1 #include "fe.h"
2 
3 /*
4 Preconditions:
5   |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
6 
7 Write p=2^255-19; q=floor(h/p).
8 Basic claim: q = floor(2^(-255)(h + 19 2^(-25)h9 + 2^(-1))).
9 
10 Proof:
11   Have |h|<=p so |q|<=1 so |19^2 2^(-255) q|<1/4.
12   Also have |h-2^230 h9|<2^231 so |19 2^(-255)(h-2^230 h9)|<1/4.
13 
14   Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9).
15   Then 0<y<1.
16 
17   Write r=h-pq.
18   Have 0<=r<=p-1=2^255-20.
19   Thus 0<=r+19(2^-255)r<r+19(2^-255)2^255<=2^255-1.
20 
21   Write x=r+19(2^-255)r+y.
22   Then 0<x<2^255 so floor(2^(-255)x) = 0 so floor(q+2^(-255)x) = q.
23 
24   Have q+2^(-255)x = 2^(-255)(h + 19 2^(-25) h9 + 2^(-1))
25   so floor(2^(-255)(h + 19 2^(-25) h9 + 2^(-1))) = q.
26 */
27 
fe_tobytes(unsigned char * s,const fe h)28 void fe_tobytes(unsigned char *s,const fe h)
29 {
30   crypto_int32 h0 = h[0];
31   crypto_int32 h1 = h[1];
32   crypto_int32 h2 = h[2];
33   crypto_int32 h3 = h[3];
34   crypto_int32 h4 = h[4];
35   crypto_int32 h5 = h[5];
36   crypto_int32 h6 = h[6];
37   crypto_int32 h7 = h[7];
38   crypto_int32 h8 = h[8];
39   crypto_int32 h9 = h[9];
40   crypto_int32 q;
41   crypto_int32 carry0;
42   crypto_int32 carry1;
43   crypto_int32 carry2;
44   crypto_int32 carry3;
45   crypto_int32 carry4;
46   crypto_int32 carry5;
47   crypto_int32 carry6;
48   crypto_int32 carry7;
49   crypto_int32 carry8;
50   crypto_int32 carry9;
51 
52   q = (19 * h9 + (((crypto_int32) 1) << 24)) >> 25;
53   q = (h0 + q) >> 26;
54   q = (h1 + q) >> 25;
55   q = (h2 + q) >> 26;
56   q = (h3 + q) >> 25;
57   q = (h4 + q) >> 26;
58   q = (h5 + q) >> 25;
59   q = (h6 + q) >> 26;
60   q = (h7 + q) >> 25;
61   q = (h8 + q) >> 26;
62   q = (h9 + q) >> 25;
63 
64   /* Goal: Output h-(2^255-19)q, which is between 0 and 2^255-20. */
65   h0 += 19 * q;
66   /* Goal: Output h-2^255 q, which is between 0 and 2^255-20. */
67 
68   carry0 = h0 >> 26; h1 += carry0; h0 -= carry0 << 26;
69   carry1 = h1 >> 25; h2 += carry1; h1 -= carry1 << 25;
70   carry2 = h2 >> 26; h3 += carry2; h2 -= carry2 << 26;
71   carry3 = h3 >> 25; h4 += carry3; h3 -= carry3 << 25;
72   carry4 = h4 >> 26; h5 += carry4; h4 -= carry4 << 26;
73   carry5 = h5 >> 25; h6 += carry5; h5 -= carry5 << 25;
74   carry6 = h6 >> 26; h7 += carry6; h6 -= carry6 << 26;
75   carry7 = h7 >> 25; h8 += carry7; h7 -= carry7 << 25;
76   carry8 = h8 >> 26; h9 += carry8; h8 -= carry8 << 26;
77   carry9 = h9 >> 25;               h9 -= carry9 << 25;
78                   /* h10 = carry9 */
79 
80   /*
81   Goal: Output h0+...+2^255 h10-2^255 q, which is between 0 and 2^255-20.
82   Have h0+...+2^230 h9 between 0 and 2^255-1;
83   evidently 2^255 h10-2^255 q = 0.
84   Goal: Output h0+...+2^230 h9.
85   */
86 
87   s[0] = h0 >> 0;
88   s[1] = h0 >> 8;
89   s[2] = h0 >> 16;
90   s[3] = (h0 >> 24) | (h1 << 2);
91   s[4] = h1 >> 6;
92   s[5] = h1 >> 14;
93   s[6] = (h1 >> 22) | (h2 << 3);
94   s[7] = h2 >> 5;
95   s[8] = h2 >> 13;
96   s[9] = (h2 >> 21) | (h3 << 5);
97   s[10] = h3 >> 3;
98   s[11] = h3 >> 11;
99   s[12] = (h3 >> 19) | (h4 << 6);
100   s[13] = h4 >> 2;
101   s[14] = h4 >> 10;
102   s[15] = h4 >> 18;
103   s[16] = h5 >> 0;
104   s[17] = h5 >> 8;
105   s[18] = h5 >> 16;
106   s[19] = (h5 >> 24) | (h6 << 1);
107   s[20] = h6 >> 7;
108   s[21] = h6 >> 15;
109   s[22] = (h6 >> 23) | (h7 << 3);
110   s[23] = h7 >> 5;
111   s[24] = h7 >> 13;
112   s[25] = (h7 >> 21) | (h8 << 4);
113   s[26] = h8 >> 4;
114   s[27] = h8 >> 12;
115   s[28] = (h8 >> 20) | (h9 << 6);
116   s[29] = h9 >> 2;
117   s[30] = h9 >> 10;
118   s[31] = h9 >> 18;
119 }
120