ed448goldilocks/test/bench.c

/* Copyright (c) 2014 Cryptography Research, Inc.
 * Released under the MIT License.  See LICENSE.txt for license information.
 */

#include "word.h"

#include <sys/time.h>
#include <sys/types.h>
#include <stdio.h>
#include <memory.h>

#include "field.h"
#include "ec_point.h"
#include "scalarmul.h"
#include "barrett_field.h"
#include "crandom.h"
#include "goldilocks.h"
#include "sha512.h"

static __inline__ void
ignore_result ( int result ) {
    (void)result;
}

static double now(void) {
  struct timeval tv;
  gettimeofday(&tv, NULL);
  
  return tv.tv_sec + tv.tv_usec/1000000.0;
}

static void field_randomize( struct crandom_state_t *crand, struct field_t *a ) {
    crandom_generate(crand, (unsigned char *)a, sizeof(*a));
    field_strong_reduce(a);
}

static void q448_randomize( struct crandom_state_t *crand, word_t sk[SCALAR_WORDS] ) {
    crandom_generate(crand, (unsigned char *)sk, SCALAR_BYTES);
}

static void field_print( const char *descr, const struct field_t *a ) {
    int j;
    unsigned char ser[FIELD_BYTES];
    field_serialize(ser,a);
    printf("%s = 0x", descr);
    for (j=FIELD_BYTES - 1; j>=0; j--) {
        printf("%02x", ser[j]);
    }
    printf("\n");
}

static void __attribute__((unused))
field_print_full (
    const char *descr,
    const struct field_t *a
) {
    int j;
    printf("%s = 0x", descr);
    for (j=15; j>=0; j--) {
        printf("%02" PRIxWORD "_" PRIxWORD56 " ",
            a->limb[j]>>28, a->limb[j]&((1<<28)-1));
    }
    printf("\n");
}

static void q448_print( const char *descr, const word_t secret[SCALAR_WORDS] ) {
    int j;
    printf("%s = 0x", descr);
    for (j=SCALAR_WORDS-1; j>=0; j--) {
        printf(PRIxWORDfull, secret[j]);
    }
    printf("\n");
}

#ifndef N_TESTS_BASE
#define N_TESTS_BASE 10000
#endif

int main(int argc, char **argv) {
    (void)argc;
    (void)argv;

    struct tw_extensible_t ext;
    struct extensible_t exta;
    struct tw_niels_t niels;
    struct tw_pniels_t pniels;
    struct affine_t affine;
    struct montgomery_t mb;
    struct field_t a,b,c,d;
    
    
    double when;
    int i;

    int nbase = N_TESTS_BASE;
    
    /* Bad randomness so we can debug. */
    char initial_seed[32];
    for (i=0; i<32; i++) initial_seed[i] = i;
    struct crandom_state_t crand;
    crandom_init_from_buffer(&crand, initial_seed);
    /* For testing the performance drop from the crandom debuffering change.
        ignore_result(crandom_init_from_file(&crand, "/dev/urandom", 10000, 1));
    */
    
    word_t sk[SCALAR_WORDS],tk[SCALAR_WORDS];
    q448_randomize(&crand, sk);
    
    memset(&a,0,sizeof(a));
    memset(&b,0,sizeof(b));
    memset(&c,0,sizeof(c));
    memset(&d,0,sizeof(d));
    when = now();
    for (i=0; i<nbase*5000; i++) {
        field_mul(&c, &b, &a);
    }
    when = now() - when;
    printf("mul:         %5.1fns\n", when * 1e9 / i);
    
    when = now();
    for (i=0; i<nbase*5000; i++) {
        field_sqr(&c, &a);
    }
    when = now() - when;
    printf("sqr:         %5.1fns\n", when * 1e9 / i);
    
    when = now();
    for (i=0; i<nbase*5000; i++) {
        field_mulw(&c, &b, 1234562);
    }
    when = now() - when;
    printf("mulw:        %5.1fns\n", when * 1e9 / i);
    
    when = now();
    for (i=0; i<nbase*500; i++) {
        field_mul(&c, &b, &a);
        field_mul(&a, &b, &c);
    }
    when = now() - when;
    printf("mul dep:     %5.1fns\n", when * 1e9 / i / 2);
    
    when = now();
    for (i=0; i<nbase*10; i++) {
        field_randomize(&crand, &a);
    }
    when = now() - when;
    printf("rand448:     %5.1fns\n", when * 1e9 / i);
    
