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Added Fips 186.4 compliance, an additional strong Lucas-Selfridge (for BPSW) and a Frobenius (Paul UNderwood) test, both optional. With documentation.

This commit is contained in:
czurnieden 2018-05-03 23:45:02 +02:00 committed by Steffen Jaeckel
parent f17d90b96d
commit a218ddce9b
3 changed files with 345 additions and 34 deletions
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260
bn_mp_prime_is_prime.c
View File

@ -13,26 +13,59 @@
* guarantee it works.
*/
/* performs a variable number of rounds of Miller-Rabin
*
* Probability of error after t rounds is no more than
// portable integer log of two with small footprint
static unsigned int floor_ilog2(int value)
{
unsigned int r = 0;
while ((value >>= 1) != 0) {
r++;
}
return r;
}
*
* Sets result to 1 if probably prime, 0 otherwise
*/
int mp_prime_is_prime(const mp_int *a, int t, int *result)
{
mp_int b;
int ix, err, res;
int ix, err, res, p_max = 0, size_a, len;
unsigned int fips_rand, mask;
/* default to no */
*result = MP_NO;
/* valid value of t? */
if ((t <= 0) || (t > PRIME_SIZE)) {
if (t > PRIME_SIZE) {
puts("t > PRIME_SIZE");
return MP_VAL;
}
/* Some shortcuts */
/* N > 3 */
if (a->used == 1) {
if (a->dp[0] == 0 || a->dp[0] == 1) {
*result = 0;
return MP_OKAY;
}
if (a->dp[0] == 2) {
*result = 1;
return MP_OKAY;
}
}
/* N must be odd */
if (mp_iseven(a) == MP_YES) {
*result = 0;
return MP_OKAY;
}
/* N is not a perfect square: floor(sqrt(N))^2 != N */
if ((err = mp_is_square(a, &res)) != MP_OKAY) {
return err;
}
if (res != 0) {
*result = 0;
return MP_OKAY;
}
/* is the input equal to one of the primes in the table? */
for (ix = 0; ix < PRIME_SIZE; ix++) {
if (mp_cmp_d(a, ltm_prime_tab[ix]) == MP_EQ) {
@ -51,22 +84,218 @@ int mp_prime_is_prime(const mp_int *a, int t, int *result)
return MP_OKAY;
}
/* now perform the miller-rabin rounds */
if ((err = mp_init(&b)) != MP_OKAY) {
/*
Run the Miller-Rabin test with base 2 for the BPSW test.
*/
if ((err = mp_init_set(&b,2)) != MP_OKAY) {
return err;
}
for (ix = 0; ix < t; ix++) {
/* set the prime */
mp_set(&b, ltm_prime_tab[ix]);
if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) {
goto LBL_B;
}
if (res == MP_NO) {
goto LBL_B;
}
/*
Rumours have it that Mathematica does a second M-R test with base 3.
Other rumours have it that their strong L-S test is slightly different.
It does not hurt, though, beside a bit of extra runtime.
*/
b.dp[0]++;
if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) {
goto LBL_B;
}
if (res == MP_NO) {
goto LBL_B;
}
if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) {
// commented out for testing purposes
//#ifdef LTM_USE_STRONG_LUCAS_SELFRIDGE_TEST
if ((err = mp_prime_strong_lucas_selfridge(a, &res)) != MP_OKAY) {
goto LBL_B;
}
if (res == MP_NO) {
goto LBL_B;
}
//#endif
//#ifdef LTM_USE_FROBENIUS_UNDERWOOD_TEST
if ((err = mp_prime_frobenius_underwood(a, &res)) != MP_OKAY) {
goto LBL_B;
}
if (res == MP_NO) {
goto LBL_B;
}
//#endif
/*
abs(t) extra rounds of M-R to extend the range of primes it can find if t < 0.
Only recommended if the input range is known to be < 3317044064679887385961981
It uses the bases for a deterministic M-R test if input < 3317044064679887385961981
The caller has to check the size.
Not for cryptographic use because with known bases strong M-R pseudoprimes can
be constructed. Use at least one MM-R test with a random base (t >= 1).
