CEILIDH

For the social dance, see Céilidh

CEILIDH is a public key cryptosystem based on the discrete logarithm problem in algebraic torus. This idea was first introduced by Alice Silverberg and Karl Rubin in 2003. The main advantage of the system is the reduced size of the keys for the same security over basic schemes.

The name CEILIDH comes from the Scots Gaelic word ceilidh which means a traditional Scottish Gathering.

Algorithms

Parameters

Let $q$ be a prime power.
An integer is chosen such that :
- The torus $T_n$ has an explicit rational parametrization.
- $\Phi_n(q)$ is divisible by a big prime $l$ where $\Phi_n$ is the $n^{th}$ Cyclotomic polynomial.
Let $m=\phi(n)$ where $\phi$ is the Euler function.
Let $\rho : T_n(\mathbb{F}_q) \rightarrow {\mathbb{F}_q}^m$ a birational map and its inverse $\psi$ .
Choose $\alpha \in T_n$ of order $l$ and let $g=\rho(\alpha))$ .

Key agreement scheme

This Scheme is based on the Diffie-Hellman key agreement.

Alice chooses a random number $a\ \pmod{\Phi_n(q)}$ .
She computes $P_A= \rho(\psi(g)^a) \in \mathbb{F}_q^m$ and sends it to Bob.

Bob chooses a random number $b\ \pmod{\Phi_n(q)}$ .
He computes $P_B= \rho(\psi(g)^b) \in \mathbb{F}_q^m$ and sends it to Alice.

Alice computes $\rho(\psi(P_B))^a) \in \mathbb{F}_q^m$
Bob computes $\rho(\psi(P_A))^b) \in \mathbb{F}_q^m$

$\psi \circ \phi$ is the identity, thus we have : $\rho(\psi(P_B))^a) = \rho(\psi(P_A))^b) = \rho(\psi(g)^{ab})$ which is the shared secret of Alice and Bob.

Encryption scheme

This scheme is based on the ElGamal encryption.

Key Generation
- Alice chooses a random number $a\ \pmod{ \Phi_n(q)}$ as her private key.
- The resulting public key is $P_A= \rho(\psi(g)^a) \in \mathbb{F}_q^m$ .

Encryption
- The message $M$ is an element of $\mathbb{F}_q^m$ .
- Bob chooses a random integer $k$ in the range $1\leq k \leq l-1$ .
- Bob computes $\gamma = \rho(\psi(g)^k) \in \mathbb{F}_q^m$ and $\delta = \rho(\psi(M)\psi(P_A)^k) \in \mathbb{F}_q^m$ .
- Bob sends the ciphertext $(\gamma,\delta)$ to Alice.

Decryption
- Alice computes $M = \rho(\psi(\delta)\psi(\gamma)^{-a})$ .

Security

The CEILIDH scheme is based on the ElGamal scheme and thus has similar security properties.

If the computational Diffie-Hellman assumption holds the underlying cyclic group $G$ , then the encryption function is one-way.^[1]

If the decisional Diffie-Hellman assumption (DDH) holds in $G$ , then CEILIDH achieves semantic security.^[1] Semantic security is not implied by the computational Diffie-Hellman assumption alone.^[2] See decisional Diffie-Hellman assumption for a discussion of groups where the assumption is believed to hold.

CEILIDH encryption is unconditionally malleable, and therefore is not secure under chosen ciphertext attack. For example, given an encryption $(c_1, c_2)$ of some (possibly unknown) message $m$ , one can easily construct a valid encryption $(c_1, 2 c_2)$ of the message $2m$ .

References

↑ 1.0 1.1 CRYPTUTOR, "Elgamal encryption scheme"
↑ M. Abdalla, M. Bellare, P. Rogaway, "DHAES, An encryption scheme based on the Diffie-Hellman Problem" (Appendix A)

Karl Rubin, Alice Silverberg: Torus-Based Cryptography. CRYPTO 2003: 349–365

External links

Torus-Based Cryptography — the paper introducing the concept (in PDF).

Public-key cryptography

Algorithms	Benaloh Blum–Goldwasser Cayley–Purser CEILIDH Cramer–Shoup Damgård–Jurik DH DSA EPOC ECDH ECDSA EdDSA EKE ElGamal signature scheme GMR Goldwasser–Micali HFE IES Lamport McEliece Merkle–Hellman MQV Naccache–Stern Naccache–Stern knapsack cryptosystem NTRUEncrypt NTRUSign Paillier Rabin RSA Okamoto–Uchiyama Schnorr Schmidt–Samoa SPEKE SRP STS Three-pass protocol XTR

Theory	Discrete logarithm Elliptic curve cryptography Non-commutative cryptography RSA problem

Standardization	CRYPTREC IEEE P1363 NESSIE NSA Suite B

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