f2f2    = (long)(f2) * (long)(f2);
            long f2f3_2  = (long)(f2_2) * (long)(f3);
            long f2f4_2  = (long)(f2_2) * (long)(f4);
            long f2f5_2  = (long)(f2_2) * (long)(f5);
            long f2f6_2  = (long)(f2_2) * (long)(f6);
            long f2f7_2  = (long)(f2_2) * (long)(f7);
            long f2f8_38 = (long)(f2_2) * (long)(f8_19);
            long f2f9_38 = (long)(f2) * (long)(f9_38);
            long f3f3_2  = (long)(f3_2) * (long)(f3);
            long f3f4_2  = (long)(f3_2) * (long)(f4);
            long f3f5_4  = (long)(f3_2) * (long)(f5_2);
            long f3f6_2  = (long)(f3_2) * (long)(f6);
            long f3f7_76 = (long)(f3_2) * (long)(f7_38);
            long f3f8_38 = (long)(f3_2) * (long)(f8_19);
            long f3f9_76 = (long)(f3_2) * (long)(f9_38);
            long f4f4    = (long)(f4) * (long)(f4);
            long f4f5_2  = (long)(f4_2) * (long)(f5);
            long f4f6_38 = (long)(f4_2) * (long)(f6_19);
            long f4f7_38 = (long)(f4) * (long)(f7_38);
            long f4f8_38 = (long)(f4_2) * (long)(f8_19);
            long f4f9_38 = (long)(f4) * (long)(f9_38);
            long f5f5_38 = (long)(f5) * (long)(f5_38);
            long f5f6_38 = (long)(f5_2) * (long)(f6_19);
            long f5f7_76 = (long)(f5_2) * (long)(f7_38);
            long f5f8_38 = (long)(f5_2) * (long)(f8_19);
            long f5f9_76 = (long)(f5_2) * (long)(f9_38);
            long f6f6_19 = (long)(f6) * (long)(f6_19);
            long f6f7_38 = (long)(f6) * (long)(f7_38);
            long f6f8_38 = (long)(f6_2) * (long)(f8_19);
            long f6f9_38 = (long)(f6) * (long)(f9_38);
            long f7f7_38 = (long)(f7) * (long)(f7_38);
            long f7f8_38 = (long)(f7_2) * (long)(f8_19);
            long f7f9_76 = (long)(f7_2) * (long)(f9_38);
            long f8f8_19 = (long)(f8) * (long)(f8_19);
            long f8f9_38 = (long)(f8) * (long)(f9_38);
            long f9f9_38 = (long)(f9) * (long)(f9_38);
            long h0      = f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38;
            long h1      = f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38;
            long h2      = f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19;
            long h3      = f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38;
            long h4      = f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38;
            long h5      = f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38;
            long h6      = f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19;
            long h7      = f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38;
            long h8      = f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38;
            long h9      = f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2;
            var  carry   = new long[10];

            carry[0] = (h0 + (1 << 25)) >> 26;
            h1      += carry[0];
            h0      -= carry[0] << 26;
            carry[4] = (h4 + (1 << 25)) >> 26;
            h5      += carry[4];
            h4      -= carry[4] << 26;

            carry[1] = (h1 + (1 << 24)) >> 25;
            h2      += carry[1];
            h1      -= carry[1] << 25;
            carry[5] = (h5 + (1 << 24)) >> 25;
            h6      += carry[5];
            h5      -= carry[5] << 25;

            carry[2] = (h2 + (1 << 25)) >> 26;
            h3      += carry[2];
            h2      -= carry[2] << 26;
            carry[6] = (h6 + (1 << 25)) >> 26;
            h7      += carry[6];
            h6      -= carry[6] << 26;

            carry[3] = (h3 + (1 << 24)) >> 25;
            h4      += carry[3];
            h3      -= carry[3] << 25;
            carry[7] = (h7 + (1 << 24)) >> 25;
            h8      += carry[7];
            h7      -= carry[7] << 25;

            carry[4] = (h4 + (1 << 25)) >> 26;
            h5      += carry[4];
            h4      -= carry[4] << 26;
            carry[8] = (h8 + (1 << 25)) >> 26;
            h9      += carry[8];
            h8      -= carry[8] << 26;

            carry[9] = (h9 + (1 << 24)) >> 25;
            h0      += carry[9] * 19;
            h9      -= carry[9] << 25;

            carry[0] = (h0 + (1 << 25)) >> 26;
            h1      += carry[0];
            h0      -= carry[0] << 26;