The 1119 bit large number
80383745745363949125707961434194210813883768828755814583748891752229742737653\
33652186502336163960045457915042023603208766569966760987284043965408232928738\
79185086916685732826776177102938969773947016708230428687109997439976544144845\
34115587245063340927902227529622941498423068816854043264575340183297861112989\
60644845216191652872597534901
has been constructed by F. Arnault (F. Arnault, "Rabin-Miller primality test:
composite numbers which pass it.", Mathematics of Computation, 1995, 64. Jg.,
Nr. 209, S. 355-361), is a semiprime with the two factors
40095821663949960541830645208454685300518816604113250877450620473800321707011\
96242716223191597219733582163165085358166969145233813917169287527980445796800\
452592031836601
20047910831974980270915322604227342650259408302056625438725310236900160853505\
98121358111595798609866791081582542679083484572616906958584643763990222898400\
226296015918301
and it is a strong pseudoprime to all forty-six prime M-R bases up to 200
It does not fail the strong Bailley-PSP test as implemented here, it is just
given as an example, if not the reason to use the BPSW-test instead of M-R-tests
with a sequence of primes 2...n.
*/
if (t < 0) {
t = -t;
/*
Sorenson, Jonathan; Webster, Jonathan (2015).
"Strong Pseudoprimes to Twelve Prime Bases".
*/
/* 318665857834031151167461 */
if ((err = mp_read_radix(&b, "437ae92817f9fc85b7e5", 16)) != MP_OKAY) {
goto LBL_B;
}
if (res == MP_NO) {
if (mp_cmp(a,&b) == MP_LT) {
p_max = 12;
}
/* 3317044064679887385961981 */
if ((err = mp_read_radix(&b, "2be6951adc5b22410a5fd", 16)) != MP_OKAY) {
goto LBL_B;
}
if (mp_cmp(a,&b) == MP_LT) {
p_max = 13;
}
// for compatibility with the current API (well, compatible within a sign's width)
if (p_max < t) {
p_max = t;
}
if(p_max > PRIME_SIZE) {
err = MP_VAL;
goto LBL_B;
}
/* we did bases 2 and 3 already, skip them */
for (ix = 2; ix < p_max; ix++) {
mp_set(&b,ltm_prime_tab[ix]);
if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) {
goto LBL_B;
}
if (res == MP_NO) {
goto LBL_B;
}
}
}
/*
Do "t" M-R tests with random bases between 3 and "a".
See Fips 186.4 p. 126ff
*/
else if (t > 0) {
// The mp_digit's have a defined bit-size but the size of the
// array a.dp is a simple 'int' and this library can not assume full
// compliance to the current C-standard (ISO/IEC 9899:2011) because
// it gets used for small embeded processors, too. Some of those MCUs
// have compilers that one cannot call standard compliant by any means.
// Hence the ugly type-fiddling in the following code.
size_a = mp_count_bits(a);
mask = (1u << floor_ilog2(size_a)) - 1u;
/*
Assuming the General Rieman hypothesis (never thought to write that in a
comment) the upper bound can be lowered to 2*(log a)^2.
E. Bach, “Explicit bounds for primality testing and related problems,”
Math. Comp. 55 (1990), 355–380.
size_a = (size_a/10) * 7;
len = 2 * (size_a * size_a);
E.g.: a number of size 2^2048 would be reduced to the upper limit
floor(2048/10)*7 = 1428
2 * 1428^2 = 4078368
(would have been ~4030331.9962 with floats and natural log instead)
That number is smaller than 2^28, the default bit-size of mp_digit.
*/
/*
How many tests, you might ask? Dana Jacobsen of Math::Prime::Util fame
does exactly 1. In words: one. Look at the end of _GMP_is_prime() in
Math-Prime-Util-GMP-0.50/primality.c if you do not believe it.
The function mp_rand() goes to some length to use a cryptographically
good PRNG. That also means that the chance to always get the same base
in the loop is non-zero, although very low.
If the BPSW test and/or the addtional Frobenious test have been
performed instead of just the Miller-Rabin test with the bases 2 and 3,
a single extra test should suffice, so such a very unlikely event
will not do much harm.
