## Friday, January 30, 2015

### Discrete Gaussian Sampling and the Quest for the Shortest Vector

The tl;dr:
When it comes to short vectors in lattices
The state-of-the-art apparatus is
Discrete Gaussian sampling
For use (for example) in
Reduced time and space cryptanalysis...
This week’s study group, presented by Joop, was on Solving the Shortest Vector Problem in 2n Time via Discrete Gaussian Sampling, a 2014 pre-print release by Aggarwal, Dadush, Regev and Stephens-Davidowitz. The lattice, to my regret, is not a creature I frequently encounter in the course of my own research, so I report here from a considerably less-than-expert perspective. But hopefully, if I stick to the high-level details then all shall be well.

The motivating theme of the paper is the Shortest Vector Problem (SVP) — an important computational problem on lattices: given a basis for a lattice, find a non-zero vector in the lattice of minimum Euclidean norm. Many cryptographic primitives (notably including breakthrough developments in fully homomorphic encryption) derive their security from the worst-case hardness of SVP and related problems.

Previous algorithms to find exact or approximate solutions to SVP fall into three main classes: enumeration, whereby basis reduction is combined with exhaustive search inside Euclidean balls (time ranging from 2O(n log n) to 2O(n2), polynomial space); sieving, whereby randomly sampled vectors are iteratively combined to generate increasingly short vectors (2cn time (c>2), exponential space);  and recent proposals using the Voronoi cell of a lattice -- the real region in which all points lie closer to the zero vector than to any other (22n time, exponential space).

The intuition behind Discrete Gaussian Sampling (DGS) is to draw from a (narrow) distribution of lattice points centred around zero. As the width of the distribution (characterised by the parameter s) decreases, the samples become more and more concentrated on short vectors; sufficiently many samples for an arbitrary s will eventually land on a solution to SVP. A simple idea but, of course, far from simple in practice. In particular, for any given lattice there is a smoothing parameter s* -- informally, the 'tipping point' at which the distribution begins to 'look like' a continuous Gaussian rather than a stack around the central point. Efficient algorithms are known for s much larger than s*, but these are inadequate to solve exact lattice problems. A key contribution of Aggarwal et al. is an algorithm that samples for any parameter s > 0; it returns 2n/2 independent discrete Gaussian distributed vectors using 2n+o(n) time and space. They add to this a second algorithm for "above smoothing" sampling with a substantially improved time and space complexity (2n/2 -- the current record for fastest provable running time of a hard lattice problem), and show that it can be used to approximate decision SVP to within a factor of 1.93.

Here's where I muster all my expertise-less enthusiasm and try to explain the (high level) details of the algorithms without saying anything too stupid…

Both algorithms operate by iteratively combining vectors drawn from carefully chosen high parameter distributions (from which it is well-known how to sample efficiently) in such a way that the dispersion parameter of the distribution of the combined vectors is progressively lowered. As an aid to intuition, consider, for a moment, the counterpart continuous case: the distribution of a (continuous) Gaussian-distributed vector divided by two is similarly Gaussian distributed with half the width. In the discrete case, though, dividing by two generally does not produce another vector in the lattice. A 'fix' would be to sample from L with parameter s', keep those that were also in 2L, and divide them by 2 -- but the "loss factor" of doing so can be high (the probability that a sample from L is also in 2L can be as small as 2-n), so one needs to be cleverer.

The 2n-time below-smoothing sampler looks for pairs of vectors sampled from lattice L with parameter s whose sum is in 2L (equivalently: vectors in the same coset mod 2L). The 'greedy' combiner which pairs as many as possible in each coset-related bucket would have a loss factor of just 2 in reducing from the sampled vectors to the shorter combined vectors. However, below the smoothing parameter the required distributional properties are lost in the combining step. The workaround to produce combined vectors with the correct distribution is very involved; I caught something about coin-flips and Poisson distributions, but the sterling efforts of our presenter to render comprehensible the technical details did not take sufficient root in my brain for me to attempt the same here -- I refer you to the paper at this point! From an input sample of order 2n, combining multiple times according to the method they devise reduces s as required with a total loss factor of 2n/2, thus outputting on the order of 2n/2 sampled vectors.

