Merge pull request #497 from czurnieden/proven_log_n

Addition of a proof for mp_log and extension of mp_log to bigint bases

Author: sjaeckel

.github/workflows/main.yml

@@ -43,10 +43,14 @@ jobs:
           - { BUILDOPTIONS: '--symbols',                                                                                         SANITIZER: '',  COMPILE_DEBUG: '0', COMPILE_LTO: '0', CONV_WARNINGS: '',        OTHERDEPS: 'libtool-bin' }
           # Run always with valgrind (no sanitizer, but debug info)
           - { BUILDOPTIONS: '--with-cc=gcc --with-m64 --with-valgrind',                                                          SANITIZER: '',  COMPILE_DEBUG: '1', COMPILE_LTO: '0', CONV_WARNINGS: '',        OTHERDEPS: '' }
+          # Alternative big-int version of mp_log(_n)
+          - { BUILDOPTIONS: '--with-cc=gcc --with-m64 --cflags=-DS_MP_WORD_TOO_SMALL_C=""  --with-valgrind',                     SANITIZER: '',  COMPILE_DEBUG: '1', COMPILE_LTO: '0', CONV_WARNINGS: '',        OTHERDEPS: '' }
           # Shared library build
           - { BUILDOPTIONS: '--with-cc=gcc --make-option=-f --make-option=makefile.shared',                                      SANITIZER: '',  COMPILE_DEBUG: '0', COMPILE_LTO: '1', CONV_WARNINGS: '',        OTHERDEPS: 'libtool-bin' }
           # GCC for the 32-bit architecture (no valgrind)
           - { BUILDOPTIONS: '--with-cc=gcc --with-m32',                                                                          SANITIZER: '',  COMPILE_DEBUG: '0', COMPILE_LTO: '0', CONV_WARNINGS: '',        OTHERDEPS: 'libc6-dev-i386 gcc-multilib' }
+          # Alternative big-int version of mp_log(_n) for the 32-bit architecture (no valgrind)
+          - { BUILDOPTIONS: '--with-cc=gcc --with-m32 --cflags=-DS_MP_WORD_TOO_SMALL_C="" ',                                     SANITIZER: '',  COMPILE_DEBUG: '0', COMPILE_LTO: '0', CONV_WARNINGS: '',        OTHERDEPS: 'libc6-dev-i386 gcc-multilib' }
           # clang for the 32-bit architecture (no valgrind)
           - { BUILDOPTIONS: '--with-cc=clang-10 --with-m32',                                                                     SANITIZER: '',  COMPILE_DEBUG: '0', COMPILE_LTO: '0', CONV_WARNINGS: '',        OTHERDEPS: 'clang-10 llvm-10 gcc-multilib' }
           # RSA superclass with tests (no sanitizer, but debug info)
@@ -103,6 +107,10 @@ jobs:
           # but testing all three in one run took to long and timed out.
           - { BUILDOPTIONS: '--with-cc=gcc --cflags=-DMP_16BIT --limit-valgrind',                                                SANITIZER: '1', COMPILE_DEBUG: '0', COMPILE_LTO: '0', CONV_WARNINGS: '',        OTHERDEPS: '' }
           - { BUILDOPTIONS: '--with-cc=gcc --cflags=-DMP_32BIT --limit-valgrind',                                                SANITIZER: '1', COMPILE_DEBUG: '0', COMPILE_LTO: '0', CONV_WARNINGS: '',        OTHERDEPS: '' }
+          # Alternative big-int version of mp_log(_n)
+          - { BUILDOPTIONS: '--with-cc=gcc --cflags=-DMP_16BIT --cflags=-DS_MP_WORD_TOO_SMALL_C="" --limit-valgrind',            SANITIZER: '1', COMPILE_DEBUG: '0', COMPILE_LTO: '0', CONV_WARNINGS: '',        OTHERDEPS: '' }
+          - { BUILDOPTIONS: '--with-cc=gcc --cflags=-DMP_32BIT --cflags=-DS_MP_WORD_TOO_SMALL_C="" --limit-valgrind',            SANITIZER: '1', COMPILE_DEBUG: '0', COMPILE_LTO: '0', CONV_WARNINGS: '',        OTHERDEPS: '' }
+
           # clang for the x86-64 architecture with restricted limb sizes
           - { BUILDOPTIONS: '--with-cc=clang --cflags=-DMP_16BIT --limit-valgrind',                                              SANITIZER: '1', COMPILE_DEBUG: '0', COMPILE_LTO: '0', CONV_WARNINGS: '',        OTHERDEPS: 'clang llvm' }
           - { BUILDOPTIONS: '--with-cc=clang --cflags=-DMP_32BIT --limit-valgrind',                                              SANITIZER: '1', COMPILE_DEBUG: '0', COMPILE_LTO: '0', CONV_WARNINGS: '',        OTHERDEPS: 'clang llvm' }