To preemptivly answer the dangling question: no, a witness does not
need to be prime.
*/
for (ix = 0; ix < t; ix++) {
// mp_rand() guarantees the first digit to be non-zero
if ((err = mp_rand(&b, 1)) != MP_OKAY) {
goto LBL_B;
}
// Reduce digit before casting because mp_digit might be bigger than
// an unsigned int and "mask" on the other side is most probably not.
fips_rand = (unsigned int) (b.dp[0] & (mp_digit) mask);
#ifdef MP_8BIT
// One 8-bit digit is too small, so concatenate two if the size of
// unsigned int allows for it.
if( (sizeof(unsigned int) * CHAR_BIT)/2 >= (sizeof(mp_digit) * CHAR_BIT) ) {
if ((err = mp_rand(&b, 1)) != MP_OKAY) {
goto LBL_B;
}
fips_rand <<= sizeof(mp_digit) * CHAR_BIT;
fips_rand |= (unsigned int) b.dp[0];
}
#endif
len = (int) ((fips_rand & mask)/ DIGIT_BIT);
// Unlikely, but still possible.
if(len < 0){
ix--;
continue;
}
if ((err = mp_rand(&b, len)) != MP_OKAY) {
goto LBL_B;
}
// Although the chance for b <= 3 is miniscule, try again.
if(mp_cmp_d(&b,3) != MP_GT) {
ix--;
continue;
}
if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) {
goto LBL_B;
}
if (res == MP_NO) {
goto LBL_B;
}
}
}
/* passed the test */
@ -75,6 +304,7 @@ LBL_B:
mp_clear(&b);
return err;
}
#endif
/* ref: $Format:%D$ */

86
doc/bn.tex
View File

@ -152,7 +152,7 @@ myprng | mtest/mtest | test
This will output a row of numbers that are increasing. Each column is a different test (such as addition, multiplication, etc)
that is being performed. The numbers represent how many times the test was invoked. If an error is detected the program
will exit with a dump of the relevent numbers it was working with.
will exit with a dump of the relevant numbers it was working with.
\section{Build Configuration}
LibTomMath can configured at build time in three phases we shall call ``depends'', ``tweaks'' and ``trims''.
@ -291,7 +291,7 @@ exponentiations. It depends largely on the processor, compiler and the moduli b
Essentially the only time you wouldn't use LibTomMath is when blazing speed is the primary concern. However,
on the other side of the coin LibTomMath offers you a totally free (public domain) well structured math library
that is very flexible, complete and performs well in resource contrained environments. Fast RSA for example can
that is very flexible, complete and performs well in resource constrained environments. Fast RSA for example can
be performed with as little as 8KB of ram for data (again depending on build options).
\chapter{Getting Started with LibTomMath}
@ -693,7 +693,7 @@ int mp_count_bits(const mp_int *a);
\section{Small Constants}
Setting mp\_ints to small constants is a relatively common operation. To accomodate these instances there are two
Setting mp\_ints to small constants is a relatively common operation. To accommodate these instances there are two
small constant assignment functions. The first function is used to set a single digit constant while the second sets
an ISO C style ``unsigned long'' constant. The reason for both functions is efficiency. Setting a single digit is quick but the
domain of a digit can change (it's always at least $0 \ldots 127$).
@ -797,7 +797,7 @@ number == 654321
int mp_set_long (mp_int * a, unsigned long b);
\end{alltt}
This will assign the value of the platform-dependant sized variable $b$ to the mp\_int $a$.
This will assign the value of the platform-dependent sized variable $b$ to the mp\_int $a$.
To get the ``unsigned long'' copy of an mp\_int the following function can be used.
@ -1222,6 +1222,15 @@ int mp_tc_xor (mp_int * a, mp_int * b, mp_int * c);
The compute $c = a \odot b$ as above if both $a$ and $b$ are positive, negative values are converted into their two-complement representation first. This can be used to implement arbitrary-precision two-complement integers together with the arithmetic right-shift at page \ref{arithrightshift}.