The 2n/2-time above-smoothing sampler is not just a tweak on the first algorithm but a surprisingly different approach supplied with its own set of clever tricks and insights. The intuitive idea is to construct a ‘tower’ of increasingly sparse lattices [L0,…,Lm] where Lm = L, the lattice one wishes to sample from. Each Li is chosen to ‘lie between’ Li+1 and 2Li+1 — that is, it is a (strict) sublattice of Li+1 which contains 2Li+1, i.e. Li+1Li ⊆ 2Li+1. Because L0 is dense, one can sample ‘easily’ from it with a small parameter (2-m/2s, in fact), and combine (by summing) over cosets of L1 to get samples over the sparser lattice 2L1 with a slightly increased parameter 2-(m-1)/2s. Repeating this process eventually produces samples from Lm = L with distribution statistically close to the discrete Gaussian with parameter s, provided s is above the smoothing parameter. They show that this method is able to approximate decision SVP to within a factor of 1.93.

That’s all we had time for in our session — and already, I admit, somewhat more than I have the capacity to fully absorb without some serious preliminary study! (Although, the bits I was able to follow, I found very interesting). The rest of the paper, as well as containing all the ‘proper maths’ (including the formal reduction from SVP to DGS), covers applications, adaptations to the closest vector and other related problems, and relevant points for discussion, all in some depth. Despite the clear theoretical value of their contribution, the authors are circumspect about the practical usages of their algorithms relative to enumeration and sieving methods. These latter perform well heuristically in relevant scenarios, whilst the run-time bounds of DGS are tight in practice, with no trade-off available even if one wants to sample fewer than 2n/2 vectors.

### 52 Things: Number 17: Describe and compare the round structure of DES and AES.

This is the latest in a series of blog posts to address the list of to do Cryptography: a set of questions compiled to give PhD candidates a sense of what they should know by the end of their first year.In this week, we describe and compare the round structure of DES and AES.

Both DES and AES are examples of iterated block ciphers. The block ciphers obtain their security by repeated use of a simple round function. The round function takes an n-bit block and returns an n-bit block, where n is the block size of the overall cipher. The number of rounds r can either be a variable or fixed. As a general rule increasing the number of rounds will increase the level of security of the block cipher. Each use of the round function employs a round key ki (where 1 ≤ i ≤ r) derived from the main secret key k, using an algorithm called a key schedule. To allow decryption, for every round key the function implementing the round must be invertible, and for decryption the round keys are used in the opposite order that they were used for encryption. In DES the functions needed to implement the round function are not invertible, but the whole round is invertible. For AES (Rijndael) not only is the whole round function invertible but every function used to create the round function is also invertible.
More particularly, the DES cipher is a variant of the basic Feistel cipher. The interesting property of a Feistel cipher is that the round function is invertible regardless of the choice of the function in the box marked F. To see this notice that each encryption round is given by:
Li = Ri-1
Ri = Li-1 ⊕ F(Ki,Ri-1).

Hence, the decryption can be performed via:
Ri-1 = Li
Li-1 = Ri ⊕ F(Ki,Li).

This way we can choose any function for the function F, and we will still obtain an encryption function which can be inverted using the secret key. The same code/circuitry can be used for the encryption and decryption functions. We only need to use the round keys in the reverse order for decryption. As a variant of the Feistel cipher design, DES includes the following distinct characteristics:
• the number of rounds r is 16,
• the block length n is 64 bits,
• the key length is 56 bits,
• the round keys K1,...,K16 are each 48 bits
• before and after the main Feistel iteration a permutation is performed.
In summary the DES cipher operates on 64 bits of plaintext in the following manner:
• Perform an initial permutation.
• Split the blocks into left and right half.
• Perform 16 rounds of identical operations (Festal cipher). In each round the, the F function consists of the following six stages:
• Expansion Permutation: The right half of 32 bits is expanded and permuted to 48 bits.
• Round Key Addition: The 48-bit output from the expansion permutation is XORed with the round key, which is also 48 bits in length.
• Splitting: The resulting 48-bit value is split into eight lots of six-bit values.
• S-Box: Each six-bit value is passed into one of eight different S-Boxes (Substitution Box) to produce a four-bit result. Each S-Box is a look-up table of four rows and sixteen columns. The six input bits specify which row and column to use. Bits 1 and 6 generate the row number, whilst bits 2, 3, 4 and 5 specify the column number. The output of each S-Box is the value held in that element in the table.
• P-Box: We now have eight lots of four-bit outputs which are then combined into a 32-bit value and permuted to form the output of the function F.
• Join the half blocks back together.
• Perform a final permutation.
The DES key schedule takes the 56-bit key, which is actually input as a bitstring of 64 bits comprising of the key and eight parity bits, for error detection. It first permutes the bits of the key (which takes a 64-bit input and produces a 56-bit output, hence discarding the parity bits). The output of this permutation, called PC-1 in the literature, is divided into a 28-bit left half C0 and a 28-bit right half D0. Now for each round we compute:
Ci=Ci−1 ≪ pi
Di=Di−1 ≪ pi

