weierstrass.c File Reference

Weierstrass elliptic curves. More...

#include <errno.h>
#include <ipxe/weierstrass.h>

Go to the source code of this file.

Macros
#define	WEIERSTRASS_zero   WEIERSTRASS_a
	Zero register (for add/subtract operations. More...
#define	WEIERSTRASS_REGISTER(name)   _C2 ( WEIERSTRASS_, name )
	Construct big integer register index. More...
#define	WEIERSTRASS_OP(opcode, dest, left, right)
	Define a bytecode operation. More...
#define	WEIERSTRASS_OPCODE(op)   ( ( (op) >> 12 ) & 0xf )
	Extract bytecode operation code. More...
#define	WEIERSTRASS_DEST(op)   ( ( (op) >> 8 ) & 0xf )
	Extract destination big integer register. More...
#define	WEIERSTRASS_LEFT(op)   ( ( (op) >> 4 ) & 0xf )
	Extract left source big integer register. More...
#define	WEIERSTRASS_RIGHT(op)   ( ( (op) >> 0 ) & 0xf )
	Extract right source big integer register. More...
#define	WEIERSTRASS_ADD3(dest, augend, addend)   WEIERSTRASS_OP ( WEIERSTRASS_OP_ADD, dest, augend, addend )
	Define a three-argument addition operation. More...
#define	WEIERSTRASS_ADD2(augend, addend)   WEIERSTRASS_ADD3 ( augend, augend, addend )
	Define a two-argument addition operation. More...
#define	WEIERSTRASS_MOV(dest, source)   WEIERSTRASS_ADD3( dest, source, zero )
	Define a move operation. More...
#define	WEIERSTRASS_SUB3(dest, minuend, subtrahend, multiple)
	Define a three-argument subtraction operation. More...
#define	WEIERSTRASS_SUB2(minuend, subtrahend, multiple)   WEIERSTRASS_SUB3 ( minuend, minuend, subtrahend, multiple )
	Define a two-argument subtraction operation. More...
#define	WEIERSTRASS_STOP   WEIERSTRASS_SUB2 ( zero, zero, 0N )
	Define a stop operation. More...
#define	WEIERSTRASS_MUL3(dest, multiplicand, multiplier)   WEIERSTRASS_OP ( WEIERSTRASS_OP_MUL, dest, multiplicand, multiplier )
	Define a three-argument multiplication operation. More...
#define	WEIERSTRASS_MUL2(multiplicand, multiplier)   WEIERSTRASS_MUL3 ( multiplicand, multiplicand, multiplier )
	Define a two-argument multiplication operation. More...
#define	weierstrass_add(curve, augend, addend, result)
	Add points on curve. More...
#define	weierstrass_verify(curve, point)
	Verify freshly initialised point is on curve. More...
#define	weierstrass_init(curve, point, temp, data)
	Initialise curve point. More...
#define	weierstrass_done(curve, point, temp, out)
	Finalise curve point. More...

Enumerations
enum	weierstrass_register {   WEIERSTRASS_a = 0, WEIERSTRASS_3b, WEIERSTRASS_x1, WEIERSTRASS_y1,   WEIERSTRASS_z1, WEIERSTRASS_x2, WEIERSTRASS_y2, WEIERSTRASS_z2,   WEIERSTRASS_Wt, WEIERSTRASS_Wxy, WEIERSTRASS_Wyz, WEIERSTRASS_Wzx,   WEIERSTRASS_Wp, WEIERSTRASS_x3, WEIERSTRASS_y3, WEIERSTRASS_z3,   WEIERSTRASS_NUM_REGISTERS = 16 }
	Big integer register names. More...
enum	weierstrass_opcode {   WEIERSTRASS_OP_SUB_0N = 0, WEIERSTRASS_OP_SUB_2N = WEIERSTRASS_2N, WEIERSTRASS_OP_SUB_4N = WEIERSTRASS_4N, WEIERSTRASS_OP_ADD,   WEIERSTRASS_OP_MUL }
	Bytecode operation codes. More...

Functions
	FILE_LICENCE (GPL2_OR_LATER_OR_UBDL)
static void	weierstrass_init_curve (struct weierstrass_curve *curve)
	Initialise curve. More...
static void	weierstrass_exec (const struct weierstrass_curve *curve, void **regs, unsigned int size, unsigned int op)
	Execute bytecode instruction. More...
static void	weierstrass_add_raw (const struct weierstrass_curve *curve, const bigint_element_t *augend0, const bigint_element_t *addend0, bigint_element_t *result0)
	Add points on curve. More...
static void	weierstrass_add_ladder (const bigint_element_t *operand0, bigint_element_t *result0, unsigned int size, const void *ctx, void *tmp __unused)
	Add points on curve as part of a Montgomery ladder. More...
static int	weierstrass_verify_raw (const struct weierstrass_curve *curve, const bigint_element_t *point0)
	Verify freshly initialised point is on curve. More...
static int	weierstrass_init_raw (struct weierstrass_curve *curve, bigint_element_t *point0, bigint_element_t *temp0, const void *data)
	Initialise curve point. More...
static void	weierstrass_done_raw (struct weierstrass_curve *curve, bigint_element_t *point0, bigint_element_t *temp0, void *out)
	Finalise curve point. More...
int	weierstrass_is_infinity (struct weierstrass_curve *curve, const void *point)
	Check if this is the point at infinity. More...
int	weierstrass_multiply (struct weierstrass_curve *curve, const void *base, const void *scalar, void *result)
	Multiply curve point by scalar. More...
int	weierstrass_add_once (struct weierstrass_curve *curve, const void *addend, const void *augend, void *result)
	Add curve points (as a one-off operation) More...

Detailed Description

Weierstrass elliptic curves.

The implementation is based upon Algorithm 1 from "Complete addition formulas for prime order elliptic curves" (Joost Renes, Craig Costello, and Lejla Batina), available from

https://www.microsoft.com/en-us/research/wp-content/uploads/2016/06/complete-2.pdf

The steps within the algorithm have been reordered and temporary variables shuffled to reduce stack usage, and calculations are carried out modulo small multiples of the field prime in order to elide reductions after intermediate addition and subtraction operations.

The algorithm is encoded using a bytecode representation, since this substantially reduces the code size compared to direct implementation of the big integer operations.

Definition in file weierstrass.c.

Macro Definition Documentation

WEIERSTRASS_zero

#define WEIERSTRASS_zero   WEIERSTRASS_a

Zero register (for add/subtract operations.

Definition at line 91 of file weierstrass.c.

WEIERSTRASS_REGISTER

#define WEIERSTRASS_REGISTER

(

name

)

_C2 ( WEIERSTRASS_, name )

Construct big integer register index.

Definition at line 94 of file weierstrass.c.

WEIERSTRASS_OP

#define WEIERSTRASS_OP	(		opcode,
			dest,
			left,
			right
	)

Value:

( ( (opcode) << 12 ) |                          \
          ( WEIERSTRASS_REGISTER ( dest ) << 8 ) |      \
          ( WEIERSTRASS_REGISTER ( left ) << 4 ) |      \
          ( WEIERSTRASS_REGISTER ( right ) << 0 ) )

Define a bytecode operation.

Parameters

opcode	Operation code
dest	Destination big integer register name
left	Left source big integer register name
right	Right source big integer register name

Definition at line 118 of file weierstrass.c.

WEIERSTRASS_OPCODE

#define WEIERSTRASS_OPCODE

(

op

)

( ( (op) >> 12 ) & 0xf )

Extract bytecode operation code.