            var final = new FieldElement();
            var h     = new int[10];

            h[0] = (int)(h0);
            h[1] = (int)(h1);
            h[2] = (int)(h2);
            h[3] = (int)(h3);
            h[4] = (int)(h4);
            h[5] = (int)(h5);
            h[6] = (int)(h6);
            h[7] = (int)(h7);
            h[8] = (int)(h8);
            h[9] = (int)(h9);
            final.SetElementsDirect(h);
            return(final);
        }
Example #3
0
        /// <summary>
        /// Calculates h = f * 121666. Can overlap h with f; I have no clue why this is a thing
        /// </summary>
        /// <remarks>
        ///   Preconditions:
        /// |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
        ///
        /// Postconditions:
        /// |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
        /// </remarks>
        /// <returns></returns>
        public FieldElement Mul121666()
        {
            long h0    = (long)(_elements[0]) * 121666;
            long h1    = (long)(_elements[1]) * 121666;
            long h2    = (long)(_elements[2]) * 121666;
            long h3    = (long)(_elements[3]) * 121666;
            long h4    = (long)(_elements[4]) * 121666;
            long h5    = (long)(_elements[5]) * 121666;
            long h6    = (long)(_elements[6]) * 121666;
            long h7    = (long)(_elements[7]) * 121666;
            long h8    = (long)(_elements[8]) * 121666;
            long h9    = (long)(_elements[9]) * 121666;
            var  carry = new long[10];

            carry[9] = (h9 + (1 << 24)) >> 25;
            h0      += carry[9] * 19;
            h9      -= carry[9] << 25;
            carry[1] = (h1 + (1 << 24)) >> 25;
            h2      += carry[1];
            h1      -= carry[1] << 25;
            carry[3] = (h3 + (1 << 24)) >> 25;
            h4      += carry[3];
            h3      -= carry[3] << 25;
            carry[5] = (h5 + (1 << 24)) >> 25;
            h6      += carry[5];
            h5      -= carry[5] << 25;
            carry[7] = (h7 + (1 << 24)) >> 25;
            h8      += carry[7];
            h7      -= carry[7] << 25;

            carry[0] = (h0 + (1 << 25)) >> 26;
            h1      += carry[0];
            h0      -= carry[0] << 26;
            carry[2] = (h2 + (1 << 25)) >> 26;
            h3      += carry[2];
            h2      -= carry[2] << 26;
            carry[4] = (h4 + (1 << 25)) >> 26;
            h5      += carry[4];
            h4      -= carry[4] << 26;
            carry[6] = (h6 + (1 << 25)) >> 26;
            h7      += carry[6];
            h6      -= carry[6] << 26;
            carry[8] = (h8 + (1 << 25)) >> 26;
            h9      += carry[8];
            h8      -= carry[8] << 26;

            var final = new FieldElement();
            var h     = new int[10];