where x ≪ pi means perform a cyclic shift on x to the left by pi positions. Finally the two portions Ci and Di are joined back together and are subject to another permutation, called PC-2, to produce the final 48-bit round key.
Note that a key length of 56 bits is insufficient for many modern applications, hence often one uses DES by using three keys and three iterations of the main cipher. Such a version is called Triple DES or 3DES. In 3DES the key length is equal to 168. There is another way of using DES three times, but using two keys instead of three giving rise to a key length of 112. In this two-key version of 3DES one uses the 3DES basic structure but with the first and third key being equal. However, two-key 3DES is not as secure as one might initially think.

More details on actual values (S-Boxes, P-Boxes and all Permutation tables) can be found in [1].

The AES (Rijndael) algorithm, unlike DES, is a block cipher that does not rely on the basic design of the Feistel cipher. However, AES does have a number of similarities with DES. It uses a repeated number of rounds to obtain security and each round consists of substitutions and permutations, plus a key addition phase. AES in addition has a strong mathematical structure, as most of its operations are based on arithmetic in the field F28 . However, unlike DES the encryption and decryption operations are distinct.
AES identifies 32-bit words with polynomials in F28[X] of degree less than four. AES is a parametrized algorithm in that it can operate on block sizes of 128, 192 or 256 bits. It can also accept keys of size 128, 192 or 256 bits. For each combination of block and key size a different number of rounds is specified.
To make our discussion simpler we shall consider the simpler, and probably more used, variant which uses a block size of 128 bits and a key size of 128 bits, in which case 10 rounds are specified. AES operates on an internal four-by-four matrix (S(4,4)) of bytes, called the state matrix, which is usually held as a vector of four 32-bit words, each word representing a column. Each round key is also held as a four-by-four matrix [1]. The AES round function operates using a set of four operations:
• SubBytes: There are two types of S-Boxes used in Rijndael: One for the encryption rounds and one for the decryption rounds, each one being the inverse of the other. For the encryption S-Box each byte s = [s7,...,s0] of the state matrix is taken in turn and considered as an element of F28. The S-Box can be mathematically described in two steps:
1. The multiplicative inverse in F28 of s is computed to produce a new byte x = [x7, . . . , x0].
2. The bit-vector x is then mapped, via an affine F2 transformation [1], to a new bit-vector y. The new byte is given by y. The decryption S-Box is obtained by first inverting the affine transformation and then taking the multiplicative inverse.
• ShiftRows: The ShiftRows operation in AES performs a cyclic shift on the state matrix. Each row is shifted by different offsets [1]. The inverse of the ShiftRows operation is simply a similar shift but in the opposite direction. The ShiftRows operation ensures that the columns of the state matrix ‘interact’ with each other over a number of rounds.
• MixColumns: The MixColumns operation ensures that the rows in the state matrix ‘interact’ with each other over a number of rounds; combined with the ShiftRows operation it ensures each byte of the output state depends on each byte of the input state [1].
• AddRoundKey: The round key addition is particularly simple. One takes the state matrix and XORs it, byte by byte, with the round key matrix. The inverse of this operation is clearly the same operation.
The AES algorithm can be described using the pseudo-code:

for i = 1 to 9 do
SubBytes(S)
ShiftRows(S)
MixColumns(S)
end
SubBytes(S)
ShiftRows(S)

The message block to encrypt is assumed to be entered into the state matrix S. The output encrypted block is also given by the state matrix S.
The AES key schedule makes use of a round constant which we shall denote by:
RCi = xi (mod x8 + x4 + x3 + x + 1)
We label the round keys as (W4i, W4i+1, W4i+2, W4i+3) where i is the round. The initial main key is first divided into four 32-bit words (k0, k1, k2, k3). The round keys are then computed as algorithm below, where RotBytes is the function which rotates a word to the left by a single byte, and SubBytes applies the Rijndael encryption S-Box to every byte in a word [1].