Definition at line 125 of file weierstrass.c.

WEIERSTRASS_DEST

#define WEIERSTRASS_DEST

(

op

)

( ( (op) >> 8 ) & 0xf )

Extract destination big integer register.

Definition at line 128 of file weierstrass.c.

WEIERSTRASS_LEFT

#define WEIERSTRASS_LEFT

(

op

)

( ( (op) >> 4 ) & 0xf )

Extract left source big integer register.

Definition at line 131 of file weierstrass.c.

WEIERSTRASS_RIGHT

#define WEIERSTRASS_RIGHT

(

op

)

( ( (op) >> 0 ) & 0xf )

Extract right source big integer register.

Definition at line 134 of file weierstrass.c.

WEIERSTRASS_ADD3

#define WEIERSTRASS_ADD3	(		dest,
			augend,
			addend
	)		WEIERSTRASS_OP ( WEIERSTRASS_OP_ADD, dest, augend, addend )

Define a three-argument addition operation.

Definition at line 137 of file weierstrass.c.

WEIERSTRASS_ADD2

#define WEIERSTRASS_ADD2	(		augend,
			addend
	)		WEIERSTRASS_ADD3 ( augend, augend, addend )

Define a two-argument addition operation.

Definition at line 141 of file weierstrass.c.

WEIERSTRASS_MOV

#define WEIERSTRASS_MOV	(		dest,
			source
	)		WEIERSTRASS_ADD3( dest, source, zero )

Define a move operation.

Definition at line 145 of file weierstrass.c.

WEIERSTRASS_SUB3

#define WEIERSTRASS_SUB3	(		dest,
			minuend,
			subtrahend,
			multiple
	)

Value:

WEIERSTRASS_OP ( _C2 ( WEIERSTRASS_OP_SUB_, multiple ),   \
                         dest, minuend, subtrahend )

Define a three-argument subtraction operation.

Definition at line 149 of file weierstrass.c.

WEIERSTRASS_SUB2

#define WEIERSTRASS_SUB2	(		minuend,
			subtrahend,
			multiple
	)		WEIERSTRASS_SUB3 ( minuend, minuend, subtrahend, multiple )

Define a two-argument subtraction operation.

Definition at line 154 of file weierstrass.c.

WEIERSTRASS_STOP

#define WEIERSTRASS_STOP   WEIERSTRASS_SUB2 ( zero, zero, 0N )

Define a stop operation.

Definition at line 158 of file weierstrass.c.

WEIERSTRASS_MUL3

#define WEIERSTRASS_MUL3	(		dest,
			multiplicand,
			multiplier
	)		WEIERSTRASS_OP ( WEIERSTRASS_OP_MUL, dest, multiplicand, multiplier )

Define a three-argument multiplication operation.

Definition at line 161 of file weierstrass.c.

WEIERSTRASS_MUL2

#define WEIERSTRASS_MUL2	(		multiplicand,
			multiplier
	)		WEIERSTRASS_MUL3 ( multiplicand, multiplicand, multiplier )

Define a two-argument multiplication operation.

Definition at line 165 of file weierstrass.c.

weierstrass_add

#define weierstrass_add	(		curve,
			augend,
			addend,
			result
	)

Value:

do {            \
        weierstrass_add_raw ( (curve), (augend)->all.element,           \
                              (addend)->all.element,                    \
                              (result)->all.element );                  \
        } while ( 0 )

Add points on curve.

Parameters

curve	Weierstrass curve
augend	Point (x1,y1,z1) to be added
addend	Point (x2,y2,z2) to be added
result0	Point (x3,y3,z3) to hold result

Definition at line 629 of file weierstrass.c.

weierstrass_verify

#define weierstrass_verify	(		curve,
			point
	)

Value:

( {                             \
        weierstrass_verify_raw ( (curve), (point)->all.element );       \
        } )

Verify freshly initialised point is on curve.

Parameters

curve	Weierstrass curve
point	Point (x,y,z) to be verified

Return values

rc	Return status code

Definition at line 766 of file weierstrass.c.

weierstrass_init

#define weierstrass_init	(		curve,
			point,
			temp,
			data
	)

Value:

( {             \
        weierstrass_init_raw ( (curve), (point)->all.element,           \
                               (temp)->all.element, (data) );           \
        } )

Initialise curve point.

Parameters

curve	Weierstrass curve
point	Point (x,y,z) to be filled in
temp	Temporary point buffer
data	Raw curve point

Return values

rc	Return status code

Definition at line 844 of file weierstrass.c.

weierstrass_done

#define weierstrass_done	(		curve,
			point,
			temp,
			out
	)

Value:

( {                     \
        weierstrass_done_raw ( (curve), (point)->all.element,           \
                               (temp)->all.element, (out) );            \
        } )

Finalise curve point.

Parameters

curve	Weierstrass curve
point	Point (x,y,z)
temp	Temporary point buffer
out	Output buffer

Return values

rc	Return status code

Definition at line 907 of file weierstrass.c.

Enumeration Type Documentation

weierstrass_register

enum weierstrass_register

Big integer register names.

Enumerator
WEIERSTRASS_a
WEIERSTRASS_3b
WEIERSTRASS_x1
WEIERSTRASS_y1
WEIERSTRASS_z1
WEIERSTRASS_x2
WEIERSTRASS_y2
WEIERSTRASS_z2
WEIERSTRASS_Wt
WEIERSTRASS_Wxy
WEIERSTRASS_Wyz
WEIERSTRASS_Wzx
WEIERSTRASS_Wp
WEIERSTRASS_x3
WEIERSTRASS_y3
WEIERSTRASS_z3
WEIERSTRASS_NUM_REGISTERS

Definition at line 51 of file weierstrass.c.

                          {

        /*
         * Read-only registers
         */

        /* Curve constant "a" (for multiply), zero (for add/subtract) */
        WEIERSTRASS_a = 0,
        /* Curve constant "3b" */
        WEIERSTRASS_3b,
        /* Augend (x,y,z) co-ordinates */
        WEIERSTRASS_x1,
        WEIERSTRASS_y1,
        WEIERSTRASS_z1,
        /* Addend (x,y,z) co-ordinates */
        WEIERSTRASS_x2,
        WEIERSTRASS_y2,
        WEIERSTRASS_z2,

        /*
         * Read-write registers
         */

        /* Temporary working registers */
        WEIERSTRASS_Wt,
        WEIERSTRASS_Wxy,
        WEIERSTRASS_Wyz,
        WEIERSTRASS_Wzx,
        /* Low half of multiplication product */
        WEIERSTRASS_Wp,
        /* Result (x,y,z) co-ordinates */
        WEIERSTRASS_x3,
        WEIERSTRASS_y3,
        WEIERSTRASS_z3,

        /* Number of registers */
        WEIERSTRASS_NUM_REGISTERS = 16
};

weierstrass_opcode

enum weierstrass_opcode

Bytecode operation codes.

Enumerator
WEIERSTRASS_OP_SUB_0N	Subtract big integers (and add nothing)
WEIERSTRASS_OP_SUB_2N	Subtract big integers (and add 2N)
WEIERSTRASS_OP_SUB_4N	Subtract big integers (and add 4N)
WEIERSTRASS_OP_ADD	Add big integers.
WEIERSTRASS_OP_MUL	Multiply big integers (and perform Montgomery reduction)

Definition at line 97 of file weierstrass.c.