\subsection{Bit Picking}
\index{mp\_get\_bit}
\begin{alltt}
int mp_get_bit(mp_int *a, int b)
\end{alltt}
Pick a bit: returns \texttt{MP\_YES} if the bit at position $b$ (0-index) is set, that is if it is 1 (one), \texttt{MP\_NO}
if the bit is 0 (zero) and \texttt{MP\_VAL} if $b < 0$.
\section{Addition and Subtraction}
To compute an addition or subtraction the following two functions can be used.
@ -1613,9 +1622,9 @@ a single final reduction to correct for the normalization and the fast reduction
For more details consider examining the file \textit{bn\_mp\_exptmod\_fast.c}.
\section{Restricted Dimminished Radix}
\section{Restricted Diminished Radix}
``Dimminished Radix'' reduction refers to reduction with respect to moduli that are ameniable to simple
``Diminished Radix'' reduction refers to reduction with respect to moduli that are amenable to simple
digit shifting and small multiplications. In this case the ``restricted'' variant refers to moduli of the
form $\beta^k - p$ for some $k \ge 0$ and $0 < p < \beta$ where $\beta$ is the radix (default to $2^{28}$).
@ -1636,8 +1645,8 @@ int mp_dr_reduce(mp_int *a, mp_int *b, mp_digit mp);
\end{alltt}
This reduces $a$ in place modulo $b$ with the pre--computed value $mp$. $b$ must be of a restricted
dimminished radix form and $a$ must be in the range $0 \le a < b^2$. Dimminished radix reductions are
much faster than both Barrett and Montgomery reductions as they have a much lower asymtotic running time.
diminished radix form and $a$ must be in the range $0 \le a < b^2$. Diminished radix reductions are
much faster than both Barrett and Montgomery reductions as they have a much lower asymptotic running time.
Since the moduli are restricted this algorithm is not particularly useful for something like Rabin, RSA or
BBS cryptographic purposes. This reduction algorithm is useful for Diffie-Hellman and ECC where fixed
@ -1646,7 +1655,7 @@ primes are acceptable.
Note that unlike Montgomery reduction there is no normalization process. The result of this function is
equal to the correct residue.
\section{Unrestricted Dimminshed Radix}
\section{Unrestricted Diminished Radix}
Unrestricted reductions work much like the restricted counterparts except in this case the moduli is of the
form $2^k - p$ for $0 < p < \beta$. In this sense the unrestricted reductions are more flexible as they
@ -1731,8 +1740,8 @@ $X$ the operation is performed as $Y \equiv (G^{-1} \mbox{ mod }P)^{\vert X \ver
$gcd(G, P) = 1$.
This function is actually a shell around the two internal exponentiation functions. This routine will automatically
detect when Barrett, Montgomery, Restricted and Unrestricted Dimminished Radix based exponentiation can be used. Generally
moduli of the a ``restricted dimminished radix'' form lead to the fastest modular exponentiations. Followed by Montgomery
detect when Barrett, Montgomery, Restricted and Unrestricted Diminished Radix based exponentiation can be used. Generally
moduli of the a ``restricted diminished radix'' form lead to the fastest modular exponentiations. Followed by Montgomery
and the other two algorithms.
\section{Modulus a Power of Two}
@ -1815,6 +1824,22 @@ require ten tests whereas a 1024-bit number would only require four tests.
You should always still perform a trial division before a Miller-Rabin test though.
\section{Strong Lucas-Selfridge Test}
\index{mp\_prime\_strong\_lucas\_selfridge}
\begin{alltt}
int mp_prime_strong_lucas_selfridge(const mp_int *a, int *result)
\end{alltt}
Performs a strong Lucas-Selfridge test. The strong Lucas-Selfridge test together with the Rabin-Miler test with bases $2$ and $3$ resemble the BPSW test. The single internal use is as a compile-time option in \texttt{mp\_prime\_is\_prime} and can be excluded
from the Libtommath build if not needed.
\section{Frobenius (Underwood) Test}
\index{mp\_prime\_frobenius\_underwood}
\begin{alltt}
int mp_prime_frobenius_underwood(const mp_int *N, int *result)
\end{alltt}
Performs the variant of the Frobenius test as described by Paul Underwood. The single internal use is as a compile-time option in
\texttt{mp\_prime\_is\_prime} and can be excluded from the Libtommath build if not needed.