    struct sha512_ctx_t sha;
    uint8_t hashout[128];
    when = now();
    for (i=0; i<nbase; i++) {
        sha512_init(&sha);
        sha512_final(&sha, hashout);
    }
    when = now() - when;
    printf("sha512 1blk: %5.1fns\n", when * 1e9 / i);
    
    when = now();
    for (i=0; i<nbase; i++) {
        sha512_update(&sha, hashout, 128);
    }
    when = now() - when;
    printf("sha512 blk:  %5.1fns (%0.2f MB/s)\n", when * 1e9 / i, 128*i/when/1e6);
    
    when = now();
    for (i=0; i<nbase; i++) {
        field_isr(&c, &a);
    }
    when = now() - when;
    printf("isr auto:    %5.1fµs\n", when * 1e6 / i);
    
    for (i=0; i<100; i++) {
        field_randomize(&crand, &a);
        field_isr(&d,&a);
        field_sqr(&b,&d);
        field_mul(&c,&b,&a);
        field_sqr(&b,&c);
        field_subw(&b,1);
        field_bias(&b,1);
        if (!field_is_zero(&b)) {
            printf("ISR validation failure!\n");
            field_print("a", &a);
            field_print("s", &d);
        }
    }
    
    when = now();
    for (i=0; i<nbase; i++) {
        elligator_2s_inject(&affine, &a);
    }
    when = now() - when;
    printf("elligator:   %5.1fµs\n", when * 1e6 / i);
    
    for (i=0; i<100; i++) {
        field_randomize(&crand, &a);
        elligator_2s_inject(&affine, &a);
        if (!validate_affine(&affine)) {
            printf("Elligator validation failure!\n");
            field_print("a", &a);
            field_print("x", &affine.x);
            field_print("y", &affine.y);
        }
    }
    
    when = now();
    for (i=0; i<nbase; i++) {
        deserialize_affine(&affine, &a);
    }
    when = now() - when;
    printf("decompress:  %5.1fµs\n", when * 1e6 / i);
    
    convert_affine_to_extensible(&exta, &affine);
    when = now();
    for (i=0; i<nbase; i++) {
        serialize_extensible(&a, &exta);
    }
    when = now() - when;
    printf("compress:    %5.1fµs\n", when * 1e6 / i);
    
    int goods = 0;
    for (i=0; i<100; i++) {
        field_randomize(&crand, &a);
        mask_t good = deserialize_affine(&affine, &a);
        if (good & !validate_affine(&affine)) {
            printf("Deserialize validation failure!\n");
            field_print("a", &a);
            field_print("x", &affine.x);
            field_print("y", &affine.y);
        } else if (good) {
            goods++;
            convert_affine_to_extensible(&exta,&affine);
            serialize_extensible(&b, &exta);
            field_sub(&c,&b,&a);
            field_bias(&c,2);
            if (!field_is_zero(&c)) {
                printf("Reserialize validation failure!\n");
                field_print("a", &a);
                field_print("x", &affine.x);
                field_print("y", &affine.y);
                deserialize_affine(&affine, &b);
                field_print("b", &b);
                field_print("x", &affine.x);
                field_print("y", &affine.y);
                printf("\n");
            }
        }
    }
    if (goods<i/3) {
        printf("Deserialization validation failure! Deserialized %d/%d points\n", goods, i);
    }
    
    word_t lsk[768/WORD_BITS];
    crandom_generate(&crand, (unsigned char *)lsk, sizeof(lsk));
    
    when = now();
    for (i=0; i<nbase*100; i++) {
        barrett_reduce(lsk,sizeof(lsk)/sizeof(word_t),0,&curve_prime_order);
    }
    when = now() - when;
    printf("barrett red: %5.1fns\n", when * 1e9 / i);
    
    when = now();
    for (i=0; i<nbase*10; i++) {
        barrett_mac(lsk,SCALAR_WORDS,lsk,SCALAR_WORDS,lsk,SCALAR_WORDS,&curve_prime_order);
    }
    when = now() - when;
    printf("barrett mac: %5.1fns\n", when * 1e9 / i);
    