                        {
        /** Subtract big integers (and add nothing)*/
        WEIERSTRASS_OP_SUB_0N = 0,
        /** Subtract big integers (and add 2N) */
        WEIERSTRASS_OP_SUB_2N = WEIERSTRASS_2N,
        /** Subtract big integers (and add 4N) */
        WEIERSTRASS_OP_SUB_4N = WEIERSTRASS_4N,
        /** Add big integers */
        WEIERSTRASS_OP_ADD,
        /** Multiply big integers (and perform Montgomery reduction) */
        WEIERSTRASS_OP_MUL,
};

Function Documentation

FILE_LICENCE()

FILE_LICENCE

(

GPL2_OR_LATER_OR_UBDL

)

weierstrass_init_curve()

static void weierstrass_init_curve ( struct weierstrass_curve *  curve )

static

Initialise curve.

Parameters

curve

Weierstrass curve

Definition at line 173 of file weierstrass.c.

                                                                       {
        unsigned int size = curve->size;
        bigint_t ( size ) __attribute__ (( may_alias )) *prime =
                ( ( void * ) curve->prime[0] );
        bigint_t ( size ) __attribute__ (( may_alias )) *fermat =
                ( ( void * ) curve->fermat );
        bigint_t ( size ) __attribute__ (( may_alias )) *square =
                ( ( void * ) curve->square );
        bigint_t ( size ) __attribute__ (( may_alias )) *one =
                ( ( void * ) curve->one );
        bigint_t ( size ) __attribute__ (( may_alias )) *a =
                ( ( void * ) curve->a );
        bigint_t ( size ) __attribute__ (( may_alias )) *b3 =
                ( ( void * ) curve->b3 );
        bigint_t ( size ) __attribute__ (( may_alias )) *mont =
                ( ( void * ) curve->mont[0] );
        bigint_t ( size ) __attribute__ (( may_alias )) *temp =
                ( ( void * ) curve->prime[1] );
        bigint_t ( size * 2 ) __attribute__ (( may_alias )) *product =
                ( ( void * ) temp );
        bigint_t ( size ) __attribute__ (( may_alias )) *two =
                ( ( void * ) temp );
        static const uint8_t one_raw[] = { 1 };
        static const uint8_t two_raw[] = { 2 };
        size_t len = curve->len;
        unsigned int i;

        /* Initialise field prime */
        bigint_init ( prime, curve->prime_raw, len );
        DBGC ( curve, "WEIERSTRASS %s   N = %s\n",
               curve->name, bigint_ntoa ( prime ) );

        /* Calculate Montgomery constant R^2 mod N */
        bigint_reduce ( prime, square );
        DBGC ( curve, "WEIERSTRASS %s R^2 = %s mod N\n",
               curve->name, bigint_ntoa ( square ) );

        /* Calculate constant "3b" */
        bigint_init ( b3, curve->b_raw, len );
        DBGC ( curve, "WEIERSTRASS %s   b = %s\n",
               curve->name, bigint_ntoa ( b3 ) );
        bigint_copy ( b3, a );
        bigint_add ( b3, b3 );
        bigint_add ( a, b3 );

        /* Initialise "a" */
        bigint_init ( a, curve->a_raw, len );
        DBGC ( curve, "WEIERSTRASS %s   a = %s\n",
               curve->name, bigint_ntoa ( a ) );

        /* Initialise "1" */
        bigint_init ( one, one_raw, sizeof ( one_raw ) );

        /* Convert relevant constants to Montgomery form
         *
         * We rely on the fact that the prime multiples have not yet
         * been calculated, and so can be used as a temporary buffer.
         */
        for ( i = 0 ; i < WEIERSTRASS_NUM_MONT ; i++ ) {
                static const char *names[] = { "  ", " a", "3b" };
                bigint_multiply ( &mont[i], square, product );
                bigint_montgomery ( prime, product, &mont[i] );
                DBGC ( curve, "WEIERSTRASS %s %sR = %s mod N\n",
                       curve->name, names[i], bigint_ntoa ( &mont[i] ) );
        }

        /* Calculate constant "N-2"
         *
         * We rely on the fact that the prime multiples have not yet
         * been calculated, and so can be used as a temporary buffer.
         */
        bigint_copy ( prime, fermat );
        bigint_init ( two, two_raw, sizeof ( two_raw ) );
        bigint_subtract ( two, fermat );
        DBGC ( curve, "WEIERSTRASS %s N-2 = %s\n",
               curve->name, bigint_ntoa ( fermat ) );

        /* Calculate multiples of field prime */
        for ( i = 1 ; i < WEIERSTRASS_NUM_MULTIPLES ; i++ ) {
                bigint_copy ( &prime[ i - 1 ], &prime[i] );
                bigint_add ( &prime[i], &prime[i] );
                DBGC ( curve, "WEIERSTRASS %s  %dN = %s\n",
                       curve->name, ( 1 << i ), bigint_ntoa ( &prime[i] ) );
        }
}

References __attribute__, weierstrass_curve::a, weierstrass_curve::a_raw, weierstrass_curve::b3, weierstrass_curve::b_raw, bigint_add, bigint_copy, bigint_init, bigint_montgomery, bigint_multiply, bigint_ntoa, bigint_reduce, bigint_subtract, bigint_t(), DBGC, weierstrass_curve::fermat, weierstrass_curve::len, len, weierstrass_curve::mont, weierstrass_curve::name, weierstrass_curve::one, weierstrass_curve::prime, weierstrass_curve::prime_raw, product, size, weierstrass_curve::size, weierstrass_curve::square, WEIERSTRASS_NUM_MONT, and WEIERSTRASS_NUM_MULTIPLES.

Referenced by weierstrass_init_raw().

weierstrass_exec()

static void weierstrass_exec ( const struct weierstrass_curve *  curve, void **  regs, unsigned int  size, unsigned int  op  )

static

Execute bytecode instruction.

Parameters

curve	Weierstrass curve
regs	Registers
size	Big integer size
op	Operation

Definition at line 267 of file weierstrass.c.

                                                 {
        const bigint_t ( size ) __attribute__ (( may_alias ))
                *prime = ( ( const void * ) curve->prime[0] );
        bigint_t ( size * 2 ) __attribute__ (( may_alias ))
                *product = regs[WEIERSTRASS_Wp];
        bigint_t ( size ) __attribute__ (( may_alias )) *dest;
        const bigint_t ( size ) __attribute__ (( may_alias )) *left;
        const bigint_t ( size ) __attribute__ (( may_alias )) *right;
        const bigint_t ( size ) __attribute__ (( may_alias )) *addend;
        const bigint_t ( size ) __attribute__ (( may_alias )) *subtrahend;
        unsigned int op_code;
        unsigned int op_dest;
        unsigned int op_left;
        unsigned int op_right;

        /* Decode instruction */
        op_code = WEIERSTRASS_OPCODE ( op );
        op_dest = WEIERSTRASS_DEST ( op );
        op_left = WEIERSTRASS_LEFT ( op );
        op_right = WEIERSTRASS_RIGHT ( op );
        dest = regs[op_dest];
        left = regs[op_left];
        right = regs[op_right];

        /* Check destination is a writable register */
        assert ( op_dest >= WEIERSTRASS_Wt );

        /* Handle multiplications */
        if ( op_code == WEIERSTRASS_OP_MUL ) {
                assert ( op_left != WEIERSTRASS_Wp );
                assert ( op_right != WEIERSTRASS_Wp );
                bigint_multiply ( left, right, product );
                bigint_montgomery_relaxed ( prime, product, dest );
                DBGCP ( curve, "WEIERSTRASS %s R%d := R%d x R%d = %s\n",
                        curve->name, op_dest, op_left, op_right,
                        bigint_ntoa ( dest ) );
                return;
        }