\section{Primality Testing}
Testing if a number is a square can be done a bit faster than just by calculating the square root. It is used by the primality testing function described below.
\index{mp\_is\_square}
@ -1827,16 +1852,28 @@ int mp_is_square(const mp_int *arg, int *ret);
\begin{alltt}
int mp_prime_is_prime (mp_int * a, int t, int *result)
\end{alltt}
This will perform a trial division followed by $t$ rounds of Miller-Rabin tests on $a$ and store the result in $result$.
If $a$ passes all of the tests $result$ is set to one, otherwise it is set to zero. Note that $t$ is bounded by
$1 \le t < PRIME\_SIZE$ where $PRIME\_SIZE$ is the number of primes in the prime number table (by default this is $256$).
This will perform a trial division followed by two rounds of Miller-Rabin with bases 2 and 3. It is possible, although only at
the compile time of this library for now, to include a strong Lucas-Selfridge test and/or a Frobenius test. See file
\texttt{bn\_mp\_prime\_is\_prime.c} for the necessary details. It shall be noted that both functions are much slower than
the Miller-Rabin test.
If $t$ is set to a positive value $t$ additional rounds of the Miller-Rabin test with random bases will be performed to allow for Fips 186.4 (vid.~p.~126ff) compliance. The function \texttt{mp\_prime\_rabin\_miller\_trials} can be used to determine the number of rounds. It is vital that the function \texttt{mp\_rand()} has a cryptographically strong random number generator available.
If $t$ is set to a negative value the test will run the deterministic Miller-Rabin test for the primes up to
$3317044064679887385961981$. That limit has to be checked by the caller. If $-t > 13$ than $-t - 13$ additional rounds of the
Miller-Rabin test will be performed but note that $-t$ is bounded by $1 \le -t < PRIME\_SIZE$ where $PRIME\_SIZE$ is the number
of primes in the prime number table (by default this is $256$) and the first 13 primes have already been used. It will return
\texttt{MP\_VAL} in case of$-t > PRIME\_SIZE$.
If $a$ passes all of the tests $result$ is set to one, otherwise it is set to zero.
\section{Next Prime}
\index{mp\_prime\_next\_prime}
\begin{alltt}
int mp_prime_next_prime(mp_int *a, int t, int bbs_style)
\end{alltt}
This finds the next prime after $a$ that passes mp\_prime\_is\_prime() with $t$ tests. Set $bbs\_style$ to one if you
This finds the next prime after $a$ that passes mp\_prime\_is\_prime() with $t$ tests but see the documentation for
mp\_prime\_is\_prime for details regarding the use of the argument $t$. Set $bbs\_style$ to one if you
want only the next prime congruent to $3 \mbox{ mod } 4$, otherwise set it to zero to find any next prime.
\section{Random Primes}
@ -1846,7 +1883,8 @@ int mp_prime_random(mp_int *a, int t, int size, int bbs,
ltm_prime_callback cb, void *dat)
\end{alltt}
This will find a prime greater than $256^{size}$ which can be ``bbs\_style'' or not depending on $bbs$ and must pass
$t$ rounds of tests. The ``ltm\_prime\_callback'' is a typedef for
$t$ rounds of tests but see the documentation for mp\_prime\_is\_prime for details regarding the use of the argument $t$.
The ``ltm\_prime\_callback'' is a typedef for
\begin{alltt}
typedef int ltm_prime_callback(unsigned char *dst, int len, void *dat);
@ -2016,7 +2054,7 @@ This finds the triple U1/U2/U3 using the Extended Euclidean algorithm such that
a \cdot U1 + b \cdot U2 = U3
\end{equation}
Any of the U1/U2/U3 paramters can be set to \textbf{NULL} if they are not desired.
Any of the U1/U2/U3 parameters can be set to \textbf{NULL} if they are not desired.
\section{Greatest Common Divisor}
\index{mp\_gcd}
@ -2042,6 +2080,14 @@ symbol. The result is stored in $c$ and can take on one of three values $\lbrac
then the result will be $-1$ when $a$ is not a quadratic residue modulo $p$. The result will be $0$ if $a$ divides $p$
and the result will be $1$ if $a$ is a quadratic residue modulo $p$.