    memset(&ext,0,sizeof(ext));
    memset(&niels,0,sizeof(niels)); /* avoid assertions in p521 even though this isn't a valid ext or niels */
    when = now();
    for (i=0; i<nbase*100; i++) {
        add_tw_niels_to_tw_extensible(&ext, &niels);
    }
    when = now() - when;
    printf("exti+niels:  %5.1fns\n", when * 1e9 / i);
    
    convert_tw_extensible_to_tw_pniels(&pniels, &ext);
    when = now();
    for (i=0; i<nbase*100; i++) {
        add_tw_pniels_to_tw_extensible(&ext, &pniels);
    }
    when = now() - when;
    printf("exti+pniels: %5.1fns\n", when * 1e9 / i);
    
    when = now();
    for (i=0; i<nbase*100; i++) {
        double_tw_extensible(&ext);
    }
    when = now() - when;
    printf("exti dbl:    %5.1fns\n", when * 1e9 / i);
    
    when = now();
    for (i=0; i<nbase*100; i++) {
        untwist_and_double(&exta, &ext);
    }
    when = now() - when;
    printf("i->a isog:   %5.1fns\n", when * 1e9 / i);
    
    when = now();
    for (i=0; i<nbase*100; i++) {
        twist_and_double(&ext, &exta);
    }
    when = now() - when;
    printf("a->i isog:   %5.1fns\n", when * 1e9 / i);
    
    memset(&mb,0,sizeof(mb));
    when = now();
    for (i=0; i<nbase*100; i++) {
        montgomery_step(&mb);
    }
    when = now() - when;
    printf("monty step:  %5.1fns\n", when * 1e9 / i);
	
    when = now();
    for (i=0; i<nbase/10; i++) {
        ignore_result(montgomery_ladder(&a,&b,sk,FIELD_BITS,0));
    }
    when = now() - when;
    printf("full ladder: %5.1fµs\n", when * 1e6 / i);
    
    when = now();
    for (i=0; i<nbase/10; i++) {
        scalarmul(&ext,sk);
    }
    when = now() - when;
    printf("edwards smz: %5.1fµs\n", when * 1e6 / i);
    
    when = now();
    for (i=0; i<nbase/10; i++) {
        scalarmul_vlook(&ext,sk);
    }
    when = now() - when;
    printf("edwards svl: %5.1fµs\n", when * 1e6 / i);
    
    when = now();
    for (i=0; i<nbase/10; i++) {
        scalarmul(&ext,sk);
        untwist_and_double_and_serialize(&a,&ext);
    }
    when = now() - when;
    printf("edwards smc: %5.1fµs\n", when * 1e6 / i);
    
    when = now();
    for (i=0; i<nbase/10; i++) {
        q448_randomize(&crand, sk);
        scalarmul_vt(&ext,sk,SCALAR_BITS);
    }
    when = now() - when;
    printf("edwards vtm: %5.1fµs\n", when * 1e6 / i);
    
    struct tw_niels_t wnaft[1<<6];
    when = now();
    for (i=0; i<nbase/10; i++) {
        ignore_result(precompute_fixed_base_wnaf(wnaft,&ext,6));
    }
    when = now() - when;
    printf("wnaf6 pre:   %5.1fµs\n", when * 1e6 / i);
    
    when = now();
    for (i=0; i<nbase/10; i++) {
        q448_randomize(&crand, sk);
        scalarmul_fixed_base_wnaf_vt(&ext,sk,SCALAR_BITS,wnaft,6);
    }
    when = now() - when;
    printf("edwards vt6: %5.1fµs\n", when * 1e6 / i);
    
    when = now();
    for (i=0; i<nbase/10; i++) {
        ignore_result(precompute_fixed_base_wnaf(wnaft,&ext,4));
    }
    when = now() - when;
    printf("wnaf4 pre:   %5.1fµs\n", when * 1e6 / i);
    
    when = now();
    for (i=0; i<nbase/10; i++) {
        q448_randomize(&crand, sk);
        scalarmul_fixed_base_wnaf_vt(&ext,sk,SCALAR_BITS,wnaft,4);
    }
    when = now() - when;
    printf("edwards vt4: %5.1fµs\n", when * 1e6 / i);

    when = now();
    for (i=0; i<nbase/10; i++) {
        ignore_result(precompute_fixed_base_wnaf(wnaft,&ext,5));
    }
    when = now() - when;
    printf("wnaf5 pre:   %5.1fµs\n", when * 1e6 / i);
    