        /* Copy left source, if required */
        if ( op_dest != op_left )
                bigint_copy ( left, dest );

        /* Do nothing more if addend/subtrahend is zero */
        if ( ! op_right ) {
                DBGCP ( curve, "WEIERSTRASS %s R%d := R%d = %s\n",
                        curve->name, op_dest, op_left, bigint_ntoa ( dest ) );
                return;
        }

        /* Determine addend and subtrahend */
        addend = NULL;
        subtrahend = NULL;
        if ( op_code == WEIERSTRASS_OP_ADD ) {
                DBGCP ( curve, "WEIERSTRASS %s R%d := R%d + R%d = ",
                        curve->name, op_dest, op_left, op_right );
                addend = ( ( const void * ) right );
        } else {
                subtrahend = ( ( const void * ) right );
                if ( op_code > WEIERSTRASS_OP_SUB_0N ) {
                        DBGCP ( curve, "WEIERSTRASS %s R%d := R%d - R%d + "
                                "%dN = ", curve->name, op_dest, op_left,
                                op_right, ( 1 << op_code ) );
                        addend = ( ( const void * ) curve->prime[op_code] );
                } else {
                        DBGCP ( curve, "WEIERSTRASS %s R%d := R%d - R%d = ",
                                curve->name, op_dest, op_left, op_right );
                }
        }

        /* Perform addition and subtraction */
        if ( addend )
                bigint_add ( addend, dest );
        if ( subtrahend )
                bigint_subtract ( subtrahend, dest );
        DBGCP ( curve, "%s\n", bigint_ntoa ( dest ) );
}

References __attribute__, assert(), bigint_add, bigint_copy, bigint_montgomery_relaxed, bigint_multiply, bigint_ntoa, bigint_subtract, bigint_t(), DBGCP, dest, weierstrass_curve::name, NULL, op, weierstrass_curve::prime, product, regs, size, WEIERSTRASS_DEST, WEIERSTRASS_LEFT, WEIERSTRASS_OP_ADD, WEIERSTRASS_OP_MUL, WEIERSTRASS_OP_SUB_0N, WEIERSTRASS_OPCODE, WEIERSTRASS_RIGHT, WEIERSTRASS_Wp, and WEIERSTRASS_Wt.

Referenced by weierstrass_add_raw(), and weierstrass_verify_raw().

weierstrass_add_raw()

static void weierstrass_add_raw ( const struct weierstrass_curve *  curve, const bigint_element_t *  augend0, const bigint_element_t *  addend0, bigint_element_t *  result0  )

static

Add points on curve.

Parameters

curve	Weierstrass curve
augend0	Element 0 of point (x1,y1,z1) to be added
addend0	Element 0 of point (x2,y2,z2) to be added
result0	Element 0 of point (x3,y3,z3) to hold result

Points are represented in projective coordinates, with all values in Montgomery form and in the range [0,4N) where N is the field prime.

The augend may have the same value as the addend (i.e. this routine may be used to perform point doubling as well as point addition), and either or both may be the point at infinity.

The result may overlap either input, since the inputs are fully consumed before the result is written.

Definition at line 366 of file weierstrass.c.

                                                              {
        unsigned int size = curve->size;
        const bigint_t ( size ) __attribute__ (( may_alias ))
                *prime = ( ( const void * ) curve->prime[0] );
        const bigint_t ( size ) __attribute__ (( may_alias ))
                *a = ( ( const void * ) curve->a );
        const bigint_t ( size ) __attribute__ (( may_alias ))
                *b3 = ( ( const void * ) curve->b3 );
        const weierstrass_t ( size ) __attribute__ (( may_alias ))
                *augend = ( ( const void * ) augend0 );
        const weierstrass_t ( size ) __attribute__ (( may_alias ))
                *addend = ( ( const void * ) addend0 );
        weierstrass_t ( size ) __attribute__ (( may_alias ))
                *result = ( ( void * ) result0 );
        struct {
                bigint_t ( size ) Wt;
                bigint_t ( size ) Wxy;
                bigint_t ( size ) Wyz;
                bigint_t ( size ) Wzx;
                bigint_t ( size * 2 ) Wp;
        } temp;
        void *regs[WEIERSTRASS_NUM_REGISTERS];
        unsigned int schedule;
        const uint16_t *op;
        unsigned int i;