\section{Kronecker Symbol}
\index{mp\_kronecker}
\begin{alltt}
int mp_kronecker (mp_int * a, mp_int * p, int *c)
\end{alltt}
Extension of the Jacoby symbol to all $\lbrace a, p \rbrace \in \mathbb{Z}$ .
\section{Modular square root}
\index{mp\_sqrtmod\_prime}
\begin{alltt}
@ -2087,6 +2133,12 @@ These work like the full mp\_int capable variants except the second parameter $b
functions fairly handy if you have to work with relatively small numbers since you will not have to allocate
an entire mp\_int to store a number like $1$ or $2$.
\index{mp\_mul\_si}
\begin{alltt}
int mp_mul_si(mp_int *a, long b, mp_int *c);
\end{alltt}
Just like the functions above but with the ability to use a signed input as the small number.
The division by three can be made faster by replacing the division with a multiplication by the multiplicative inverse of three.
\index{mp\_div\_3}

33
tommath.h
View File

@ -298,6 +298,11 @@ int mp_or(const mp_int *a, const mp_int *b, mp_int *c);
/* c = a AND b */
int mp_and(const mp_int *a, const mp_int *b, mp_int *c);
/* Checks the bit at position b and returns MP_YES
if the bit is 1, MP_NO if it is 0 and MP_VAL
in case of error */
int mp_get_bit(const mp_int *a, int b);
/* c = a XOR b (two complement) */
int mp_tc_xor(const mp_int *a, const mp_int *b, mp_int *c);
@ -359,6 +364,10 @@ int mp_sub_d(const mp_int *a, mp_digit b, mp_int *c);
/* c = a * b */
int mp_mul_d(const mp_int *a, mp_digit b, mp_int *c);
/* multiply bigint a with int d and put the result in c
Like mp_mul_d() but with a signed long as the small input */
int mp_mul_si(const mp_int *a, long d, mp_int *c);
/* a/b => cb + d == a */
int mp_div_d(const mp_int *a, mp_digit b, mp_int *c, mp_digit *d);
@ -417,6 +426,9 @@ int mp_is_square(const mp_int *arg, int *ret);
/* computes the jacobi c = (a | n) (or Legendre if b is prime) */
int mp_jacobi(const mp_int *a, const mp_int *n, int *c);
/* computes the Kronecker symbol c = (a | p) (like jacobi() but with {a,p} in Z */
int mp_kronecker(const mp_int *a, const mp_int *p, int *c);
/* used to setup the Barrett reduction for a given modulus b */
int mp_reduce_setup(mp_int *a, const mp_int *b);
@ -498,10 +510,27 @@ int mp_prime_miller_rabin(const mp_int *a, const mp_int *b, int *result);
*/
int mp_prime_rabin_miller_trials(int size);
/* performs t rounds of Miller-Rabin on "a" using the first
* t prime bases. Also performs an initial sieve of trial
/* performs one strong Lucas-Selfridge test of "a".
* Sets result to 0 if composite or 1 if probable prime
*/
int mp_prime_strong_lucas_selfridge(const mp_int *a, int *result);
/* performs one Frobenius test of "a" as described by Paul Underwood.
* Sets result to 0 if composite or 1 if probable prime
*/
int mp_prime_frobenius_underwood(const mp_int *N, int *result);
/* performs t random rounds of Miller-Rabin on "a" additional to
* bases 2 and 3. Also performs an initial sieve of trial
* division. Determines if "a" is prime with probability
* of error no more than (1/4)**t.
* Both a strong Lucas-Selfridge to complete the BPSW test
* and a separate Frobenius test are available at compile time.
* With t<0 a deterministic test is run for primes up to
* 318665857834031151167461. With t<13 (abs(t)-13) additional
* tests with sequential small primes are run starting at 43.
* Is Fips 186.4 compliant if called with t as computed by
* mp_prime_rabin_miller_trials();
*
* Sets result to 1 if probably prime, 0 otherwise
*/