    when = now();
    for (i=0; i<nbase/10; i++) {
        q448_randomize(&crand, sk);
        scalarmul_fixed_base_wnaf_vt(&ext,sk,SCALAR_BITS,wnaft,5);
    }
    when = now() - when;
    printf("edwards vt5: %5.1fµs\n", when * 1e6 / i);
    
    when = now();
    for (i=0; i<nbase/10; i++) {
        q448_randomize(&crand, sk);
        q448_randomize(&crand, tk);
        linear_combo_var_fixed_vt(&ext,sk,FIELD_BITS,tk,FIELD_BITS,wnaft,5);
    }
    when = now() - when;
    printf("vt vf combo: %5.1fµs\n", when * 1e6 / i);
    
    when = now();
    for (i=0; i<nbase/10; i++) {
        deserialize_affine(&affine, &a);
        convert_affine_to_extensible(&exta,&affine);
        twist_and_double(&ext,&exta);
        scalarmul(&ext,sk);
        untwist_and_double(&exta,&ext);
        serialize_extensible(&b, &exta);
    }
    when = now() - when;
    printf("edwards sm:  %5.1fµs\n", when * 1e6 / i);
    
    struct fixed_base_table_t t_5_5_18, t_3_5_30, t_8_4_14, t_5_3_30, t_15_3_10;

    while (1) {
        field_randomize(&crand, &a);
        if (deserialize_affine(&affine, &a)) break;
    }
    convert_affine_to_extensible(&exta,&affine);
    twist_and_double(&ext,&exta);
    when = now();
    for (i=0; i<nbase/10; i++) {
        if (i) destroy_fixed_base(&t_5_5_18);
        ignore_result(precompute_fixed_base(&t_5_5_18, &ext, 5, 5, 18, NULL));
    }
    when = now() - when;
    printf("pre(5,5,18): %5.1fµs\n", when * 1e6 / i);
    
    when = now();
    for (i=0; i<nbase/10; i++) {
        if (i) destroy_fixed_base(&t_3_5_30);
        ignore_result(precompute_fixed_base(&t_3_5_30, &ext, 3, 5, 30, NULL));
    }
    when = now() - when;
    printf("pre(3,5,30): %5.1fµs\n", when * 1e6 / i);
    
    when = now();
    for (i=0; i<nbase/10; i++) {
        if (i) destroy_fixed_base(&t_5_3_30);
        ignore_result(precompute_fixed_base(&t_5_3_30, &ext, 5, 3, 30, NULL));
    }
    when = now() - when;
    printf("pre(5,3,30): %5.1fµs\n", when * 1e6 / i);
    
    when = now();
    for (i=0; i<nbase/10; i++) {
        if (i) destroy_fixed_base(&t_15_3_10);
        ignore_result(precompute_fixed_base(&t_15_3_10, &ext, 15, 3, 10, NULL));
    }
    when = now() - when;
    printf("pre(15,3,10):%5.1fµs\n", when * 1e6 / i);
    
    when = now();
    for (i=0; i<nbase/10; i++) {
        if (i) destroy_fixed_base(&t_8_4_14);
        ignore_result(precompute_fixed_base(&t_8_4_14, &ext, 8, 4, 14, NULL));
    }
    when = now() - when;
    printf("pre(8,4,14): %5.1fµs\n", when * 1e6 / i);
	
    when = now();
    for (i=0; i<nbase; i++) {
        scalarmul_fixed_base(&ext, sk, FIELD_BITS, &t_5_5_18);
    }
    when = now() - when;
    printf("com(5,5,18): %5.1fµs\n", when * 1e6 / i);
    
    when = now();
    for (i=0; i<nbase; i++) {
        scalarmul_fixed_base(&ext, sk, FIELD_BITS, &t_3_5_30);
    }
    when = now() - when;
    printf("com(3,5,30): %5.1fµs\n", when * 1e6 / i);

    when = now();
    for (i=0; i<nbase; i++) {
        scalarmul_fixed_base(&ext, sk, FIELD_BITS, &t_8_4_14);
    }
    when = now() - when;
    printf("com(8,4,14): %5.1fµs\n", when * 1e6 / i);

    when = now();
    for (i=0; i<nbase; i++) {
        scalarmul_fixed_base(&ext, sk, FIELD_BITS, &t_5_3_30);
    }
    when = now() - when;
    printf("com(5,3,30): %5.1fµs\n", when * 1e6 / i);