.gitignore

@@ -85,6 +85,16 @@ etc/tune
 tommath.tex
 libtommath.pc
 
+# ignore files generated by bibtex/biber
+83aea75e0c59d1de027747003151d53dcb2a51c9.bib
+*.bbl
+*.bcf
+*.blg
+*.loa
+*.lol
+*.run.xml
+
+
 # ignore files generated by testme.sh
 gcc_errors_*.txt
 test_*.txt

demo/test.c

@@ -1395,8 +1395,7 @@ static int test_mp_reduce_2k_l(void)
    return EXIT_SUCCESS;
 #   endif /* LTM_DEMO_TEST_REDUCE_2K_L */
 }
-/* stripped down version of mp_radix_size. The faster version can be off by up t
-o +3  */
+/* stripped down version of mp_radix_size. The faster version can be off by up to +3  */
 static mp_err s_rs(const mp_int *a, int radix, int *size)
 {
    mp_err res;
@@ -1425,13 +1424,21 @@ static mp_err s_rs(const mp_int *a, int radix, int *size)
    *size = digs + 1;
    return MP_OKAY;
 }
+
+
 static int test_mp_log_n(void)
 {
    mp_int a;
    mp_digit d;
-   int base, lb, size;
+   int base, lb, size, i;
    const int max_base = MP_MIN(INT_MAX, MP_DIGIT_MAX);
 
+   if (MP_HAS(S_MP_WORD_TOO_SMALL)) {
+      fprintf(stderr, "Testing mp_log_n with restricted size of mp_word.\n");
+   } else {
+      fprintf(stderr, "Testing mp_log_n with normal size of mp_word.\n");
+   }
+
    DOR(mp_init(&a));
 
    /*
@@ -1484,25 +1491,32 @@ static int test_mp_log_n(void)
    DO(mp_rand(&a, 10));
    for (base = 2; base < 65; base++) {
       DO(mp_log_n(&a, base, &lb));
-      DO(s_rs(&a,(int)base, &size));
+      DO(s_rs(&a,base, &size));
       /* radix_size includes the memory needed for '\0', too*/
       size -= 2;
       EXPECT(lb == size);
    }
 
    /*
-     bases 2..64 with "a" a random small constant to
-     test the part of mp_ilogb that uses native types.
+     bases 2..64 with "a" a small constant and a small exponent "n" to test
+     in the range a^n - 10 .. a^n + 10. That will check the correction loops
+     and the test for perfect power.
+     For simplicity a = base and n = 23 (64^23 == 2^138 > 2^128)
    */
-   DO(mp_rand(&a, 1));
    for (base = 2; base < 65; base++) {
-      DO(mp_log_n(&a, base, &lb));
-      DO(s_rs(&a,(int)base, &size));
-      size -= 2;
-      EXPECT(lb == size);
+      mp_set(&a,(mp_digit)base);
+      DO(mp_expt_n(&a, 23, &a));
+      DO(mp_sub_d(&a, 10u, &a));
+      for (i = 0; i < 20; i++) {
+         DO(mp_log_n(&a, base, &lb));
+         DO(s_rs(&a, base, &size));
+         size -= 2;
+         EXPECT(lb == size);
+         DO(mp_add_d(&a, 1u, &a));
+      }
    }
 
-   /*Test upper edgecase with base UINT32_MAX and number (UINT32_MAX/2)*UINT32_MAX^10  */
+   /*Test base upper edgecase with base = UINT32_MAX and number = (UINT32_MAX/2)*UINT32_MAX^10  */
    mp_set(&a, max_base);
    DO(mp_expt_n(&a, 10uL, &a));
    DO(mp_add_d(&a, max_base / 2, &a));
@@ -1516,6 +1530,76 @@ static int test_mp_log_n(void)
    return EXIT_FAILURE;
 }
 
