Ristretto255
Ristretto is a new unified point compression format for curves over large-characteristic fields, which divides the curve’s cofactor by 4 or 8 at very little cost of performance, efficiently implementing a prime-order group.
libsodium 1.0.18+ implements ristreto255: ristretto on top of the Curve25519 curve.
Compared to Curve25519 points encoded as their coordinates, ristretto makes it easier to safely implement protocols originally designed for prime-order groups.
Example
Perform a secure two-party computation of f(x) = p(x)^k. x is the input sent to the second party by the first party after blinding it using a random invertible scalar r, and k is a secret key only known by the second party. p(x) is a hash-to-group function.
// -------- First party -------- Send blinded p(x)
unsigned char x[crypto_core_ristretto255_HASHBYTES];
randombytes_buf(x, sizeof x);
// Compute px = p(x), a group element derived from x
unsigned char px[crypto_core_ristretto255_BYTES];
crypto_core_ristretto255_from_hash(px, x);
// Compute a = p(x) * g^r
unsigned char r[crypto_core_ristretto255_SCALARBYTES];
unsigned char gr[crypto_core_ristretto255_BYTES];
unsigned char a[crypto_core_ristretto255_BYTES];
crypto_core_ristretto255_scalar_random(r);
crypto_scalarmult_ristretto255_base(gr, r);
crypto_core_ristretto255_add(a, px, gr);
// -------- Second party -------- Send g^k and a^k
unsigned char k[crypto_core_ristretto255_SCALARBYTES];
randombytes_buf(k, sizeof k);
// Compute v = g^k
unsigned char v[crypto_core_ristretto255_BYTES];
crypto_scalarmult_ristretto255_base(v, k);
// Compute b = a^k
unsigned char b[crypto_core_ristretto255_BYTES];
if (crypto_scalarmult_ristretto255(b, k, a) != 0) {
return -1;
}
// -------- First party -------- Unblind f(x)
// Compute vir = v^(-r)
unsigned char ir[crypto_core_ristretto255_SCALARBYTES];
unsigned char vir[crypto_core_ristretto255_BYTES];
crypto_core_ristretto255_scalar_negate(ir, r);
crypto_scalarmult_ristretto255(vir, ir, v);
// Compute f(x) = b * v^(-r) = (p(x) * g^r)^k * (g^k)^(-r)
// = (p(x) * g)^k * g^(-k) = p(x)^k
unsigned char fx[crypto_core_ristretto255_BYTES];
crypto_core_ristretto255_add(fx, b, vir);
Encoded element validation
int crypto_core_ristretto255_is_valid_point(const unsigned char *p);
The crypto_core_ristretto255_is_valid_point() function checks that p is a valid ristretto255-encoded element.
This operation only checks that p is in canonical form.
The function returns 1 on success, and 0 if the checks didn't pass.
Random group element
void crypto_core_ristretto255_random(unsigned char *p);
Fills p with the representation of a random group element.
Hash-to-group
int crypto_core_ristretto255_from_hash(unsigned char *p, const unsigned char *r);
The crypto_core_ristretto255_from_hash() function maps a 64 bytes vector r (usually the output of a hash function) to a group element, and stores its representation into p.
Scalar multiplication
int crypto_scalarmult_ristretto255(unsigned char *q, const unsigned char *n,
const unsigned char *p);
The crypto_scalarmult_ristretto255() function multiplies an element represented by p by a scalar n (in the [0..L[ range) and puts the resulting element into q.
q should not be used as a shared key prior to hashing.
The function returns 0 on success, or -1 if p is the zero element.
int crypto_scalarmult_ristretto255_base(unsigned char *q, const unsigned char *n);
The crypto_scalarmult_ristretto255_base() function multiplies the generator by a scalar n ([0..L[ range) and puts the resulting element into q.
The function returns -1 if n is 0, and 0 otherwise.
Element addition/substraction
int crypto_core_ristretto255_add(unsigned char *r,
const unsigned char *p, const unsigned char *q);
The crypto_core_ristretto255_add() function adds the element represented by p to the element q and stores the resulting element into r.
The function returns 0 on success, or -1 if p and/or q are not valid encoded elements.
int crypto_core_ristretto255_sub(unsigned char *r,
const unsigned char *p, const unsigned char *q);
The crypto_core_ristretto255_sub() function substracts the element represented by p to the element q and stores the resulting element into r.
The function returns 0 on success, or -1 if p and/or q are not valid encoded elements.
Scalar arithmetic over L
Scalars should ideally be randomly chosen in the [0..L[ interval, L being the order of the group (2^252 + 27742317777372353535851937790883648493).
This can be achieved with the following function:
void crypto_core_ristretto255_scalar_random(unsigned char *r);
crypto_core_ristretto255_scalar_random() fills r with a crypto_core_ristretto255_SCALARBYTES bytes representation of the scalar in the ]0..L[ interval.
A scalar in the [0..L[ interval can also be obtained by reducing a possibly larger value:
void crypto_core_ristretto255_scalar_reduce(unsigned char *r, const unsigned char *s);
The crypto_core_ristretto255_scalar_reduce() function reduces s to s mod L and puts the crypto_core_ristretto255_SCALARBYTES integer into r.
Note that s is much larger than r (64 bytes vs 32 bytes). Bits of s can be left to 0, but the interval s is sampled from should be at least 317 bits to ensure almost uniformity of r over L.
int crypto_core_ristretto255_scalar_invert(unsigned char *recip, const unsigned char *s);
The crypto_core_ristretto255_scalar_invert() function computes the multiplicative inverse of s over L, and puts it into recip.
void crypto_core_ristretto255_scalar_negate(unsigned char *neg, const unsigned char *s);
The crypto_core_ristretto255_scalar_negate() function returns neg so that s + neg = 0 (mod L).
void crypto_core_ristretto255_scalar_complement(unsigned char *comp, const unsigned char *s);
The crypto_core_ristretto255_scalar_complement() function returns comp so that s + comp = 1 (mod L).
void crypto_core_ristretto255_scalar_add(unsigned char *z,
const unsigned char *x, const unsigned char *y);
The crypto_core_ristretto255_scalar_add() function stores x + y (mod L) into z.
void crypto_core_ristretto255_scalar_sub(unsigned char *z,
const unsigned char *x, const unsigned char *y);
The crypto_core_ristretto255_scalar_sub() function stores x - y (mod L) into z.
void crypto_core_ristretto255_scalar_mul(unsigned char *z,
const unsigned char *x, const unsigned char *y);
The crypto_core_ristretto255_scalar_mul() function stores x * y (mod L) into z.
Constants
crypto_scalarmult_ristretto255_BYTEScrypto_scalarmult_ristretto255_SCALARBYTEScrypto_core_ristretto255_BYTEScrypto_core_ristretto255_HASHBYTEScrypto_core_ristretto255_SCALARBYTEScrypto_core_ristretto255_NONREDUCEDSCALARBYTES
Notes
As Ristretto encodes a field element that is always smaller than 2^255, the top bit is not used.
It can be thus used by applications to encode additional data.
Forcing the top bit to zero in order to ignore it:
s[31] &= 0x7f;
Rejecting a key that has the top bit set:
if ((s[31] & 0x80) != 0) {
return;
}
Algorithms
ristretto255