W0 =K0,W1 =K1,W2 =K2,W3 =K3
for i = 1 to 10 do ￼￼￼￼￼￼
T = RotBytes(W4i−1
T = SubBytes(T)
T = T ⊕ RCi
W4i = W4i−4 ⊕ T
W4i+1 = W4i−3 ⊕ W4i
W4i+2 = W4i−2 ⊕ W4i+1
W4i+3 = W4i−1 ⊕ W4i+2
end
References: [1] http://www.cs.bris.ac.uk/~nigel/Crypto_Book/

## Friday, January 23, 2015

### 52 Things: Number 16: Describe the key generation, signature and verification algorithms for DSA, Schnorr and RSA-FDH.

This is the latest in a series of blog posts to address the list of '52 Things Every PhD Student Should Know To Do Cryptography': a set of questions compiled to give PhD candidates a sense of what they should know by the end of their first year. In this week, we describe the key generation, signing and verification algorithms of DSA, Schnorr and RSA-FDH.

1. DSA
The Digital Signature Scheme (DSA), also known as the Digital Signature Standard (DSS), was proposed by the National Institute of Standards and Technology (NIST) in 1991 [1]. Security of DSA is based on the difficult of computing discrete logarithms. But there is no known proof of its security under a standard assumption (like DL), even in the random oracle model

Domain Parameter Generation
1. Select a prime number $p$, where $2^{L-1}<p<2^L$ and $L$ is a multiple of 64 and $512 \leq L \leq 1024$.
2. Select a prime divisor $q$ of $p-1$, where $2^{159}<q<2^{160}$.
3. Compute a generator $g$ of the subgroup of order $q$: choose a random integer $r$, where $1<r<p-1$ such that $g=r^{(p-1)/q} \ mod \ p$ and $g \neq 1$.
Key Generation
1. Select a random integer $x$, where $0<x<q$.
2. Compute $y = g^x \ mod \ p$.
Then the public key is $y$ and the private key is $x$.

Signing
1. Select a random integer $k$, where $0<k<q$.
2. Compute $r = (g^k \ mod \ p)\ mod \ q$.
3. Compute $s = (h(m)+x\cdot r)\cdot k^{-1}\ mod \ q$, where $h(m)$ is hash of $m$ using SHA-1
The signature on $m$ is the pair $(r, s)$.

Verification
1. Compute $u_1 = h(m) \cdot s^{-1}\ mod \ q$.
2. Compute $u_2 = r \cdot s^{-1}\ mod \ q$.
3. Compute $v = (g^{u_1} \cdot y^{u_2}\ mod \ p)\ mod \ q$.
4. Output $1$ if $v = r$, otherwise output $0$.
Correctness
If $(r,s)$ is a valid signature on $m$, then we have
$v = g^{u_1}\cdot y^{u_2}=g^{h(m)\cdot (h(m)+x\cdot r)^{-1}\cdot k}\cdot {g^{x \cdot r \cdot (h(m)+x\cdot r)^{-1}\cdot k}} \ mod \ p$
$= g^{(h(m)+x\cdot r)\cdot (h(m)+x \cdot r)^{-1}\cdot k}\ mod \ p$
$= g^k \ mod \ p$
Thus the verification succeeds.

2. Schnorr
The Schnorr signature is an important DLP-based signature scheme. It works in any prime order group and its security is proven in the random oracle model under DL assumption [2].

Domain Parameter Generation
1. Select a prime number $p$.
2. Select a prime divisor $q$ of $p-1$.
3. Select a generator $g$ of the subgroup of order $q$.
Key Generation
1. Select a random integer $x$, where $0<x<q$.
2. Compute $y=g^x \ mod \ p$.
The public key is $y$ and the private key is $x$.

Signing
1. Select a random integer $k$, where $0<x<q$.
2. Compute $a = g^k \ mod \ p$.
3. Compute $r = h(m \| a)$, where $m$ is the message to be signed and $h:\{0,1\}^* \rightarrow \mathcal{Z}_{q}$ is a hash function.
4. Compute $s = (k + r\cdot x)\ mod \ q$
The signature on $m$ is the pair $(r,s)$.

Verification
1. Compute $v = g^s \cdot y^{-r}\ mod \ p$
2. Output $1$ if $v = r$, otherwise output $0$.
Correctness
If $(r,s)$ is a valid signature on $m$, then we have
$v=g^s\cdot y^{-r}=g^{k+r\cdot x}\cdot g^{-r\cdot x}=g^k=r$
Thus the verification succeeds.