+static int test_mp_log(void)
+{
+   mp_int a, base, bn, t;
+   int lb, lb2, i, j;
+
+   if (MP_HAS(S_MP_WORD_TOO_SMALL)) {
+      fprintf(stdout, "Testing mp_log with restricted size of mp_word.\n");
+   } else {
+      fprintf(stdout, "Testing mp_log with normal size of mp_word.\n");
+   }
+
+   DOR(mp_init_multi(&a, &base, &bn, &t, NULL));
+
+   /*
+      The small values got tested above for mp_log_n already, leaving the big stuff
+      with bases larger than INT_MAX.
+   */
+
+   /* Edgecases a^b and -1+a^b (floor(log_2(256^129)) = 1032) */
+   for (i = 2; i < 256; i++) {
+      mp_set_i32(&a,i);
+      for (j = 2; j < ((i/2)+1); j++) {
+         DO(mp_expt_n(&a, j, &bn));
+         mp_set_i32(&base,j);
+         /* i^j a perfect power */
+         DO(mp_log(&bn, &a, &lb));
+         DO(mp_expt_n(&a, lb, &t));
+         if (mp_cmp(&t, &bn) != MP_EQ) {
+            fprintf(stderr,"FAILURE mp_log for perf. power at i = %d, j = %d\n", i, j);
+            goto LBL_ERR;
+         }
+         /* -1 + i^j */
+         DO(mp_decr(&bn));
+         DO(mp_log(&bn, &a, &lb2));
+         if (lb != (lb2+1)) {
+            fprintf(stderr,"FAILURE mp_log for -1 + i^j at i = %d, j = %d\n", i, j);
+            goto LBL_ERR;
+         }
+      }
+   }
+
+   /* Random a, base */
+   for (i = 1; i < 256; i++) {
+      DO(mp_rand(&a, i));
+      for (j = 1; j < ((i/2)+1); j++) {
+         DO(mp_rand(&base, j));
+         DO(mp_log(&a, &base, &lb));
+         DO(mp_expt_n(&base, lb, &bn));
+         /* "bn" must be smaller than or equal to "a" at this point. */
+         if (mp_cmp(&bn, &a) == MP_GT) {
+            fprintf(stderr,"FAILURE mp_log random in GT check");
+            goto LBL_ERR;
+         }
+         DO(mp_mul(&bn, &base, &bn));
+         /* "bn" must be bigger than "a" at this point. */
+         if (mp_cmp(&bn, &a) != MP_GT) {
+            fprintf(stderr,"FAILURE mp_log random in NOT GT check");
+            goto LBL_ERR;
+         }
+      }
+   }
+
+   mp_clear_multi(&a, &base, &bn, &t, NULL);
+   return EXIT_SUCCESS;
+LBL_ERR:
+   mp_clear_multi(&a, &base, &bn, &t, NULL);
+   return EXIT_FAILURE;
+}
+
+
 static int test_mp_incr(void)
 {
    mp_int a, b;
@@ -2373,6 +2457,7 @@ static int unit_tests(int argc, char **argv)
       T1(mp_get_u64, MP_GET_I64),
       T1(mp_get_ul, MP_GET_L),
       T1(mp_log_n, MP_LOG_N),
+      T1(mp_log, MP_LOG),
       T1(mp_incr, MP_ADD_D),
       T1(mp_invmod, MP_INVMOD),
       T1(mp_is_square, MP_IS_SQUARE),

doc/bn.tex

@@ -1946,18 +1946,24 @@ \section{Root Finding}
 \begin{alltt}
 mp_err mp_sqrt(const mp_int *arg, mp_int *ret)
 \end{alltt}
-
 \chapter{Logarithm}
 \section{Integer Logarithm}
 A logarithm function for positive integer input \texttt{a, base} computing  $\floor{\log_bx}$ such
-that $(\log_b x)^b \le x$.
+that $(\log_b x)^b \le x$. The function \texttt{mp\_log\_n} is just a wrapper that converts \texttt{base}
+to a bigint and calls \texttt{mp\_log}.
+
+\index{mp\_log}
+\begin{alltt}
+mp_err mp_log(const mp_int *a, const mp_int *base, int *c)
+\end{alltt}
 
 \index{mp\_log\_n}
 \begin{alltt}
 mp_err mp_log_n(const mp_int *a, int base, int *c)
 \end{alltt}
 