            h[0] = (int)(h0);
            h[1] = (int)(h1);
            h[2] = (int)(h2);
            h[3] = (int)(h3);
            h[4] = (int)(h4);
            h[5] = (int)(h5);
            h[6] = (int)(h6);
            h[7] = (int)(h7);
            h[8] = (int)(h8);
            h[9] = (int)(h9);
            final.SetElementsDirect(h);
            return(final);
        }
Example #4
0
        /// <summary>
        /// Calculates this * g
        /// </summary>
        /// <param name="g"></param>
        /// <remarks>
        /// Can overlap h with f or g.
        ///
        /// Preconditions:
        /// |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
        /// |g| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
        ///
        /// Postconditions:
        /// |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
        ///
        /// Notes on implementation strategy:
        ///
        /// Using schoolbook multiplication.
        /// Karatsuba would save a little in some cost models.
        ///
        /// Most multiplications by 2 and 19 are 32-bit precomputations;
        /// cheaper than 64-bit postcomputations.
        ///
        /// There is one remaining multiplication by 19 in the carry chain;
        /// one *19 precomputation can be merged into this,
        /// but the resulting data flow is considerably less clean.
        ///
        /// There are 12 carries below.
        /// 10 of them are 2-way parallelizable and vectorizable.
        /// Can get away with 11 carries, but then data flow is much deeper.
        ///
        /// With tighter constraints on inputs can squeeze carries into int32.
        /// </remarks>
        /// <returns></returns>
        public FieldElement Multiply(FieldElement g)
        {
            int  f0      = _elements[0];
            int  f1      = _elements[1];
            int  f2      = _elements[2];
            int  f3      = _elements[3];
            int  f4      = _elements[4];
            int  f5      = _elements[5];
            int  f6      = _elements[6];
            int  f7      = _elements[7];
            int  f8      = _elements[8];
            int  f9      = _elements[9];
            int  g0      = g[0];
            int  g1      = g[1];
            int  g2      = g[2];
            int  g3      = g[3];
            int  g4      = g[4];
            int  g5      = g[5];
            int  g6      = g[6];
            int  g7      = g[7];
            int  g8      = g[8];
            int  g9      = g[9];
            int  g1_19   = 19 * g1; // 1.4*2^29
            int  g2_19   = 19 * g2; // 1.4*2^30; still ok
            int  g3_19   = 19 * g3;
            int  g4_19   = 19 * g4;
            int  g5_19   = 19 * g5;
            int  g6_19   = 19 * g6;
            int  g7_19   = 19 * g7;
            int  g8_19   = 19 * g8;
            int  g9_19   = 19 * g9;
            int  f1_2    = 2 * f1;
            int  f3_2    = 2 * f3;
            int  f5_2    = 2 * f5;
            int  f7_2    = 2 * f7;
            int  f9_2    = 2 * f9;
            long f0g0    = (long)f0 * g0;
            long f0g1    = (long)(f0) * (long)(g1);
            long f0g2    = (long)(f0) * (long)(g2);
            long f0g3    = (long)(f0) * (long)(g3);
            long f0g4    = (long)(f0) * (long)(g4);
            long f0g5    = (long)(f0) * (long)(g5);
            long f0g6    = (long)(f0) * (long)(g6);
            long f0g7    = (long)(f0) * (long)(g7);
            long f0g8    = (long)(f0) * (long)(g8);
            long f0g9    = (long)(f0) * (long)(g9);
            long f1g0    = (long)(f1) * (long)(g0);
            long f1g1_2  = (long)(f1_2) * (long)(g1);
            long f1g2    = (long)(f1) * (long)(g2);
            long f1g3_2  = (long)(f1_2) * (long)(g3);
            long f1g4    = (long)(f1) * (long)(g4);
            long f1g5_2  = (long)(f1_2) * (long)(g5);
            long f1g6    = (long)(f1) * (long)(g6);
            long f1g7_2  = (long)(f1_2) * (long)(g7);
            long f1g8    = (long)(f1) * (long)(g8);
            long f1g9_38 = (long)(f1_2) * (long)(g9_19);
            long f2g0    = (long)(f2) * (long)(g0);
            long f2g1    = (long)(f2) * (long)(g1);
            long f2g2    = (long)(f2) * (long)(g2);
            long f2g3    = (long)(f2) * (long)(g3);
            long f2g4    = (long)(f2) * (long)(g4);
            long f2g5    = (long)(f2) * (long)(g5);
            long f2g6    = (long)(f2) * (long)(g6);
            long f2g7    = (long)(f2) * (long)(g7);
            long f2g8_19 = (long)(f2) * (long)(g8_19);
            long f2g9_19 = (long)(f2) * (long)(g9_19);
            long f3g0    = (long)(f3) * (long)(g0);
            long f3g1_2  = (long)(f3_2) * (long)(g1);
            long f3g2    = (long)(f3) * (long)(g2);