3. RSA-FDH
The RSA-FDH (full domain hash) scheme was introduced by Bellare and Rogaway in [3]. It is a RSA-based signature scheme and follows the hash-then-sign paradigm. It makes use of the hash function (the image size of the hash function equals to RSA modulus) to generate random-looking output for the plain RSA signature scheme. Thus it prevents the algebraic attacks on the plain RSA signature scheme and it is able to sign messages of arbitrary length. But it is hard to create such hash function in practice. RSA-FDH can be proven EU-CMA secure in the random oracle model

Key Generation
1. Select two large primes $p$ and $q$.
2. Compute $N=p\cdot q$.
3. Select a random integer $e$, where $1<e<\phi(N)$, such that $gcd(e,\phi(N))=1$.
4. Compute the integer $d$, where $1<d<\phi(N)$, such that $e\cdot d = 1\ mod \ \phi(N)$.
The public key is $(N,e)$ and the private key is $(d,p,q)$.

Signing
1. Compute $s=h(m)^d\ mod \ N$, where $m$ is the message to be signed and $h:\{0,1\}^* \rightarrow \mathcal{Z}_N$ is a hash function.
The signature on $m$ is $s$.

Verification
1. Output $1$ if $s^e = h(m)\ mod \ N$, otherwise output $0$.
Correctness
If $s$ is a valid signature on $m$, then we have
$s^e=h(m)^{d\cdot e} mod \ N=h(m)\ mod \ N$
Thus the verification succeeds.

[1]http://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf
[2]http://eprint.iacr.org/2012/029.pdf
[3]http://web.cs.ucdavis.edu/~rogaway/papers/exact.pdf

## Friday, January 16, 2015

### 52 Things: Number 15: Key generation, encryption and decryption algorithms for RSA-OAEP and ECIES.

This is the latest in a series of blog posts to address the list of   to do Cryptography: a set of questions compiled to give PhD candidates a sense of what they should know by the end of their first year. We came back to "more crypto" staff by describing the key generation, encryption and decryption algorithms for RSA-OAEP and ECIES.

1. RSA-OAEP
RSA-OAEP  stands for RSA encryption/decryption scheme and OAEP padding scheme respectively. They are often used conjointly in real world.

1.1 RSA[1]
RSA is one of the earliest public key encryption scheme that has been deployed widely in real world. It is based on the assumption of the hardness of RSA problem which has been described in previous blog (here).

Key Generation:
1.  Generate two large primes $p$, $q$, and compute the modulo $N = pq$.
2.  Selects a random number $e \in \mathbb{Z}_N$ s.t. $gcd(\phi(N),e)=1$ where $gcd$ stands for Greatest Common Divisor
3.  Since $\phi(N)$ and $e$ are co-prime ($gcd(\phi(N),e) = 1$), we can use XGCD to find the multiplicative inverse $d$ of $e$ over modulo $\phi(N)$: $d = e^{-1} \mod \phi(N)$.
4. We distribute $(N,e)$ as our public key and hold $(p,q,d)$ as our secret key.
Encryption:
1. Parse the message to an integer $m \in \mathbb{Z}_N$.
2. Compute $c = m^e \mod N$.
3. Output $c$ as our ciphertext.
Decryption:
Before we receive the ciphertext, we precompute some values: $d \mod p-1$, $q^{-1} \mod p$, $d \mod q-1$ and $p^{-1} \mod q$.
Then upon receiving the ciphertext $c$, we
1. Compute$$m = ((c ^{d \mod p-1}\mod p)q({q^{-1} \mod p})\\ + (c^{d \mod q-1}\mod q)p({p^{-1} \mod q})) \mod N$$
2. Output $m$ as our plaintext.
Notice that the computation of $m$ is in fact $m=c^d \mod N$ using CRT. The reason is that performing exponentiations over small modulo($p$ and $q$) are faster than doing it over large modulos($N$). The decryption works by applying  Fermat's Little Theorem and we will have $c^d =m^{ed}= m ^{1 \mod \phi(N)} = m (\mod N)$ .

1.2 OAEP[2]
OAEP stands for Optimal Asymmetric Encryption Padding. It is a padding scheme used together with asymmetric encryption (usually RSA). It can bring some randomness to a deterministic encryption scheme. When used with RSA, the combined scheme is proven to be IND-CCA secure.

