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This software is distributed without any warranty. .\" .\" You should have received a copy of the CC0 Public Domain Dedication along .\" with this software. If not, see .\" .\" .Dd March 6, 2023 .Dt CRYPTO_ELLIGATOR_MAP 3MONOCYPHER .Os .Sh NAME .Nm crypto_elligator_map , .Nm crypto_elligator_rev , .Nm crypto_elligator_key_pair .Nd hiding of X25519 public keys .Sh SYNOPSIS .In monocypher.h .Ft void .Fo crypto_elligator_map .Fa "uint8_t curve[32]" .Fa "const uint8_t hidden[32]" .Fc .Ft int .Fo crypto_elligator_rev .Fa "uint8_t hidden[32]" .Fa "const uint8_t curve[32]" .Fa "uint8_t tweak" .Fc .Ft void .Fo crypto_elligator_key_pair .Fa "uint8_t hidden[32]" .Fa "uint8_t secret_key[32]" .Fa "uint8_t seed[32]" .Fc .Sh DESCRIPTION These functions allow obfuscating X25519 public keys by making them appear effectively indistinguishable from random noise. This is of interest for key exchange protocols that require indistinguishability from randomness, such as padded uniform random blobs (PURBs). They are intended for ephemeral (short-lived, possibly just one-time) X25519 keys, not for long-term public keys. After an initial key exchange involving hidden keys, subsequent key exchange messages should be encrypted instead; see, for example, the Noise Protocol Framework. This is an .Em advanced feature . Unless you are implementing an protocol that requires indistinguishability of all communications from random noise, consider .Xr crypto_x25519 3monocypher instead. Both this family of functions and .Xr crypto_x25519 3monocypher should be used as a building block to implement a key exchange protocol. .Pp For understanding what these functions do, it is important to note that a .Dq public key in this context refers to a .Em point on Curve25519 . This also means that these functions are not compatible with .Xr crypto_eddsa_sign 3monocypher and related functions. .Pp .Fn crypto_elligator_rev takes a public key .Fa curve and a .Fa tweak , hiding the public key so that it is effectively indistinguishable from random noise. Note that only .Xr crypto_x25519_dirty_fast 3monocypher or .Xr crypto_x25519_dirty_small 3monocypher can generate a suitable public key; the .Xr crypto_x25519 3monocypher function is insufficient. The .Fa tweak must be chosen at random. Even then, this operation .Em may fail because not all curve points are capable of being hidden. In this case, .Fn crypto_elligator_rev must be tried again with a new key pair, though .Fa tweak does not need to be changed. On average, two attempts are needed. Once a suitable public key has been found, .Fn crypto_elligator_rev always succeeds for it. Given the same values for .Fa tweak and .Fa curve , .Fn crypto_elligator_rev yields the same output value .Fa hidden . .Pp .Fn crypto_elligator_map performs the inverse operation: It decodes a hidden point to a curve point on Curve25519. .Pp .Fn crypto_elligator_key_pair is a convenience function that generates a secret key and its corresponding public key, which is effectively indistinguishable from random noise, from a random seed. .Em The execution time of this function is unpredictable because it may take many failures until a key pair could be generated successfully. .Fn crypto_elligator_key_pair uses .Xr crypto_x25519_dirty_fast 3monocypher internally; if code size is an important concern, its functionality can be replicated with .Xr crypto_x25519_dirty_small 3monocypher instead. .Pp The arguments are: .Bl -tag -width Ds .It Fa curve A point on the curve which is a Curve25519 public key generated with either .Xr crypto_x25519_dirty_fast 3monocypher or .Xr crypto_x25519_dirty_small 3monocypher . .It Fa hidden The hidden encoding of a point on the curve which is effectively indistinguishable from random. .It Fa secret_key The secret key that was generated from the given .Fa seed . .It Fa seed A 32-byte random number from which to derive a key pair. See .Xr intro 3monocypher for advice about generating random bytes (use the operating system's random number generator). The .Fa seed is wiped automatically. .It Fa tweak A 1-byte random number, which influences the final output of .Fn crypto_elligator_rev . .El .Pp The .Fa hidden and .Fa curve arguments may overlap or point at the same buffer. .Sh RETURN VALUES .Fn crypto_elligator_rev returns 0 on success and -1 if the given .Fa curve argument is unsuitable for hiding. .Pp .Fn crypto_elligator_map and .Fn crypto_elligator_key_pair return nothing. They cannot fail. .Sh EXAMPLES Generate a key pair manually using .Xr crypto_x25519_dirty_small 3monocypher instead of its fast variant: .Bd -literal -offset indent uint8_t sk [32]; /* Secret key output */ uint8_t pk [32]; /* Hidden public key output */ uint8_t tweak; /* Random tweak input */ arc4random_buf(&tweak, 1); for (;;) { arc4random_buf(sk, 32); crypto_x25519_dirty_small(pk, sk); if (crypto_elligator_rev(pk, pk, tweak) == 0) break; } /* Now save the secret key and send the hidden public key. */ .Ed .Pp Performing a key exchange with the other party's public key having been hidden: .Bd -literal -offset indent uint8_t hidden_pk [32]; /* Their hidden public key */ uint8_t their_pk [32]; /* Their unhidden public key */ uint8_t your_sk [32]; /* Your secret key */ uint8_t shared_key[32]; /* Shared session key */ crypto_elligator_map(their_pk, hidden_pk); crypto_x25519(shared_key, your_sk, their_pk); /* Wipe secrets if they are no longer needed */ crypto_wipe(your_sk, 32); .Ed .Sh SEE ALSO .Xr crypto_x25519 3monocypher , .Xr crypto_x25519_dirty_small 3monocypher , .Xr intro 3monocypher .Sh STANDARDS These functions implement the Elligator 2 mapping for Curve25519. This mapping is incompatible with both the hash-to-curve Internet draft and the implementation of Elligator 2 in libsodium. Elligator 2 was described in: .Rs .%A Daniel J. Bernstein .%A Mike Hamburg .%A Anna Krasnova .%A Tanja Lange .%T Elligator: Elliptic-curve points indistinguishable from uniform random strings .%J CCS '13: Proceedings of the 2013 ACM SIGSAC conference on Computer & communications security .%I Association for Computing Machinery .%D 2013 .%P pp. 967\(en980 .Re .Pp Monocypher's Elligator 2 representatives are encoded as little-endian 254-bit numbers. The two most significant bits (254 and 255) are not used. .Fn crypto_elligator_map ignores them, and .Fn crypto_elligator_rev sets them at random. The mapping uses 2 as the non-square .Va Z , and [0..2^254-10] as the set of non-negative field elements. See .Lk https://elligator.org/map for more details. .Sh HISTORY The .Fn crypto_curve_to_hidden , .Fn crypto_hidden_to_curve , and .Fn crypto_hidden_key_pair functions first appeared in Monocypher 3.1.0. In Monocypher 4.1.0, they were renamed .Fn crypto_elligator_rev , .Fn crypto_elligator_map , and .Fn crypto_elligator_key_pair respectively. .Sh CAVEATS Monocypher does not perform any input validation. Any deviation from the specified input and output length ranges results in .Sy undefined behaviour . Make sure your inputs are correct. .Sh SECURITY CONSIDERATIONS The secret keys for the public keys fed into .Fn crypto_elligator_rev .Sy must be chosen randomly rather than deterministically. Otherwise, the timing information given by the required number of retries also leaks information on the secret keys. .Pp These functions .Em help build highly difficult-to-analyse protocols but are insufficient by themselves: Other metadata, such as the number of bytes sent in a packet or the size of the 32-byte random looking string that represents the curve point itself, can be very strong indicators of the use of cryptography. Consider using appropriate padding algorithms, such as PADME, and obscure other metadata as much as possible.