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-== Tutorial ==
-
-This chapter walks through how one might implement the
-Boneh-Lynn-Shacham (BLS) signature scheme using the PBC library.
-It is based on the file `example/bls.c`.
-
-We have three groups 'G1', 'G2', 'GT' of prime order 'r', and a bilinear map
-'e' that takes an element from 'G1' and an element from 'G2', and outputs an
-element of 'GT'. We publish these along with the system parameter 'g', which is
-a randomly chosen element of 'G2'.
-
-Alice wishes to sign a message.  She generates her public and private keys as
-follows.  Her private key is a random element 'x' of 'Zr', and her corresponding
-public key is 'g'^'x'^.
-
-To sign a message, Alice hashes the message to some element
-'h' of 'G1', and then outputs the signature 'h'^'x'^.
-
-To verify a signature sigma, Bob checks that
-'e'('h','g'^'x'^) = 'e'(sigma, 'g').
-
-We now translate the above to C code using the PBC library.
-
-=== BLS signatures ===
-
-First we include `pbc/pbc.h`:
-
-  #include <pbc.h>
-
-Next we initialize a pairing:
-
-  pairing_t pairing;
-  char param[1024];
-  size_t count = fread(param, 1, 1024, stdin);
-  if (!count) pbc_die("input error");
-  pairing_init_set_buf(pairing, param, count);
-
-Later we give pairing parameters to our program on standard input. Any file in
-the `param` subdirectory will suffice, for example:
-
- $ bls < param/a.param
-
-We shall need several +element_t+ variables to hold the system parameters, keys
-and other quantities. We declare them and initialize them,
-....
-element_t g, h;
-element_t public_key, secret_key;
-element_t sig;
-element_t temp1, temp2;
-
-element_init_G2(g, pairing);
-element_init_G2(public_key, pairing);
-element_init_G1(h, pairing);
-element_init_G1(sig, pairing);
-element_init_GT(temp1, pairing);
-element_init_GT(temp2, pairing);
-element_init_Zr(secret_key, pairing);
-....
-generate system parameters,
-
-    element_random(g);
-
-generate a private key,
-
-    element_random(secret_key);
-
-and the corresponding public key.
-
-    element_pow_zn(public_key, g, secret_key);
-
-When given a message to sign, we first compute its hash, using some standard
-hash algorithm.  Many libraries can do this, and this operation does not
-involve pairings, so PBC does not provide functions for this step. For this
-example, and our message has already been hashed, possibly using another
-library.
-
-Say the message hash is "ABCDEF" (a 48-bit hash).  We map these bytes to an
-element h of G1,
-
-    element_from_hash(h, "ABCDEF", 6);
-
-then sign it:
-
-    element_pow_zn(sig, h, secret_key);
-
-To verify this signature, we compare the
-outputs of the pairing applied to the signature and system parameter,
-and the pairing applied to the message hash and public key.
-If the pairing outputs match then the signature is valid.
-
-....
-pairing_apply(temp1, sig, g, pairing);
-pairing_apply(temp2, h, public_key, pairing);
-if (!element_cmp(temp1, temp2)) {
-    printf("signature verifies\n");
-} else {
-    printf("signature does not verify\n");
-}
-....
-
-=== Import/export ===
-
-To be useful, at some stage the signature must be converted
-to bytes for storage or transmission:
-
-    int n = pairing_length_in_bytes_compressed_G1(pairing);
-    // Alternatively:
-    // int n = element_length_in_bytes_compressed(sig);
-    unsigned char *data = malloc(n);
-    element_to_bytes_compressed(data, sig);
-
-On the other end, the signature must be decompressed:
-
-    element_from_bytes_compressed(sig, data);
-
-Eliding +_compressed+ in the above code
-will also work but the buffer 'data' will be roughly twice as large.
-
-We can save more space by using the 'x'-coordinate of the signature only
-
-    int n = pairing_length_in_bytes_x_only_G1(pairing);
-    // Alternative:
-    //   int n = element_length_in_bytes_x_only(sig);
-    unsigned char *data = malloc(n);
-    element_to_bytes_compressed(data, sig);
-
-but then there is a complication during verification since two different
-points have the same 'x'-coordinate. One way to solve this problem is to
-guess one point and try to verify. If that fails, we try the other.
-It can be shown that the pairing outputs of the two points are inverses
-of each other, avoiding the need to compute a pairing the second time.
-(In fact, there are even better ways to handle this.)
-....
-int n = pairing_length_in_bytes_x_only_G1(pairing);
-//int n = element_length_in_bytes_x_only(sig);
-unsigned char *data = malloc(n);
-
-element_to_bytes_x_only(data, sig);
-
-element_from_bytes_x_only(sig, data)
-
-pairing_apply(temp1, sig, g, pairing);
-pairing_apply(temp2, h, public_key, pairing);
-
-if (!element_cmp(temp1, temp2)) {
-    printf("signature verifies on first guess\n");
-} else {
-    element_invert(temp1, temp1);
-    if (!element_cmp(temp1, temp2)) {
-	printf("signature verifies on second guess\n");
-    } else {
-	printf("signature does not verify\n");
-    }
-}
-....