        /* On entry, we assume that x1, x2, y1, y2, z1, z2 are all in
         * the range [0,4N).  Additions will extend the range.
         * Subtractions will extend the range (and require an addition
         * of a suitable multiple of the modulus to ensure that the
         * result is a positive value).  Relaxed Montgomery
         * multiplications will reduce the range to [0,2N).  The
         * outputs x3, y3, z3 will be in the range [0,4N) and
         * therefore usable as subsequent inputs.
         */
        static const uint16_t ops[] = {
                /* [Wxy] Qxy = (x1+y1)*(x2+y2)                      (mod 2N) */
                WEIERSTRASS_ADD3 ( Wt, x1, y1 ),
                WEIERSTRASS_ADD3 ( Wxy, x2, y2 ),
                WEIERSTRASS_MUL2 ( Wxy, Wt ),
                /* [Wyz] Qyz = (y1+z1)*(y2+z2)                      (mod 2N) */
                WEIERSTRASS_ADD3 ( Wt, y1, z1 ),
                WEIERSTRASS_ADD3 ( Wyz, y2, z2 ),
                WEIERSTRASS_MUL2 ( Wyz, Wt ),
                /* [Wzx] Qzx = (z1+x1)*(z2+x2)                      (mod 2N) */
                WEIERSTRASS_ADD3 ( Wt, z1, x1 ),
                WEIERSTRASS_ADD3 ( Wzx, z2, x2 ),
                WEIERSTRASS_MUL2 ( Wzx, Wt ),
                /* [x3] Px = x1*x2                                  (mod 2N) */
                WEIERSTRASS_MUL3 ( x3, x1, x2 ),
                /* [y3] Py = y1*y2                                  (mod 2N) */
                WEIERSTRASS_MUL3 ( y3, y1, y2 ),
                /* [z3] Pz = z1*z2                                  (mod 2N) */
                WEIERSTRASS_MUL3 ( z3, z1, z2 ),
                /* [Wxy] Rxy = Qxy - Px - Py                        (mod 6N)
                 *           = (x1+y1)*(x2+y2) - x1*x2 - y1*y2      (mod 6N)
                 *           = x1*y2 + x2*y1                        (mod 6N)
                 */
                WEIERSTRASS_SUB2 ( Wxy, x3, 0N ),
                WEIERSTRASS_SUB2 ( Wxy, y3, 4N ),
                /* [Wyz] Ryz = Qyz - Py - Pz                        (mod 6N)
                 *           = (y1+z1)*(y2+z2) - y1*y2 - z1*z2      (mod 6N)
                 *           = y1*z2 + y2*z1                        (mod 6N)
                 */
                WEIERSTRASS_SUB2 ( Wyz, y3, 0N ),
                WEIERSTRASS_SUB2 ( Wyz, z3, 4N ),
                /* [Wzx] Rzx = Qzx - Pz - Px                        (mod 6N)
                 *           = (z1+x1)*(z2+x2) - z1*z2 - x1*x2      (mod 6N)
                 *           = x1*z2 + x2*z1                        (mod 6N)
                 */
                WEIERSTRASS_SUB2 ( Wzx, z3, 0N ),
                WEIERSTRASS_SUB2 ( Wzx, x3, 4N ),
                /* [Wt] aRzx = a * Rzx                              (mod 2N)
                 *           = a * (x1*z2 + x2*z1)                  (mod 2N)
                 */
                WEIERSTRASS_MUL3 ( Wt, a, Wzx ),
                /* [Wp] 3bPz = 3b * Pz                              (mod 2N)
                 *           = 3b*z1*z2                             (mod 2N)
                 */
                WEIERSTRASS_MUL3 ( Wp, 3b, z3 ),
                /* [Wp] Sy = aRzx + 3bPz                            (mod 4N)
                 *         = a*(x1*z2 + x2*z1) + 3b*z1*z2           (mod 4N)
                 */
                WEIERSTRASS_ADD2 ( Wp, Wt ),
                /* [Wt] Syz = Py + Sy                               (mod 6N)
                 *          = y1*y2 + a*(x1*z2 + x2*z1) + 3b*z1*z2  (mod 6N)
                 */
                WEIERSTRASS_ADD3 ( Wt, y3, Wp ),
                /* [y3] Sxy = Py - Sy                               (mod 6N)
                 *          = y1*y2 - a*(x1*z2 + x2*z1) - 3b*z1*z2  (mod 6N)
                 */
                WEIERSTRASS_SUB2 ( y3, Wp, 4N ),
                /* [z3] aPz = a * Pz                                (mod 2N)
                 *          = a * z1*z2                             (mod 2N)
                 */
                WEIERSTRASS_MUL2 ( z3, a ),
                /* [Wzx] 3bRzx = 3b * Rzx                           (mod 2N)
                 *             = 3b * (x1*z2 + x2*z1)               (mod 2N)
                 */
                WEIERSTRASS_MUL2 ( Wzx, 3b ),
                /* [x3] aPzx' = Px - aPz                            (mod 4N)
                 *            = x1*x2 - a*z1*z2                     (mod 4N)
                 */
                WEIERSTRASS_SUB2 ( x3, z3, 2N ),
                /* [Wp] Szx = a * aPzx'                             (mod 2N)
                 *          = a * (x1*x2 - a*z1*z2)                 (mod 2N)
                 *          = a*x1*x2 - (a^2)*z1*z2                 (mod 2N)
                 */
                WEIERSTRASS_MUL3 ( Wp, a, x3 ),
                /* [x3] Px = aPzx' + aPz                            (mod 6N)
                 *         = x1*x2 - a*z1*z2 + a*z1*z2              (mod 6N)
                 *         = x1*x2                                  (mod 6N)
                 */
                WEIERSTRASS_ADD2 ( x3, z3 ),
                /* [Wzx] Tzx = 3bRzx + Szx                          (mod 4N)
                 *           = a*x1*x2 + 3b*(x1*z2 + x2*z1) -
                 *             (a^2)*z1*z2                          (mod 4N)
                 */
                WEIERSTRASS_ADD2 ( Wzx, Wp ),
                /* [z3] aPzx = Px + aPz                             (mod 8N)
                 *           = x1*x2 + a*z1*z2                      (mod 8N)
                 */
                WEIERSTRASS_ADD2 ( z3, x3 ),
                /* [x3] 2Px = Px + Px                               (mod 12N)
                 *          = 2*x1*x2                               (mod 12N)
                 */
                WEIERSTRASS_ADD2 ( x3, x3 ),
                /* [x3] Tyz = 2Px + aPzx                            (mod 20N)
                 *          = 2*x1*x2 + x1*x2 + a*z1*z2             (mod 20N)
                 *          = 3*x1*x2 + a*z1*z2                     (mod 20N)
                 */
                WEIERSTRASS_ADD2 ( x3, z3 ),
                /* [z3] Syz = Syz                                   (mod 6N)
                 *          = y1*y2 + a*(x1*z2 + x2*z1) + 3b*z1*z2  (mod 6N)
                 */
                WEIERSTRASS_MOV ( z3, Wt ),
                /* [Wt] Tyz = Tyz                                   (mod 20N)
                 *          = 3*x1*x2 + a*z1*z2                     (mod 20N)
                 */
                WEIERSTRASS_MOV ( Wt, x3 ),
                /* [x3] Ux = Rxy * Sxy                              (mod 2N)
                 *         = (x1*y2 + x2*y1) *
                 *           (y1*y2 - a*(x1*z2 + x2*z1) - 3b*z1*z2) (mod 2N)
                 */
                WEIERSTRASS_MUL3 ( x3, Wxy, y3 ),
                /* [y3] Uy = Syz * Sxy                              (mod 2N)
                 *         = (y1*y2 + a*(x1*z2 + x2*z1) + 3b*z1*z2) *
                 *           (y1*y2 - a*(x1*z2 + x2*z1) - 3b*z1*z2) (mod 2N)
                 */
                WEIERSTRASS_MUL2 ( y3, z3 ),
                /* [z3] Uz = Ryz * Syz                              (mod 2N)
                 *         = (y1*z2 + y2*z1) *
                 *           (y1*y2 + a*(x1*z2 + x2*z1) + 3b*z1*z2) (mod 2N)
                 */
                WEIERSTRASS_MUL2 ( z3, Wyz ),
                /* [Wp] Vx = Ryz * Tzx                              (mod 2N)
                 *         = (y1*z2 + y2*z1) *
                 *           (a*x1*x2 + 3b*(x1*z2 + x2*z1) - (a^2)*z1*z2)
                 *                                                  (mod 2N)
                 */
                WEIERSTRASS_MUL3 ( Wp, Wyz, Wzx ),
                /* [x3] x3 = Ux - Vx                                (mod 4N)
                 *         = ((x1*y2 + x2*y1) *
                 *            (y1*y2 - a*(x1*z2 + x2*z1) - 3b*z1*z2)) -
                 *           ((y1*z2 + y2*z1) *
                 *            (a*x1*x2 + 3b*(x1*z2 + x2*z1) - (a^2)*z1*z2))
                 *                                                  (mod 4N)
                 */
                WEIERSTRASS_SUB2 ( x3, Wp, 2N ),
                /* [Wp] Vy = Tyz * Tzx                              (mod 2N)
                 *         = (3*x1*x2 + a*z1*z2) *
                 *           (a*x1*x2 + 3b*(x1*z2 + x2*z1) - (a^2)*z1*z2)
                 *                                                  (mod 2N)
                 */
                WEIERSTRASS_MUL3 ( Wp, Wt, Wzx ),
                /* [y3] y3 = Vy + Uy                                (mod 4N)
                 *         = ((3*x1*x2 + a*z1*z2) *
                 *            (a*x1*x2 + 3b*(x1*z2 + x2*z1) - (a^2)*z1*z2)) +
                 *           ((y1*y2 + a*(x1*z2 + x2*z1) + 3b*z1*z2) *
                 *            (y1*y2 - a*(x1*z2 + x2*z1) - 3b*z1*z2))
                 *                                                  (mod 4N)
                 */
                WEIERSTRASS_ADD2 ( y3, Wp ),
                /* [Wp] Vz = Rxy * Tyz                              (mod 2N)
                 *         = (x1*y2 + x2*y1) * (3*x1*x2 + a*z1*z2)  (mod 2N)
                 */
                WEIERSTRASS_MUL3 ( Wp, Wxy, Wt ),
                /* [z3] z3 = Uz + Vz                                (mod 4N)
                 *         = ((y1*z2 + y2*z1) *
                 *            (y1*y2 + a*(x1*z2 + x2*z1) + 3b*z1*z2)) +
                 *           ((x1*y2 + x2*y1) * (3*x1*x2 + a*z1*z2))
                 *                                                  (mod 4N)
                 */
                WEIERSTRASS_ADD2 ( z3, Wp ),
                /* Stop */
                WEIERSTRASS_STOP
        };