Let
• $f$ be a $k$-bit trapdoor one-way permutation. $f:\{0,1\}^k \rightarrow \{0,1\}^k$
• $m$ be the $n$-bit message
• $G$, $H$ be two psudorandom functions: $G: \{0,1\}^s \rightarrow \{0,1\}^{n+t}$ and $H:\{0,1\}^{n+t} \rightarrow \{0,1\}^s$, where $k = n + t + s$
• $R$ be $s$-bit random number: $R \leftarrow \{0,1\}^s$
Encryption:
We compute the $k$-bit ciphertext as follows:

Encrypt(m) = f_{pk}(\{(m||0^t) \oplus G(R) \}||\{R \oplus H((m||0^t) \oplus G(R))\})

Decryption:
By using the trapdoor, we can recover the following value:

f_{sk}(c) =  \{(m||0^t) \oplus G(R) \}||\{R \oplus H((m||0^t) \oplus G(R))\}

Then
1.  Let the first $n+t$ bits be $T$: $T = (m||0^t) \oplus G(R)$, and the other $s$ bits be $S$: $S = R \oplus H((m||0^t) \oplus G(R))$
2.  Compute $R$ as $R=H(T) \oplus S$
3.  Compute $m||0^t = T \oplus G(R)$
4. Verify if there is exactly $t$ 0s following the $n$-bit message $m$. If validated, remove the $t$ 0 bits and output $m$.
In practice, we replace $f_{pk}$ and $f_{sk}$ by RSA encryption and decryption function respectively.

ECIES

Elliptic Curve Integrated Encryption Scheme is a variation of ElGamal public key encryption scheme based on Elliptic Curve Cryptography (click here to find more about elliptic curve).

For simplicity, we define an Elliptic Curve in the form:

E: y^2 = x^3 + ax +b

To further simplify our problem, we only discuss a curve $E$ on a prime field $\mathbb{F}_q$ with a base point $P$ having a prime order $n$. Then we can define a simplified domain parameter: $D = (q, a, b, P, n)$ where:
• $q$ is the prime field order. i.e. $q$ is a prime and $x, y, a, b$ are reduced to $\{0, 1, 2, ..., q-1\}$
• $a,b$ are the coefficients of the curve.
• $P$ is a point on the curve.
• $n$ is the prime order of $P$. i.e. additions of $P$ yields $n$ points on the curve where $n$ is a prime.
The domain parameter are made public.

ECIES are always associated with a symmetric encryption scheme and a MAC scheme. We denote them as $\{Enc_k(m)=c, Dec_k(c)=m\}$ and $\{MAC_k(m)=t, Very(t,m)=T/F\}$ respectively.

We also denote $KDF(s_1, s_2) = (k_{enc}, k_{MAC})$ as the Key Derivation Function which takes two seeds $s_1, s_2$ and outputs a pair of symmetric encryption key and MAC key.

Then we describe the scheme as:

Key Generation:
1. Pick a random integer $d \in [1, n - 1]$.
2. Compute a new point $Q = dP$.
3.  Output $Q$ as the public key and $d$ as the secret key.
Then the encryption of a message $m$ is done as follows:

Encryption:
1.  Pick a random integer $k \in [1, n-1]$.
2.  Compute $R=kP, Z=kQ$. If $Z=\infty$ then we restart the process and pick a different $k$.
3.  Generate $(k_1, k_2) = KDF(x_Z, R)$ where $x_Z$ is the $x$-coordinate of $Z$.
4. Compute $c = Enc_{k_1}(m)$ and $t=MAC_{k_2}(c)$.
5. Output $(R, c, t)$ as the ciphertext.
On receiving a ciphertext $(R,c,t)$,

Decryption:
1.  Verify if $R$ is valid. This can be easily done by substituting $R$ in to the curve.
2. Compute $Z'=dR$.
3. Generate $(k'_1, k'_2) = KDF(x_{Z'}, R)$, where $x_{Z'}$ is the $x$-coordinate of $Z'$.
4. Verify if MAC is correct by calling $Very(t, c)$.
5. Decrypt $m' = Dec_{k'_1}$.
6. Output $m'$ as the plaintext.
Since $Z' = dR = dkP = kQ = Z$; therefore seeds fed to KDF() are in fact same. Hence receiver can generate keys as same as the sender's and decrypt the message.

However, my knowledge to ECC is very limited. For those who are interested, you can find more in [4].