 \subsection{Example}
+Example given for \texttt{mp\_log\_n} only because the single difference is the type of \texttt{base}.
 \begin{small}
   \begin{alltt}
 #include <stdlib.h>
@@ -1968,15 +1974,15 @@ \subsection{Example}
 
 int main(int argc, char **argv)
 {
-   mp_int x, output;
-   int base;
+   mp_int x;
+   int base, output;
    mp_err e;
 
    if (argc != 3) {
       fprintf(stderr,"Usage %s base x\textbackslash{}n", argv[0]);
       exit(EXIT_FAILURE);
    }
-   if ((e = mp_init_multi(&x, &output, NULL)) != MP_OKAY) {
+   if ((e = mp_init(&x)) != MP_OKAY) {
       fprintf(stderr,"mp_init failed: \textbackslash{}"%s\textbackslash{}"\textbackslash{}n",
                      mp_error_to_string(e));
               exit(EXIT_FAILURE);
@@ -1998,20 +2004,18 @@ \subsection{Example}
                       mp_error_to_string(e));
       exit(EXIT_FAILURE);
    }
+   printf("%d\n",output);
 
-   if ((e = mp_fwrite(&output, 10, stdout)) != MP_OKAY) {
-      fprintf(stderr,"mp_fwrite failed: \textbackslash{}"%s\textbackslash{}"\textbackslash{}n",
-                      mp_error_to_string(e));
-      exit(EXIT_FAILURE);
-   }
-   putchar('\textbackslash{}n');
-
-   mp_clear_multi(&x, &output, NULL);
+   mp_clear(&x);
    exit(EXIT_SUCCESS);
 }
 \end{alltt}
 \end{small}
 
+
+
+
+
 \chapter{Prime Numbers}
 
 \section{Fermat Test}

libtommath_VS2008.vcproj

@@ -544,6 +544,10 @@
 			RelativePath="mp_lcm.c"
 			>
 		</File>
+		<File
+			RelativePath="mp_log.c"
+			>
+		</File>
 		<File
 			RelativePath="mp_log_n.c"
 			>
@@ -829,27 +833,27 @@
 			>
 		</File>
 		<File
-			RelativePath="s_mp_get_bit.c"
+			RelativePath="s_mp_fp_log.c"
 			>
 		</File>
 		<File
-			RelativePath="s_mp_invmod.c"
+			RelativePath="s_mp_fp_log_d.c"
 			>
 		</File>
 		<File
-			RelativePath="s_mp_invmod_odd.c"
+			RelativePath="s_mp_get_bit.c"
 			>
 		</File>
 		<File
-			RelativePath="s_mp_log.c"
+			RelativePath="s_mp_invmod.c"
 			>
 		</File>
 		<File
-			RelativePath="s_mp_log_2expt.c"
+			RelativePath="s_mp_invmod_odd.c"
 			>
 		</File>
 		<File
-			RelativePath="s_mp_log_d.c"
+			RelativePath="s_mp_log_2expt.c"
 			>
 		</File>
 		<File

makefile

@@ -33,7 +33,7 @@ mp_error_to_string.o mp_exch.o mp_expt_n.o mp_exptmod.o mp_exteuclid.o mp_fread.
 mp_from_ubin.o mp_fwrite.o mp_gcd.o mp_get_double.o mp_get_i32.o mp_get_i64.o mp_get_l.o mp_get_mag_u32.o \
 mp_get_mag_u64.o mp_get_mag_ul.o mp_grow.o mp_hash.o mp_init.o mp_init_copy.o mp_init_i32.o mp_init_i64.o \
 mp_init_l.o mp_init_multi.o mp_init_set.o mp_init_size.o mp_init_u32.o mp_init_u64.o mp_init_ul.o \
-mp_invmod.o mp_is_square.o mp_kronecker.o mp_lcm.o mp_log_n.o mp_lshd.o mp_mod.o mp_mod_2d.o \
+mp_invmod.o mp_is_square.o mp_kronecker.o mp_lcm.o mp_log.o mp_log_n.o mp_lshd.o mp_mod.o mp_mod_2d.o \
 mp_montgomery_calc_normalization.o mp_montgomery_reduce.o mp_montgomery_setup.o mp_mul.o mp_mul_2.o \
 mp_mul_2d.o mp_mul_d.o mp_mulmod.o mp_neg.o mp_or.o mp_pack.o mp_pack_count.o mp_prime_fermat.o \
 mp_prime_frobenius_underwood.o mp_prime_is_prime.o mp_prime_miller_rabin.o mp_prime_next_prime.o \
@@ -44,8 +44,8 @@ mp_reduce_setup.o mp_root_n.o mp_rshd.o mp_sbin_size.o mp_set.o mp_set_double.o
 mp_set_l.o mp_set_u32.o mp_set_u64.o mp_set_ul.o mp_shrink.o mp_signed_rsh.o mp_sqrmod.o mp_sqrt.o \
 mp_sqrtmod_prime.o mp_sub.o mp_sub_d.o mp_submod.o mp_to_radix.o mp_to_sbin.o mp_to_ubin.o mp_ubin_size.o \
 mp_unpack.o mp_xor.o mp_zero.o s_mp_add.o s_mp_copy_digs.o s_mp_div_3.o s_mp_div_recursive.o \
-s_mp_div_school.o s_mp_div_small.o s_mp_exptmod.o s_mp_exptmod_fast.o s_mp_get_bit.o s_mp_invmod.o \
-s_mp_invmod_odd.o s_mp_log.o s_mp_log_2expt.o s_mp_log_d.o s_mp_montgomery_reduce_comba.o s_mp_mul.o \
+s_mp_div_school.o s_mp_div_small.o s_mp_exptmod.o s_mp_exptmod_fast.o s_mp_fp_log.o s_mp_fp_log_d.o \
+s_mp_get_bit.o s_mp_invmod.o s_mp_invmod_odd.o s_mp_log_2expt.o s_mp_montgomery_reduce_comba.o s_mp_mul.o \
 s_mp_mul_balance.o s_mp_mul_comba.o s_mp_mul_high.o s_mp_mul_high_comba.o s_mp_mul_karatsuba.o \
 s_mp_mul_toom.o s_mp_prime_is_divisible.o s_mp_prime_tab.o s_mp_radix_map.o \
 s_mp_radix_size_overestimate.o s_mp_rand_platform.o s_mp_sqr.o s_mp_sqr_comba.o s_mp_sqr_karatsuba.o \