        /* Initialise register list */
        regs[WEIERSTRASS_a] = ( ( void * ) a );
        regs[WEIERSTRASS_3b] = ( ( void * ) b3 );
        regs[WEIERSTRASS_x1] = ( ( void * ) &augend->x );
        regs[WEIERSTRASS_x2] = ( ( void * ) &addend->x );
        regs[WEIERSTRASS_x3] = ( ( void * ) &result->x );
        regs[WEIERSTRASS_Wt] = &temp;
        schedule = ( ( ( 1 << WEIERSTRASS_NUM_REGISTERS ) - 1 )
                     - ( 1 << WEIERSTRASS_a )
                     - ( 1 << WEIERSTRASS_3b )
                     - ( 1 << WEIERSTRASS_x1 )
                     - ( 1 << WEIERSTRASS_x2 )
                     - ( 1 << WEIERSTRASS_x3 )
                     - ( 1 << WEIERSTRASS_Wt ) );
        for ( i = 0 ; schedule ; i++, schedule >>= 1 ) {
                if ( schedule & 1 )
                        regs[i] = ( regs[ i - 1 ] + sizeof ( *prime ) );
        }
        DBGC2 ( curve, "WEIERSTRASS %s augend (%s,",
                curve->name, bigint_ntoa ( &augend->x ) );
        DBGC2 ( curve, "%s,", bigint_ntoa ( &augend->y ) );
        DBGC2 ( curve, "%s)\n", bigint_ntoa ( &augend->z ) );
        DBGC2 ( curve, "WEIERSTRASS %s addend (%s,",
                curve->name, bigint_ntoa ( &addend->x ) );
        DBGC2 ( curve, "%s,", bigint_ntoa ( &addend->y ) );
        DBGC2 ( curve, "%s)\n", bigint_ntoa ( &addend->z ) );

        /* Sanity checks */
        assert ( regs[WEIERSTRASS_a] == a );
        assert ( regs[WEIERSTRASS_3b] == b3 );
        assert ( regs[WEIERSTRASS_x1] == &augend->x );
        assert ( regs[WEIERSTRASS_y1] == &augend->y );
        assert ( regs[WEIERSTRASS_z1] == &augend->z );
        assert ( regs[WEIERSTRASS_x2] == &addend->x );
        assert ( regs[WEIERSTRASS_y2] == &addend->y );
        assert ( regs[WEIERSTRASS_z2] == &addend->z );
        assert ( regs[WEIERSTRASS_x3] == &result->x );
        assert ( regs[WEIERSTRASS_y3] == &result->y );
        assert ( regs[WEIERSTRASS_z3] == &result->z );
        assert ( regs[WEIERSTRASS_Wt] == &temp.Wt );
        assert ( regs[WEIERSTRASS_Wxy] == &temp.Wxy );
        assert ( regs[WEIERSTRASS_Wyz] == &temp.Wyz );
        assert ( regs[WEIERSTRASS_Wzx] == &temp.Wzx );
        assert ( regs[WEIERSTRASS_Wp] == &temp.Wp );

        /* Execute bytecode instruction sequence */
        for ( op = ops ; *op != WEIERSTRASS_STOP ; op++ )
                weierstrass_exec ( curve, regs, size, *op );
        DBGC2 ( curve, "WEIERSTRASS %s result (%s,",
                curve->name, bigint_ntoa ( &result->x ) );
        DBGC2 ( curve, "%s,", bigint_ntoa ( &result->y ) );
        DBGC2 ( curve, "%s)\n", bigint_ntoa ( &result->z ) );
}

References __attribute__, weierstrass_curve::a, assert(), weierstrass_curve::b3, bigint_ntoa, bigint_t(), DBGC2, weierstrass_curve::name, op, weierstrass_curve::prime, regs, result, size, weierstrass_curve::size, WEIERSTRASS_3b, WEIERSTRASS_a, WEIERSTRASS_ADD2, WEIERSTRASS_ADD3, weierstrass_exec(), WEIERSTRASS_MOV, WEIERSTRASS_MUL2, WEIERSTRASS_MUL3, WEIERSTRASS_NUM_REGISTERS, WEIERSTRASS_STOP, WEIERSTRASS_SUB2, weierstrass_t, WEIERSTRASS_Wp, WEIERSTRASS_Wt, WEIERSTRASS_Wxy, WEIERSTRASS_Wyz, WEIERSTRASS_Wzx, WEIERSTRASS_x1, WEIERSTRASS_x2, WEIERSTRASS_x3, WEIERSTRASS_y1, WEIERSTRASS_y2, WEIERSTRASS_y3, WEIERSTRASS_z1, WEIERSTRASS_z2, and WEIERSTRASS_z3.

weierstrass_add_ladder()

static void weierstrass_add_ladder ( const bigint_element_t *  operand0, bigint_element_t *  result0, unsigned int  size, const void *  ctx, void *tmp  __unused  )

static

Add points on curve as part of a Montgomery ladder.

Parameters

operand	Element 0 of first input operand (may overlap result)
result	Element 0 of second input operand and result
size	Number of elements in operands and result
ctx	Operation context
tmp	Temporary working space (not used)

Definition at line 644 of file weierstrass.c.

                                                          {
        const struct weierstrass_curve *curve = ctx;
        const weierstrass_t ( curve->size ) __attribute__ (( may_alias ))
                *operand = ( ( const void * ) operand0 );
        weierstrass_t ( curve->size ) __attribute__ (( may_alias ))
                *result = ( ( void * ) result0 );

        /* Add curve points */
        assert ( size == bigint_size ( &operand->all ) );
        assert ( size == bigint_size ( &result->all ) );
        weierstrass_add ( curve, operand, result, result );
}

References __attribute__, assert(), bigint_size, ctx, result, size, weierstrass_curve::size, weierstrass_add, and weierstrass_t.

Referenced by weierstrass_multiply().

weierstrass_verify_raw()

static int weierstrass_verify_raw ( const struct weierstrass_curve *  curve, const bigint_element_t *  point0  )

static

Verify freshly initialised point is on curve.

Parameters

curve	Weierstrass curve
point0	Element 0 of point (x,y,z) to be verified

Return values

rc	Return status code

As with point addition, points are represented in projective coordinates, with all values in Montgomery form and in the range [0,4N) where N is the field prime.

This verification logic is valid only for points that have been freshly constructed via weierstrass_init() (i.e. must either have z=1 or be the point at infinity (0,1,0)).

Definition at line 675 of file weierstrass.c.