References:
[1] http://people.csail.mit.edu/rivest/Rsapaper.pdf
[2] http://tools.ietf.org/html/rfc2437
[3] http://www.shoup.net/papers/iso-2_1.pdf
[4] http://www.springer.com/computer/security+and+cryptology/book/978-0-387-95273-4

## Sunday, January 11, 2015

### Real World Crypto 2015: Are you there Bob? It's me, Alice.

Social media are all about the three F's: friends, fans and followers. But what if you want to keep your list of F-buddies private? On the final day of the Real World Crypto workshop Ian Goldberg spoke about how we can advertise our presence online to our friends without revealing our relationship graph.

In the 90s, people would fire up ICQ instant messenger and an ear-splitting foghorn sound effect would announce their presence online to everyone on their street. Friends further away relied on notification from the ICQ server, with the server knowing when users are online and who their friends are. In today's more privacy-sensitive climate this is undesirable, as the service operator may be legally compelled to surrender this data to governments on request.

The Dagstuhl Privacy Preserving Presence Protocol P (DP5 - the extra P is for patter) allows a service operator to provide this online presence information without learning the information itself, and so freely comply with search warrants without compromising user privacy.

The DP5 protocol keeps a friendship database and a presence database, and assumes two parties wishing to communicate already a hold shared key. Time is divided into long-term epochs T over which friendship data can be updated, and short-term epochs t over which status data can be updated.

In each long-term epoch T, Alice evaluates two PRFs at the point T under the key she shares with Bob in order to generate a public identifier ID and an epoch key K. She generates a public key P and encrypts it under the epoch key. This ciphertext C and the ID are stored in the friendship database.

In each short-term epoch t, Alice evaluates a third PRF at the point t under a key derived through hashing her public key P, generating an encryption key k. She then encrypts a status message under key k. This ciphertext c and an identifier derived from P are stored in the presence database.

When Bob wants to know if Alice is online, he evaluates the appropriate PRFs at point T under their shared key to recover the epoch key K and identifier ID. He pulls the corresponding record from the friendship database and decrypts the ciphertext C to recover Alice's public key P. He then computes from P the key k and the identifier for the presence database, pulling the corresponding record and decrypting the ciphertext under k. If decryption is successful then Alice is online and her status message is revealed, otherwise he concludes that Alice is offline.

This is a simplified explanation of the DP5 protocol, which makes use of Private Information Retrieval to ensure the server is unaware of the nature of the database queries. PIR is a beast, so the protocol doesn't scale too well, but it does provide a private way to tell your friends you're online and let them know your status.

***** X_Br1sT0L_B10gGeR_X is offline *****

## Friday, January 9, 2015

### Real World Crypto 2015: “Designers: ask not what your implementer can do for you, but what you can do for your implementer” Dan Bernstein, Real World Crypto 2015

Last year at Real World Crypto you may remember my blog post about how the conference went about the business of lamented the downfall of theoretically secure cryptographic primitives based on practical implementation issues. Well this year at real world crypto things have gone in a similar direction. An example of this is Bernstein's talk on Wednesday entitled “Error-prone cryptographic designs.” Much of the talk has already been described in a previous post by David B here which helpfully gives us the “Bernstein principle” which I'll follow up on, perhaps from a more applied point of view.

As the title suggests the talk was about error prone cryptography, but not as you might expect. The talk was not a frustrated outburst at how the practical implementation side of things was letting the side down and needed to get it's act together, the tone of which you may be familiar with from other talks with similar titles. The talk did contain many of the ingredients of this sort of talk. There was a look at how the problems with cryptosystems seem always to come from the implementation of secure primitives not the primitives themselves, backed up by a number of horror stories showing this for instance. But this talk had one major twist.

The main point of the presentation was made near the beginning in one of the horror stories (see blog post), when he referred to the breaking of the encryption used by Sony Playstation. Relating to this he observed how the blame for the attack got distributed. The attack was a practical attack and so naturally the designers of the primitives remarked how their secure designs had been made vulnerable by the terrible implementers who seem to keep getting things wrong and making all their hard work meaningless. But wait a minute, asked the implementers, if we keep getting what you designs wrong, perhaps the problem is with you? Perhaps the designs of the primitives are so hard to implement properly that the designers are really to blame for not taking implementation practicalities seriously enough in their design choices.

So who was to blame for this and many other security slip ups that have occurred through implementation errors? Well the answer to the question still remains unresolved, but the fact that the discussion is taking place throws a different kind of light on the practice of designing crypto systems. The current system of designer make something secure; implementer implement the design securely, may well be where many of the problems are coming from and should be replaced by something more like designer make something that's secure and easy to implement; implementer … you can't really go wrong!