            long f3g3_2  = (long)(f3_2) * (long)(g3);
            long f3g4    = (long)(f3) * (long)(g4);
            long f3g5_2  = (long)(f3_2) * (long)(g5);
            long f3g6    = (long)(f3) * (long)(g6);
            long f3g7_38 = (long)(f3_2) * (long)(g7_19);
            long f3g8_19 = (long)(f3) * (long)(g8_19);
            long f3g9_38 = (long)(f3_2) * (long)(g9_19);
            long f4g0    = (long)(f4) * (long)(g0);
            long f4g1    = (long)(f4) * (long)(g1);
            long f4g2    = (long)(f4) * (long)(g2);
            long f4g3    = (long)(f4) * (long)(g3);
            long f4g4    = (long)(f4) * (long)(g4);
            long f4g5    = (long)(f4) * (long)(g5);
            long f4g6_19 = (long)(f4) * (long)(g6_19);
            long f4g7_19 = (long)(f4) * (long)(g7_19);
            long f4g8_19 = (long)(f4) * (long)(g8_19);
            long f4g9_19 = (long)(f4) * (long)(g9_19);
            long f5g0    = (long)(f5) * (long)(g0);
            long f5g1_2  = (long)(f5_2) * (long)(g1);
            long f5g2    = (long)(f5) * (long)(g2);
            long f5g3_2  = (long)(f5_2) * (long)(g3);
            long f5g4    = (long)(f5) * (long)(g4);
            long f5g5_38 = (long)(f5_2) * (long)(g5_19);
            long f5g6_19 = (long)(f5) * (long)(g6_19);
            long f5g7_38 = (long)(f5_2) * (long)(g7_19);
            long f5g8_19 = (long)(f5) * (long)(g8_19);
            long f5g9_38 = (long)(f5_2) * (long)(g9_19);
            long f6g0    = (long)(f6) * (long)(g0);
            long f6g1    = (long)(f6) * (long)(g1);
            long f6g2    = (long)(f6) * (long)(g2);
            long f6g3    = (long)(f6) * (long)(g3);
            long f6g4_19 = (long)(f6) * (long)(g4_19);
            long f6g5_19 = (long)(f6) * (long)(g5_19);
            long f6g6_19 = (long)(f6) * (long)(g6_19);
            long f6g7_19 = (long)(f6) * (long)(g7_19);
            long f6g8_19 = (long)(f6) * (long)(g8_19);
            long f6g9_19 = (long)(f6) * (long)(g9_19);
            long f7g0    = (long)(f7) * (long)(g0);
            long f7g1_2  = (long)(f7_2) * (long)(g1);
            long f7g2    = (long)(f7) * (long)(g2);
            long f7g3_38 = (long)(f7_2) * (long)(g3_19);
            long f7g4_19 = (long)(f7) * (long)(g4_19);
            long f7g5_38 = (long)(f7_2) * (long)(g5_19);
            long f7g6_19 = (long)(f7) * (long)(g6_19);
            long f7g7_38 = (long)(f7_2) * (long)(g7_19);
            long f7g8_19 = (long)(f7) * (long)(g8_19);
            long f7g9_38 = (long)(f7_2) * (long)(g9_19);
            long f8g0    = (long)(f8) * (long)(g0);
            long f8g1    = (long)(f8) * (long)(g1);
            long f8g2_19 = (long)(f8) * (long)(g2_19);
            long f8g3_19 = (long)(f8) * (long)(g3_19);
            long f8g4_19 = (long)(f8) * (long)(g4_19);
            long f8g5_19 = (long)(f8) * (long)(g5_19);
            long f8g6_19 = (long)(f8) * (long)(g6_19);
            long f8g7_19 = (long)(f8) * (long)(g7_19);
            long f8g8_19 = (long)(f8) * (long)(g8_19);
            long f8g9_19 = (long)(f8) * (long)(g9_19);
            long f9g0    = (long)(f9) * (long)(g0);
            long f9g1_38 = (long)(f9_2) * (long)(g1_19);
            long f9g2_19 = (long)(f9) * (long)(g2_19);
            long f9g3_38 = (long)(f9_2) * (long)(g3_19);
            long f9g4_19 = (long)(f9) * (long)(g4_19);
            long f9g5_38 = (long)(f9_2) * (long)(g5_19);
            long f9g6_19 = (long)(f9) * (long)(g6_19);
            long f9g7_38 = (long)(f9_2) * (long)(g7_19);
            long f9g8_19 = (long)(f9) * (long)(g8_19);
            long f9g9_38 = (long)(f9_2) * (long)(g9_19);
            long h0      = f0g0 + f1g9_38 + f2g8_19 + f3g7_38 + f4g6_19 + f5g5_38 + f6g4_19 + f7g3_38 + f8g2_19 + f9g1_38;
            long h1      = f0g1 + f1g0 + f2g9_19 + f3g8_19 + f4g7_19 + f5g6_19 + f6g5_19 + f7g4_19 + f8g3_19 + f9g2_19;
            long h2      = f0g2 + f1g1_2 + f2g0 + f3g9_38 + f4g8_19 + f5g7_38 + f6g6_19 + f7g5_38 + f8g4_19 + f9g3_38;
            long h3      = f0g3 + f1g2 + f2g1 + f3g0 + f4g9_19 + f5g8_19 + f6g7_19 + f7g6_19 + f8g5_19 + f9g4_19;
            long h4      = f0g4 + f1g3_2 + f2g2 + f3g1_2 + f4g0 + f5g9_38 + f6g8_19 + f7g7_38 + f8g6_19 + f9g5_38;
            long h5      = f0g5 + f1g4 + f2g3 + f3g2 + f4g1 + f5g0 + f6g9_19 + f7g8_19 + f8g7_19 + f9g6_19;
            long h6      = f0g6 + f1g5_2 + f2g4 + f3g3_2 + f4g2 + f5g1_2 + f6g0 + f7g9_38 + f8g8_19 + f9g7_38;
            long h7      = f0g7 + f1g6 + f2g5 + f3g4 + f4g3 + f5g2 + f6g1 + f7g0 + f8g9_19 + f9g8_19;
            long h8      = f0g8 + f1g7_2 + f2g6 + f3g5_2 + f4g4 + f5g3_2 + f6g2 + f7g1_2 + f8g0 + f9g9_38;
            long h9      = f0g9 + f1g8 + f2g7 + f3g6 + f4g5 + f5g4 + f6g3 + f7g2 + f8g1 + f9g0;
            var  carry   = new long[10];