                                                                     {
        unsigned int size = curve->size;
        const bigint_t ( size ) __attribute__ (( may_alias ))
                *prime = ( ( const void * ) curve->prime[0] );
        const bigint_t ( size ) __attribute__ (( may_alias ))
                *a = ( ( const void * ) curve->a );
        const bigint_t ( size ) __attribute__ (( may_alias ))
                *b3 = ( ( const void * ) curve->b3 );
        const weierstrass_t ( size ) __attribute__ (( may_alias ))
                *point = ( ( const void * ) point0 );
        struct {
                bigint_t ( size ) Wt;
                bigint_t ( size * 2 ) Wp;
        } temp;
        void *regs[WEIERSTRASS_NUM_REGISTERS];
        const uint16_t *op;

        /* Calculate 3*(x^3 + a*x + b - y^2) */
        static const uint16_t ops[] = {
                /* [Wt] Tx = x^2                                    (mod 2N) */
                WEIERSTRASS_MUL3 ( Wt, x1, x1 ),
                /* [Wt] Txa = Tx + a                                (mod 3N)
                 *          = x^2 + a                               (mod 3N)
                 */
                WEIERSTRASS_MOV ( Wp, a ),
                WEIERSTRASS_ADD2 ( Wt, Wp ),
                /* [Wt] Txax = Txa * x                              (mod 2N)
                 *           = (x^2 + a)*x                          (mod 2N)
                 *           = x^3 + a*x                            (mod 2N)
                 */
                WEIERSTRASS_MUL2 ( Wt, x1 ),
                /* [Wp] Ty = y^2                                    (mod 2N) */
                WEIERSTRASS_MUL3 ( Wp, y1, y1 ),
                /* [Wt] Txaxy = Txax - Ty                           (mod 4N)
                 *            = x^3 + a*x - y^2                     (mod 4N)
                 */
                WEIERSTRASS_SUB2 ( Wt, Wp, 2N ),
                /* [Wp] 2Txaxy = Txaxy + Txaxy                      (mod 8N)
                 *             = 2*(x^3 + a*x - y^2)                (mod 8N)
                 */
                WEIERSTRASS_ADD3 ( Wp, Wt, Wt ),
                /* [Wt] 3Txaxy = 2Txaxy + Txaxy                     (mod 12N)
                 *             = 3*(x^3 + a*x - y^2)                (mod 12N)
                 */
                WEIERSTRASS_ADD2 ( Wt, Wp ),
                /* [Wt] 3Txaxyb = 3Txaxy + 3b                       (mod 13N)
                 *              = 3*(x^3 + a*x - y^2) + 3b          (mod 13N)
                 *              = 3*(x^3 + a*x + b - y^2)           (mod 13N)
                 */
                WEIERSTRASS_ADD2 ( Wt, 3b ),
                /* [Wt] check   = 3Txaxyb * z                       (mod 2N)
                 *              = 3*(x^3 + a*x + b - y^2) * z       (mod 2N)
                 */
                WEIERSTRASS_MUL2 ( Wt, z1 ),
                /* Stop */
                WEIERSTRASS_STOP
        };

        /* Initialise register list */
        regs[WEIERSTRASS_a] = ( ( void * ) a );
        regs[WEIERSTRASS_3b] = ( ( void * ) b3 );
        regs[WEIERSTRASS_x1] = ( ( void * ) &point->x );
        regs[WEIERSTRASS_y1] = ( ( void * ) &point->y );
        regs[WEIERSTRASS_z1] = ( ( void * ) &point->z );
        regs[WEIERSTRASS_Wt] = &temp.Wt;
        regs[WEIERSTRASS_Wp] = &temp.Wp;

        /* Execute bytecode instruction sequence */
        for ( op = ops ; *op != WEIERSTRASS_STOP ; op++ )
                weierstrass_exec ( curve, regs, size, *op );

        /* Check that result is zero (modulo the field prime) */
        bigint_grow ( &temp.Wt, &temp.Wp );
        bigint_montgomery ( prime, &temp.Wp, &temp.Wt );
        if ( ! bigint_is_zero ( &temp.Wt ) ) {
                DBGC ( curve, "WEIERSTRASS %s base point is not on curve\n",
                       curve->name );
                return -EINVAL;
        }

        return 0;
}

References __attribute__, weierstrass_curve::a, weierstrass_curve::b3, bigint_grow, bigint_is_zero, bigint_montgomery, bigint_t(), DBGC, EINVAL, weierstrass_curve::name, op, weierstrass_curve::prime, regs, size, weierstrass_curve::size, WEIERSTRASS_3b, WEIERSTRASS_a, WEIERSTRASS_ADD2, WEIERSTRASS_ADD3, weierstrass_exec(), WEIERSTRASS_MOV, WEIERSTRASS_MUL2, WEIERSTRASS_MUL3, WEIERSTRASS_NUM_REGISTERS, WEIERSTRASS_STOP, WEIERSTRASS_SUB2, weierstrass_t, WEIERSTRASS_Wp, WEIERSTRASS_Wt, WEIERSTRASS_x1, WEIERSTRASS_y1, and WEIERSTRASS_z1.

weierstrass_init_raw()

static int weierstrass_init_raw ( struct weierstrass_curve *  curve, bigint_element_t *  point0, bigint_element_t *  temp0, const void *  data  )

static

Initialise curve point.

Parameters

curve	Weierstrass curve
point0	Element 0 of point (x,y,z) to be filled in
temp0	Element 0 of temporary point buffer
data	Raw curve point

Return values

rc	Return status code

Definition at line 779 of file weierstrass.c.

                                                                              {
        unsigned int size = curve->size;
        size_t len = curve->len;
        const bigint_t ( size ) __attribute__ (( may_alias )) *prime =
                ( ( const void * ) curve->prime[0] );
        const bigint_t ( size ) __attribute__ (( may_alias )) *prime2 =
                ( ( const void * ) curve->prime[WEIERSTRASS_2N] );
        const bigint_t ( size ) __attribute__ (( may_alias )) *square =
                ( ( const void * ) curve->square );
        const bigint_t ( size ) __attribute__ (( may_alias )) *one =
                ( ( const void * ) curve->one );
        weierstrass_t ( size ) __attribute__ (( may_alias ))
                *point = ( ( void * ) point0 );
        union {
                bigint_t ( size * 2 ) product;
                weierstrass_t ( size ) point;
        } __attribute__ (( may_alias )) *temp = ( ( void * ) temp0 );
        size_t offset;
        int is_infinite;
        unsigned int i;
        int rc;

        /* Initialise curve, if not already done
         *
         * The least significant element of the field prime must be
         * odd, and so the least significant element of the
         * (initialised) first multiple of the field prime must be
         * non-zero.
         */
        if ( ! prime2->element[0] )
                weierstrass_init_curve ( curve );

        /* Convert input to projective coordinates in Montgomery form */
        DBGC ( curve, "WEIERSTRASS %s point (", curve->name );
        for ( i = 0, offset = 0 ; i < WEIERSTRASS_AXES ; i++, offset += len ) {
                bigint_init ( &point->axis[i], ( data + offset ), len );
                DBGC ( curve, "%s%s", ( i ? "," : "" ),
                       bigint_ntoa ( &point->axis[i] ) );
                bigint_multiply ( &point->axis[i], square, &temp->product );
                bigint_montgomery_relaxed ( prime, &temp->product,
                                            &point->axis[i] );
        }
        memset ( &point->z, 0, sizeof ( point->z ) );
        is_infinite = bigint_is_zero ( &point->xy );
        bigint_copy ( one, &point->axis[ is_infinite ? 1 : 2 ] );
        DBGC ( curve, ")\n" );

        /* Verify point is on curve */
        if ( ( rc = weierstrass_verify ( curve, point ) ) != 0 )
                return rc;

        return 0;
}

References __attribute__, bigint_copy, bigint_init, bigint_is_zero, bigint_montgomery_relaxed, bigint_multiply, bigint_ntoa, bigint_t(), data, DBGC, weierstrass_curve::len, len, memset(), weierstrass_curve::name, offset, weierstrass_curve::one, weierstrass_curve::prime, product, rc, size, weierstrass_curve::size, weierstrass_curve::square, WEIERSTRASS_2N, WEIERSTRASS_AXES, weierstrass_init_curve(), weierstrass_t, and weierstrass_verify.

weierstrass_done_raw()

static void weierstrass_done_raw ( struct weierstrass_curve *  curve, bigint_element_t *  point0, bigint_element_t *  temp0, void *  out  )

static

Finalise curve point.