Although unlikely to have drawn the agreement of the whole audience, the talk didn't have the feel of a dig at designers but an appeal to them to wake up to their responsibility to the implementers in making designs that are easy to implement. It neither had the air of trying to get implementers off the hook, but sought rather to look at the question as to why in a world where secure systems seem to break because of practical attacks against implementations rather than the security of the primitives aren't the primitives ever singled out to blame for being hard to implement rather than the people given the task of implementing them.

It's true, designers have a hard problem on their hands making sure what they design is secure, but often being from a more mathematical, theoretical background, have they tended to overlook the practical aspect of what they are concocting? Many designers are highly skilled in the art of theoretical design but is this at the expense of not knowing the real engineering world in which they work as well as they should?

The talk examined these and other similar questions and was summarised with an appeal to designers to “think of the children, think of the implementers” and ended with the words given as the title for this post.

### Real World Crypto 2015: CAESAR came, you see, but who will win?

Cryptographers are humans, and humans are competitive, which might explain the popularity of cryptographic competition. The most visible competition running at this point in time is CAESAR, which hopes to recommend a (portfolio of) authenticated encryption. Elena Andreeva gave a wonderful talk giving a very brief overview of modern thinking regarding what security and functionality to aim for; and how different schemes try to achieve it.

The traditional lesson that encryption should be probabilistic to be secure, has been replaced by a nonce-based view, with further granularity depending on the security when nonces are somehow reused. Even prior to CAESAR there were an increasing number of dedicated, nonce-based schemes suggested, several of which were entered into the CAESAR competition. Taking into account nonces also changes how one would go about using a MAC to add authenticity to a mode-of-operation such as CBC. This problem, known as generic composition, has recently been revisited by Namprempre, Rogaway, and Shrimpton.

There has also been increased interest in the security ramifications of using online encryption and decryption. For efficiency purposes (e.g. to enable a single pass over the data), it can be beneficial to output plaintext as part of the decryption process before the ciphertext has been verified. At last year's Asiacrypt, Elena and her coauthors established a framework to capture the release of unverified plaintext (RUP). Only a handful of CAESAR candidates achieves this security notion.

Finally, Elena went into more detail of the current state of play of the CAESAR competition. Given the large number of candidates (57) and the large number of criteria based on which one can classify an authenticated encryption scheme, the information provided by the CAESAR zoo can be hard to interpret. Luckily for us, Elena has created a wonderful, interactive visualization tool to zoom in on whichever property takes our current fancy (for instance, geographical spread of submissions). The tool is available from her home page so have a go at it yourself!

The first round of CAESAR is drawing to a close and it is expected that next month DJB (J for Julius?) will announce which candidates the CAESAR committee has progressed to round 2. Eventually the winner or winners will be crowned in another two years' time. After her presentation, there was a question from the audience whether this winner will be fast tracked into the TLS standard.

## Thursday, January 8, 2015

### 52 Things: Number 14: What is a cryptographic pairing?

This is the latest in a series of blog posts to address the list of '52 Things Every PhD Student Should Know' to do Cryptography: a set of questions compiled to give PhD candidates a sense of what they should know by the end of their first year. We build on the previous few weeks by introducing the notion of a pairing.

Pairing definition: Given 3 cyclic groups $\mathbb{G}_1,\mathbb{G}_2,\mathbb{G}_3$ of order $q$ with generators $g_1,g_2,g_3$ respectively. We say a function $e:\mathbb{G}_1\times\mathbb{G}_2\rightarrow\mathbb{G}_3$ is a pairing if the following hold:

1. [bilinearity] $\forall A,B\in\mathbb{G}_1,C,D\in\mathbb{G}_2$: $e(A+B,C)=e(A,C)\cdot e(B,C)$ and $e(A,C+D)=e(A,C)\cdot e(A,D)$
2. [non-dengeneracy] $e(g_1,g_2)\neq 1$
3. [efficiency] $e$ is efficiently computable

Types of pairing: There are 3 type of pairings that will be described below:

1. $\mathbb{G}_1=\mathbb{G}_2$
2. $\mathbb{G}_1\neq\mathbb{G}_2$ but there is an efficiently computable isomorphism from $\mathbb{G}_2$ to $\mathbb{G}_1$ and maps the generator $g_2$ to $g_1$
3. $\mathbb{G}_1\neq\mathbb{G}_2$ and there is no efficiently computable isomorphism

The last two are asymmetric pairings while the first is a symmetric pairing.