commit acbaad8f4cfdaa62e03ca6847189d37575e12201
Author: Mayeul@Zama <mayeul.debellabre@zama.ai>
Date:   Thu Dec 8 12:18:53 2022 +0100

    add concrete-cpu-noise-model crate

diff --git a/Cargo.toml b/Cargo.toml
new file mode 100644
index 000000000..c2215fb7a
--- /dev/null
+++ b/Cargo.toml
@@ -0,0 +1,11 @@
+[package]
+# see https://doc.rust-lang.org/cargo/reference/manifest.html
+name = "concrete-cpu-noise-model"
+version = "0.1.0"
+authors = [""]
+edition = "2021"
+
+[dependencies]
+
+[dev-dependencies]
+approx = "0.5"
diff --git a/src/gaussian_noise.rs b/src/gaussian_noise.rs
new file mode 100644
index 000000000..d955f49af
--- /dev/null
+++ b/src/gaussian_noise.rs
@@ -0,0 +1,3 @@
+pub mod conversion;
+pub mod noise;
+pub mod security;
diff --git a/src/gaussian_noise/conversion.rs b/src/gaussian_noise/conversion.rs
new file mode 100644
index 000000000..548cd6d0c
--- /dev/null
+++ b/src/gaussian_noise/conversion.rs
@@ -0,0 +1,15 @@
+fn modular_variance_variance_ratio(ciphertext_modulus_log: u32) -> f64 {
+    2_f64.powi(2 * ciphertext_modulus_log as i32)
+}
+
+pub fn modular_variance_to_variance(modular_variance: f64, ciphertext_modulus_log: u32) -> f64 {
+    modular_variance / modular_variance_variance_ratio(ciphertext_modulus_log)
+}
+
+pub fn variance_to_modular_variance(variance: f64, ciphertext_modulus_log: u32) -> f64 {
+    variance * modular_variance_variance_ratio(ciphertext_modulus_log)
+}
+
+pub fn variance_to_std_dev(variance: f64) -> f64 {
+    variance.sqrt()
+}
diff --git a/src/gaussian_noise/noise.rs b/src/gaussian_noise/noise.rs
new file mode 100644
index 000000000..0c4b5976e
--- /dev/null
+++ b/src/gaussian_noise/noise.rs
@@ -0,0 +1,7 @@
+pub mod blind_rotate;
+pub mod cmux;
+pub mod external_product_glwe;
+pub mod keyswitch;
+pub mod keyswitch_one_bit;
+pub mod modulus_switching;
+pub mod private_packing_keyswitch;
diff --git a/src/gaussian_noise/noise/blind_rotate.rs b/src/gaussian_noise/noise/blind_rotate.rs
new file mode 100644
index 000000000..18f1cc780
--- /dev/null
+++ b/src/gaussian_noise/noise/blind_rotate.rs
@@ -0,0 +1,100 @@
+use super::cmux::variance_cmux;
+
+pub const FFT_SCALING_WEIGHT: f64 = -2.577_224_94;
+
+/// Final reduced noise generated by the final bootstrap step.
+/// Note that it does not depends from input noise, assuming the bootstrap is successful
+pub fn variance_blind_rotate(
+    in_lwe_dimension: u64,
+    out_glwe_dimension: u64,
+    out_polynomial_size: u64,
+    log2_base: u64,
+    level: u64,
+    ciphertext_modulus_log: u32,
+    variance_bsk: f64,
+) -> f64 {
+    in_lwe_dimension as f64
+        * variance_cmux(
+            out_glwe_dimension,
+            out_polynomial_size,
+            log2_base,
+            level,
+            ciphertext_modulus_log,
+            variance_bsk,
+        )
+}
+
+#[cfg(test)]
+mod tests {
+    use crate::gaussian_noise::conversion::variance_to_modular_variance;
+    use crate::gaussian_noise::security::minimal_variance_glwe;
+
+    use super::*;
+
+    #[test]
+    fn security_variance_bootstrap_1() {
+        let ref_modular_variance = 4.078_296_369_990_673e31;
+
+        let polynomial_size = 1 << 12;
+        let glwe_dimension = 2;
+
+        let ciphertext_modulus_log = 64;
+        let security = 128;
+        let variance_bsk = minimal_variance_glwe(
+            glwe_dimension,
+            polynomial_size,
+            ciphertext_modulus_log,
+            security,
+        );
+
+        let actual = variance_blind_rotate(
+            2048,
+            glwe_dimension,
+            polynomial_size,
+            24,
+            2,
+            ciphertext_modulus_log,
+            variance_bsk,
+        );
+
+        approx::assert_relative_eq!(
+            variance_to_modular_variance(actual, ciphertext_modulus_log),
+            ref_modular_variance,
+            max_relative = 1e-8
+        );
+    }
+
+    #[test]
+    fn golden_python_prototype_security_variance_bootstrap_2() {
+        // golden value include fft correction
+        let golden_modular_variance = 3.269_722_907_894_341e55;
+
+        let polynomial_size = 1 << 12;
+        let glwe_dimension = 4;
+
+        let ciphertext_modulus_log = 128;
+        let security = 128;
+        let variance_bsk = minimal_variance_glwe(
+            glwe_dimension,
+            polynomial_size,
+            ciphertext_modulus_log,
+            security,
+        );
+