    when = now();
    for (i=0; i<nbase; i++) {
        scalarmul_fixed_base(&ext, sk, FIELD_BITS, &t_15_3_10);
    }
    when = now() - when;
    printf("com(15,3,10):%5.1fµs\n", when * 1e6 / i);
    
    printf("\nGoldilocks:\n");
    
    int res = goldilocks_init();
    assert(!res);
    
    struct goldilocks_public_key_t gpk,hpk;
    struct goldilocks_private_key_t gsk,hsk;
    
    when = now();
    for (i=0; i<nbase; i++) {
        if (i&1) {
            res = goldilocks_keygen(&gsk,&gpk);
        } else {
            res = goldilocks_keygen(&hsk,&hpk);
        }
        assert(!res);
    }
    when = now() - when;
    printf("keygen:      %5.1fµs\n", when * 1e6 / i);
    
    uint8_t ss1[64],ss2[64];
    int gres1=0,gres2=0;
    when = now();
    for (i=0; i<nbase; i++) {
        if (i&1) {
            gres1 = goldilocks_shared_secret(ss1,&gsk,&hpk);
        } else {
            gres2 = goldilocks_shared_secret(ss2,&hsk,&gpk);
        }
    }
    when = now() - when;
    printf("ecdh:        %5.1fµs\n", when * 1e6 / i);
    if (gres1 || gres2 || memcmp(ss1,ss2,64)) {
        printf("[FAIL] %d %d\n",gres1,gres2);
        
        printf("sk1 = ");
        for (i=0; i<SCALAR_BYTES; i++) {
            printf("%02x", gsk.opaque[i]);
        }
        printf("\nsk2 = ");
        for (i=0; i<SCALAR_BYTES; i++) {
            printf("%02x", hsk.opaque[i]);
        }
        printf("\nss1 = ");
        for (i=0; i<64; i++) {
            printf("%02x", ss1[i]);
        }
        printf("\nss2 = ");
        for (i=0; i<64; i++) {
            printf("%02x", ss2[i]);
        }
        printf("\n");
    }
    
    uint8_t sout[FIELD_BYTES*2];
    const char *message = "hello world";
    size_t message_len = strlen(message);
    when = now();
    for (i=0; i<nbase; i++) {
        res = goldilocks_sign(sout,(const unsigned char *)message,message_len,&gsk);
        (void)res;
        assert(!res);
    }
    when = now() - when;
    printf("sign:        %5.1fµs\n", when * 1e6 / i);
    
    when = now();
    for (i=0; i<nbase; i++) {
        int ver = goldilocks_verify(sout,(const unsigned char *)message,message_len,&gpk);
        (void)ver;
        assert(!ver);
    }
    when = now() - when;
    printf("verify:      %5.1fµs\n", when * 1e6 / i);
    
    struct goldilocks_precomputed_public_key_t *pre = NULL;
    when = now();
    for (i=0; i<nbase; i++) {
        goldilocks_destroy_precomputed_public_key(pre);
        pre = goldilocks_precompute_public_key(&gpk);
    }
    when = now() - when;
    printf("precompute:  %5.1fµs\n", when * 1e6 / i);
    
    when = now();
    for (i=0; i<nbase; i++) {
        int ver = goldilocks_verify_precomputed(sout,(const unsigned char *)message,message_len,pre);
        (void)ver;
        assert(!ver);
    }
    when = now() - when;
    printf("verify pre:  %5.1fµs\n", when * 1e6 / i);
    
    when = now();
    for (i=0; i<nbase; i++) {
        int ret = goldilocks_shared_secret_precomputed(ss1,&gsk,pre);
        (void)ret;
        assert(!ret);
    }
    when = now() - when;
    printf("ecdh pre:    %5.1fµs\n", when * 1e6 / i);
    
    printf("\nTesting...\n");
    
    
    int failures=0, successes = 0;
    for (i=0; i<nbase/10; i++) {
        ignore_result(goldilocks_keygen(&gsk,&gpk));
        goldilocks_sign(sout,(const unsigned char *)message,message_len,&gsk);
        res = goldilocks_verify(sout,(const unsigned char *)message,message_len,&gpk);
        if (res) failures++;
    }
    if (failures) {
        printf("FAIL %d/%d signature checks!\n", failures, i);
    }
    