            // |h0| <= (1.1*1.1*2^52*(1+19+19+19+19)+1.1*1.1*2^50*(38+38+38+38+38))
            //   i.e. |h0| <= 1.2*2^59; narrower ranges for h2, h4, h6, h8
            // |h1| <= (1.1*1.1*2^51*(1+1+19+19+19+19+19+19+19+19))
            //   i.e. |h1| <= 1.5*2^58; narrower ranges for h3, h5, h7, h9

            carry[0] = (h0 + (1 << 25)) >> 26;
            h1      += carry[0];
            h0      -= carry[0] << 26;
            carry[4] = (h4 + (1 << 25)) >> 26;
            h5      += carry[4];
            h4      -= carry[4] << 26;
            // |h0| <= 2^25
            // |h4| <= 2^25
            // |h1| <= 1.51*2^58
            // |h5| <= 1.51*2^58

            carry[1] = (h1 + (1 << 24)) >> 25;
            h2      += carry[1];
            h1      -= carry[1] << 25;
            carry[5] = (h5 + (1 << 24)) >> 25;
            h6      += carry[5];
            h5      -= carry[5] << 25;
            // |h1| <= 2^24; from now on fits into int32
            // |h5| <= 2^24; from now on fits into int32
            // |h2| <= 1.21*2^59
            // |h6| <= 1.21*2^59

            carry[2] = (h2 + (1 << 25)) >> 26;
            h3      += carry[2];
            h2      -= carry[2] << 26;
            carry[6] = (h6 + (1 << 25)) >> 26;
            h7      += carry[6];
            h6      -= carry[6] << 26;
            // |h2| <= 2^25; from now on fits into int32 unchanged
            // |h6| <= 2^25; from now on fits into int32 unchanged
            // |h3| <= 1.51*2^58
            // |h7| <= 1.51*2^58

            carry[3] = (h3 + (1 << 24)) >> 25;
            h4      += carry[3];
            h3      -= carry[3] << 25;
            carry[7] = (h7 + (1 << 24)) >> 25;
            h8      += carry[7];
            h7      -= carry[7] << 25;
            // |h3| <= 2^24; from now on fits into int32 unchanged
            // |h7| <= 2^24; from now on fits into int32 unchanged
            // |h4| <= 1.52*2^33
            // |h8| <= 1.52*2^33

            carry[4] = (h4 + (1 << 25)) >> 26;
            h5      += carry[4];
            h4      -= carry[4] << 26;
            carry[8] = (h8 + (1 << 25)) >> 26;
            h9      += carry[8];
            h8      -= carry[8] << 26;
            // |h4| <= 2^25; from now on fits into int32 unchanged
            // |h8| <= 2^25; from now on fits into int32 unchanged
            // |h5| <= 1.01*2^24
            // |h9| <= 1.51*2^58

            carry[9] = (h9 + (1 << 24)) >> 25;
            h0      += carry[9] * 19;
            h9      -= carry[9] << 25;
            // |h9| <= 2^24; from now on fits into int32 unchanged
            // |h0| <= 1.8*2^37

            carry[0] = (h0 + (1 << 25)) >> 26;
            h1      += carry[0];
            h0      -= carry[0] << 26;
            // |h0| <= 2^25; from now on fits into int32 unchanged
            // |h1| <= 1.01*2^24

            int[] h = new int[10];
            h[0] = (int)(h0);
            h[1] = (int)(h1);
            h[2] = (int)(h2);
            h[3] = (int)(h3);
            h[4] = (int)(h4);
            h[5] = (int)(h5);
            h[6] = (int)(h6);
            h[7] = (int)(h7);
            h[8] = (int)(h8);
            h[9] = (int)(h9);

            var final = new FieldElement();

            final.SetElementsDirect(h);
            return(final);
        }