makefile.mingw

@@ -35,7 +35,7 @@ mp_error_to_string.o mp_exch.o mp_expt_n.o mp_exptmod.o mp_exteuclid.o mp_fread.
 mp_from_ubin.o mp_fwrite.o mp_gcd.o mp_get_double.o mp_get_i32.o mp_get_i64.o mp_get_l.o mp_get_mag_u32.o \
 mp_get_mag_u64.o mp_get_mag_ul.o mp_grow.o mp_hash.o mp_init.o mp_init_copy.o mp_init_i32.o mp_init_i64.o \
 mp_init_l.o mp_init_multi.o mp_init_set.o mp_init_size.o mp_init_u32.o mp_init_u64.o mp_init_ul.o \
-mp_invmod.o mp_is_square.o mp_kronecker.o mp_lcm.o mp_log_n.o mp_lshd.o mp_mod.o mp_mod_2d.o \
+mp_invmod.o mp_is_square.o mp_kronecker.o mp_lcm.o mp_log.o mp_log_n.o mp_lshd.o mp_mod.o mp_mod_2d.o \
 mp_montgomery_calc_normalization.o mp_montgomery_reduce.o mp_montgomery_setup.o mp_mul.o mp_mul_2.o \
 mp_mul_2d.o mp_mul_d.o mp_mulmod.o mp_neg.o mp_or.o mp_pack.o mp_pack_count.o mp_prime_fermat.o \
 mp_prime_frobenius_underwood.o mp_prime_is_prime.o mp_prime_miller_rabin.o mp_prime_next_prime.o \
@@ -46,8 +46,8 @@ mp_reduce_setup.o mp_root_n.o mp_rshd.o mp_sbin_size.o mp_set.o mp_set_double.o
 mp_set_l.o mp_set_u32.o mp_set_u64.o mp_set_ul.o mp_shrink.o mp_signed_rsh.o mp_sqrmod.o mp_sqrt.o \
 mp_sqrtmod_prime.o mp_sub.o mp_sub_d.o mp_submod.o mp_to_radix.o mp_to_sbin.o mp_to_ubin.o mp_ubin_size.o \
 mp_unpack.o mp_xor.o mp_zero.o s_mp_add.o s_mp_copy_digs.o s_mp_div_3.o s_mp_div_recursive.o \
-s_mp_div_school.o s_mp_div_small.o s_mp_exptmod.o s_mp_exptmod_fast.o s_mp_get_bit.o s_mp_invmod.o \
-s_mp_invmod_odd.o s_mp_log.o s_mp_log_2expt.o s_mp_log_d.o s_mp_montgomery_reduce_comba.o s_mp_mul.o \
+s_mp_div_school.o s_mp_div_small.o s_mp_exptmod.o s_mp_exptmod_fast.o s_mp_fp_log.o s_mp_fp_log_d.o \
+s_mp_get_bit.o s_mp_invmod.o s_mp_invmod_odd.o s_mp_log_2expt.o s_mp_montgomery_reduce_comba.o s_mp_mul.o \
 s_mp_mul_balance.o s_mp_mul_comba.o s_mp_mul_high.o s_mp_mul_high_comba.o s_mp_mul_karatsuba.o \
 s_mp_mul_toom.o s_mp_prime_is_divisible.o s_mp_prime_tab.o s_mp_radix_map.o \
 s_mp_radix_size_overestimate.o s_mp_rand_platform.o s_mp_sqr.o s_mp_sqr_comba.o s_mp_sqr_karatsuba.o \