Parameters

curve	Weierstrass curve
point0	Element 0 of point (x,y,z)
temp0	Element 0 of temporary point buffer
out	Output buffer

Definition at line 857 of file weierstrass.c.

                                                                        {
        unsigned int size = curve->size;
        size_t len = curve->len;
        const bigint_t ( size ) __attribute__ (( may_alias )) *prime =
                ( ( const void * ) curve->prime[0] );
        const bigint_t ( size ) __attribute__ (( may_alias )) *fermat =
                ( ( const void * ) curve->fermat );
        const bigint_t ( size ) __attribute__ (( may_alias )) *one =
                ( ( const void * ) curve->one );
        weierstrass_t ( size ) __attribute__ (( may_alias ))
                *point = ( ( void * ) point0 );
        union {
                bigint_t ( size * 2 ) product;
                weierstrass_t ( size ) point;
        } __attribute__ (( may_alias )) *temp = ( ( void * ) temp0 );
        size_t offset;
        unsigned int i;

        /* Invert result Z co-ordinate (via Fermat's little theorem) */
        bigint_copy ( one, &temp->point.z );
        bigint_ladder ( &temp->point.z, &point->z, fermat,
                        bigint_mod_exp_ladder, prime, &temp->product );

        /* Convert result back to affine co-ordinates */
        DBGC ( curve, "WEIERSTRASS %s result (", curve->name );
        for ( i = 0, offset = 0 ; i < WEIERSTRASS_AXES ; i++, offset += len ) {
                bigint_multiply ( &point->axis[i], &temp->point.z,
                                  &temp->product );
                bigint_montgomery_relaxed ( prime, &temp->product,
                                            &point->axis[i] );
                bigint_grow ( &point->axis[i], &temp->product );
                bigint_montgomery ( prime, &temp->product, &point->axis[i] );
                DBGC ( curve, "%s%s", ( i ? "," : "" ),
                       bigint_ntoa ( &point->axis[i] ) );
                bigint_done ( &point->axis[i], ( out + offset ), len );
        }
        DBGC ( curve, ")\n" );
}

References __attribute__, bigint_copy, bigint_done, bigint_grow, bigint_ladder, bigint_mod_exp_ladder(), bigint_montgomery, bigint_montgomery_relaxed, bigint_multiply, bigint_ntoa, bigint_t(), DBGC, weierstrass_curve::fermat, weierstrass_curve::len, len, weierstrass_curve::name, offset, weierstrass_curve::one, out, weierstrass_curve::prime, product, size, weierstrass_curve::size, WEIERSTRASS_AXES, and weierstrass_t.

weierstrass_is_infinity()

int weierstrass_is_infinity	(	struct weierstrass_curve *	curve,
		const void *	point
	)

Check if this is the point at infinity.

Parameters

point

Curve point

Return values

is_infinity

This is the point at infinity

Definition at line 918 of file weierstrass.c.

                                                  {
        unsigned int size = curve->size;
        size_t len = curve->len;
        struct {
                bigint_t ( size ) axis;
        } temp;
        size_t offset;
        int is_finite = 0;
        unsigned int i;

        /* We use all zeroes to represent the point at infinity */
        DBGC ( curve, "WEIERSTRASS %s point (", curve->name );
        for ( i = 0, offset = 0 ; i < WEIERSTRASS_AXES ; i++, offset += len ) {
                bigint_init ( &temp.axis, ( point + offset ), len );
                DBGC ( curve, "%s%s", ( i ? "," : "" ),
                       bigint_ntoa ( &temp.axis ) );
                is_finite |= ( ! bigint_is_zero ( &temp.axis ) );
        }
        DBGC ( curve, ") is%s infinity\n", ( is_finite ? " not" : "" ) );

        return ( ! is_finite );
}

References bigint_init, bigint_is_zero, bigint_ntoa, bigint_t(), DBGC, weierstrass_curve::len, len, weierstrass_curve::name, offset, size, weierstrass_curve::size, and WEIERSTRASS_AXES.

weierstrass_multiply()

int weierstrass_multiply	(	struct weierstrass_curve *	curve,
		const void *	base,
		const void *	scalar,
		void *	result
	)

Multiply curve point by scalar.

Parameters

curve	Weierstrass curve
base	Base point
scalar	Scalar multiple
result	Result point to fill in

Return values

rc	Return status code

Definition at line 951 of file weierstrass.c.

                                                              {
        unsigned int size = curve->size;
        size_t len = curve->len;
        const bigint_t ( size ) __attribute__ (( may_alias )) *one =
                ( ( const void * ) curve->one );
        struct {
                weierstrass_t ( size ) result;
                weierstrass_t ( size ) multiple;
                bigint_t ( bigint_required_size ( len ) ) scalar;
        } temp;
        int rc;

        /* Convert input to projective coordinates in Montgomery form */
        if ( ( rc = weierstrass_init ( curve, &temp.multiple, &temp.result,
                                       base ) ) != 0 ) {
                return rc;
        }

        /* Construct identity element (the point at infinity) */
        memset ( &temp.result, 0, sizeof ( temp.result ) );
        bigint_copy ( one, &temp.result.y );

        /* Initialise scalar */
        bigint_init ( &temp.scalar, scalar, len );
        DBGC ( curve, "WEIERSTRASS %s scalar %s\n",
               curve->name, bigint_ntoa ( &temp.scalar ) );

        /* Perform multiplication via Montgomery ladder */
        bigint_ladder ( &temp.result.all, &temp.multiple.all, &temp.scalar,
                        weierstrass_add_ladder, curve, NULL );

        /* Convert result back to affine co-ordinates */
        weierstrass_done ( curve, &temp.result, &temp.multiple, result );

        return 0;
}

References __attribute__, base, bigint_copy, bigint_init, bigint_ladder, bigint_ntoa, bigint_required_size, bigint_t(), DBGC, weierstrass_curve::len, len, memset(), weierstrass_curve::name, NULL, weierstrass_curve::one, rc, result, size, weierstrass_curve::size, weierstrass_add_ladder(), weierstrass_done, weierstrass_init, and weierstrass_t.

weierstrass_add_once()

int weierstrass_add_once	(	struct weierstrass_curve *	curve,
		const void *	addend,
		const void *	augend,
		void *	result
	)

Add curve points (as a one-off operation)

Parameters

curve	Weierstrass curve
addend	Curve point to add
augend	Curve point to add
result	Curve point to hold result

Return values

rc	Return status code

Definition at line 998 of file weierstrass.c.

                                          {
        unsigned int size = curve->size;
        struct {
                weierstrass_t ( size ) addend;
                weierstrass_t ( size ) augend;
                weierstrass_t ( size ) result;
        } temp;
        int rc;

        /* Convert inputs to projective coordinates in Montgomery form */
        if ( ( rc = weierstrass_init ( curve, &temp.addend, &temp.result,
                                       addend ) ) != 0 ) {
                return rc;
        }
        if ( ( rc = weierstrass_init ( curve, &temp.augend, &temp.result,
                                       augend ) ) != 0 ) {
                return rc;
        }

        /* Add curve points */
        weierstrass_add ( curve, &temp.augend, &temp.addend, &temp.result );

        /* Convert result back to affine co-ordinates */
        weierstrass_done ( curve, &temp.result, &temp.addend, result );

        return 0;
}

References rc, result, size, weierstrass_curve::size, weierstrass_add, weierstrass_done, weierstrass_init, and weierstrass_t.