    failures=0; successes = 0;
    for (i=0; i<nbase/10; i++) {
        field_randomize(&crand, &a);
		word_t two = 2;
        mask_t good = montgomery_ladder(&b,&a,&two,2,0);
		if (!good) continue;
		
		word_t x,y;
        crandom_generate(&crand, (unsigned char *)&x, sizeof(x));
        crandom_generate(&crand, (unsigned char *)&y, sizeof(y));
        x = (hword_t)x;
        y = (hword_t)y;
        word_t z=x*y;
        
    	ignore_result(montgomery_ladder(&b,&a,&x,WORD_BITS,0));
        ignore_result(montgomery_ladder(&c,&b,&y,WORD_BITS,0));
        ignore_result(montgomery_ladder(&b,&a,&z,WORD_BITS,0));
        
        field_sub(&d,&b,&c);
        field_bias(&d,2);
		if (!field_is_zero(&d)) {
            printf("Odd ladder validation failure %d!\n", ++failures);
            field_print("a", &a);
            printf("x=%"PRIxWORD", y=%"PRIxWORD", z=%"PRIxWORD"\n", x,y,z);
            field_print("c", &c);
            field_print("b", &b);
			printf("\n");
		}
	}
    
    failures = 0;
    for (i=0; i<nbase/10; i++) {
        mask_t good;
        do {
            field_randomize(&crand, &a);
            good = deserialize_affine(&affine, &a);
        } while (!good);
        
        convert_affine_to_extensible(&exta,&affine);
        twist_and_double(&ext,&exta);
        untwist_and_double(&exta,&ext);
        serialize_extensible(&b, &exta);
        untwist_and_double_and_serialize(&c, &ext);
        
        field_sub(&d,&b,&c);
        field_bias(&d,2);
        
        if (good && !field_is_zero(&d)){
            printf("Iso+serial validation failure %d!\n", ++failures);
            field_print("a", &a);
            field_print("b", &b);
            field_print("c", &c);
            printf("\n");
        } else if (good) {
            successes ++;
        }
    }
    if (successes < i/3) {
        printf("Iso+serial variation: only %d/%d successful.\n", successes, i);
    }
    
    successes = failures = 0;
    for (i=0; i<nbase/10; i++) {
        struct field_t aa;
        struct tw_extensible_t exu,exv,exw;
        
        mask_t good;
        do {
            field_randomize(&crand, &a);
            good = deserialize_affine(&affine, &a);
            convert_affine_to_extensible(&exta,&affine);
            twist_and_double(&ext,&exta);
        } while (!good);
        do {
            field_randomize(&crand, &aa);
            good = deserialize_affine(&affine, &aa);
            convert_affine_to_extensible(&exta,&affine);
            twist_and_double(&exu,&exta);
        } while (!good);
        field_randomize(&crand, &aa);
        
        q448_randomize(&crand, sk);
		if (i==0 || i==2) memset(&sk, 0, sizeof(sk));
        q448_randomize(&crand, tk);
		if (i==0 || i==1) memset(&tk, 0, sizeof(tk));
        
        copy_tw_extensible(&exv, &ext);
        copy_tw_extensible(&exw, &exu);
        scalarmul(&exv,sk);
        scalarmul(&exw,tk);
        convert_tw_extensible_to_tw_pniels(&pniels, &exw);
        add_tw_pniels_to_tw_extensible(&exv,&pniels);
        untwist_and_double(&exta,&exv);
        serialize_extensible(&b, &exta);

        ignore_result(precompute_fixed_base_wnaf(wnaft,&exu,5));
        linear_combo_var_fixed_vt(&ext,sk,FIELD_BITS,tk,FIELD_BITS,wnaft,5);
        untwist_and_double(&exta,&exv);
        serialize_extensible(&c, &exta);
        
        field_sub(&d,&b,&c);
        field_bias(&d,2);
        
        if (!field_is_zero(&d)){
            printf("PreWNAF combo validation failure %d!\n", ++failures);
            field_print("a", &a);
            field_print("A", &aa);
            q448_print("s", sk);
            q448_print("t", tk);
            field_print("c", &c);
            field_print("b", &b);
            printf("\n\n");
        } else if (good) {
            successes ++;
        }
    }
    if (successes < i) {
        printf("PreWNAF combo variation: only %d/%d successful.\n", successes, i);
    }
    
    return 0;
}