makefile.msvc

@@ -31,7 +31,7 @@ mp_error_to_string.obj mp_exch.obj mp_expt_n.obj mp_exptmod.obj mp_exteuclid.obj
 mp_from_ubin.obj mp_fwrite.obj mp_gcd.obj mp_get_double.obj mp_get_i32.obj mp_get_i64.obj mp_get_l.obj mp_get_mag_u32.obj \
 mp_get_mag_u64.obj mp_get_mag_ul.obj mp_grow.obj mp_hash.obj mp_init.obj mp_init_copy.obj mp_init_i32.obj mp_init_i64.obj \
 mp_init_l.obj mp_init_multi.obj mp_init_set.obj mp_init_size.obj mp_init_u32.obj mp_init_u64.obj mp_init_ul.obj \
-mp_invmod.obj mp_is_square.obj mp_kronecker.obj mp_lcm.obj mp_log_n.obj mp_lshd.obj mp_mod.obj mp_mod_2d.obj \
+mp_invmod.obj mp_is_square.obj mp_kronecker.obj mp_lcm.obj mp_log.obj mp_log_n.obj mp_lshd.obj mp_mod.obj mp_mod_2d.obj \
 mp_montgomery_calc_normalization.obj mp_montgomery_reduce.obj mp_montgomery_setup.obj mp_mul.obj mp_mul_2.obj \
 mp_mul_2d.obj mp_mul_d.obj mp_mulmod.obj mp_neg.obj mp_or.obj mp_pack.obj mp_pack_count.obj mp_prime_fermat.obj \
 mp_prime_frobenius_underwood.obj mp_prime_is_prime.obj mp_prime_miller_rabin.obj mp_prime_next_prime.obj \
@@ -42,8 +42,8 @@ mp_reduce_setup.obj mp_root_n.obj mp_rshd.obj mp_sbin_size.obj mp_set.obj mp_set
 mp_set_l.obj mp_set_u32.obj mp_set_u64.obj mp_set_ul.obj mp_shrink.obj mp_signed_rsh.obj mp_sqrmod.obj mp_sqrt.obj \
 mp_sqrtmod_prime.obj mp_sub.obj mp_sub_d.obj mp_submod.obj mp_to_radix.obj mp_to_sbin.obj mp_to_ubin.obj mp_ubin_size.obj \
 mp_unpack.obj mp_xor.obj mp_zero.obj s_mp_add.obj s_mp_copy_digs.obj s_mp_div_3.obj s_mp_div_recursive.obj \
-s_mp_div_school.obj s_mp_div_small.obj s_mp_exptmod.obj s_mp_exptmod_fast.obj s_mp_get_bit.obj s_mp_invmod.obj \
-s_mp_invmod_odd.obj s_mp_log.obj s_mp_log_2expt.obj s_mp_log_d.obj s_mp_montgomery_reduce_comba.obj s_mp_mul.obj \
+s_mp_div_school.obj s_mp_div_small.obj s_mp_exptmod.obj s_mp_exptmod_fast.obj s_mp_fp_log.obj s_mp_fp_log_d.obj \
+s_mp_get_bit.obj s_mp_invmod.obj s_mp_invmod_odd.obj s_mp_log_2expt.obj s_mp_montgomery_reduce_comba.obj s_mp_mul.obj \
 s_mp_mul_balance.obj s_mp_mul_comba.obj s_mp_mul_high.obj s_mp_mul_high_comba.obj s_mp_mul_karatsuba.obj \
 s_mp_mul_toom.obj s_mp_prime_is_divisible.obj s_mp_prime_tab.obj s_mp_radix_map.obj \
 s_mp_radix_size_overestimate.obj s_mp_rand_platform.obj s_mp_sqr.obj s_mp_sqr_comba.obj s_mp_sqr_karatsuba.obj \