+        let actual = variance_blind_rotate(
+            1024,
+            glwe_dimension,
+            polynomial_size,
+            5,
+            9,
+            ciphertext_modulus_log,
+            variance_bsk,
+        );
+
+        approx::assert_relative_eq!(
+            variance_to_modular_variance(actual, ciphertext_modulus_log),
+            golden_modular_variance,
+            max_relative = 1e-8
+        );
+    }
+}
diff --git a/src/gaussian_noise/noise/cmux.rs b/src/gaussian_noise/noise/cmux.rs
new file mode 100644
index 000000000..23791d7fc
--- /dev/null
+++ b/src/gaussian_noise/noise/cmux.rs
@@ -0,0 +1,20 @@
+use super::external_product_glwe::variance_external_product_glwe;
+
+// only valid in the blind rotate case
+pub fn variance_cmux(
+    glwe_dimension: u64,
+    polynomial_size: u64,
+    log2_base: u64,
+    level: u64,
+    ciphertext_modulus_log: u32,
+    variance_ggsw: f64,
+) -> f64 {
+    variance_external_product_glwe(
+        glwe_dimension,
+        polynomial_size,
+        log2_base,
+        level,
+        ciphertext_modulus_log,
+        variance_ggsw,
+    )
+}
diff --git a/src/gaussian_noise/noise/external_product_glwe.rs b/src/gaussian_noise/noise/external_product_glwe.rs
new file mode 100644
index 000000000..fc76923e0
--- /dev/null
+++ b/src/gaussian_noise/noise/external_product_glwe.rs
@@ -0,0 +1,84 @@
+use crate::gaussian_noise::conversion::modular_variance_to_variance;
+use crate::utils::square;
+
+pub fn variance_external_product_glwe(
+    glwe_dimension: u64,
+    polynomial_size: u64,
+    log2_base: u64,
+    level: u64,
+    ciphertext_modulus_log: u32,
+    variance_ggsw: f64,
+) -> f64 {
+    theoretical_variance_external_product_glwe(
+        glwe_dimension,
+        polynomial_size,
+        log2_base,
+        level,
+        ciphertext_modulus_log,
+        variance_ggsw,
+    ) + fft_noise_variance_external_product_glwe(
+        glwe_dimension,
+        polynomial_size,
+        log2_base,
+        level,
+        ciphertext_modulus_log,
+    )
+}
+
+fn theoretical_variance_external_product_glwe(
+    glwe_dimension: u64,
+    polynomial_size: u64,
+    log2_base: u64,
+    level: u64,
+    ciphertext_modulus_log: u32,
+    variance_ggsw: f64,
+) -> f64 {
+    let variance_key_coefficient_binary: f64 =
+        modular_variance_to_variance(1. / 4., ciphertext_modulus_log);
+
+    let square_expectation_key_coefficient_binary: f64 =
+        modular_variance_to_variance(square(1. / 2.), ciphertext_modulus_log);
+
+    let k = glwe_dimension as f64;
+    let b = 2_f64.powi(log2_base as i32);
+    let b2l = 2_f64.powi((log2_base * 2 * level) as i32);
+    let l = level as f64;
+    let big_n = polynomial_size as f64;
+    let q_square = 2_f64.powi(2 * ciphertext_modulus_log as i32);
+
+    let res_1 = l * (k + 1.) * big_n * (square(b) + 2.) / 12. * variance_ggsw;
+    let res_2 = (q_square - b2l) / (24. * b2l)
+        * (modular_variance_to_variance(1., ciphertext_modulus_log)
+            + k * big_n
+                * (variance_key_coefficient_binary + square_expectation_key_coefficient_binary))
+        + k * big_n / 8. * variance_key_coefficient_binary
+        + 1. / 16. * square(1. - k * big_n) * square_expectation_key_coefficient_binary;
+
+    res_1 + res_2
+}
+
+const FFT_SCALING_WEIGHT: f64 = -2.577_224_94;
+
+/// Additional noise generated by fft computation
+
+fn fft_noise_variance_external_product_glwe(
+    glwe_dimension: u64,
+    polynomial_size: u64,
+    log2_base: u64,
+    level: u64,
+    ciphertext_modulus_log: u32,
+) -> f64 {
+    // https://github.com/zama-ai/concrete-optimizer/blob/prototype/python/optimizer/noise_formulas/bootstrap.py#L25
+    let b = 2_f64.powi(log2_base as i32);
+    let l = level as f64;
+    let big_n = polynomial_size as f64;
+    let k = glwe_dimension;
+    assert!(k > 0, "k = {k}");
+    assert!(k < 7, "k = {k}");
+
+    // 22 = 2 x 11, 11 = 64 -53
+    let scale_margin = (1_u64 << 22) as f64;
+    let res =
+        f64::exp2(FFT_SCALING_WEIGHT) * scale_margin * l * b * b * big_n.powi(2) * (k as f64 + 1.);
+    modular_variance_to_variance(res, ciphertext_modulus_log)
+}
diff --git a/src/gaussian_noise/noise/keyswitch.rs b/src/gaussian_noise/noise/keyswitch.rs
new file mode 100644
index 000000000..b8170eb18
--- /dev/null
+++ b/src/gaussian_noise/noise/keyswitch.rs
@@ -0,0 +1,76 @@
+use super::keyswitch_one_bit::variance_keyswitch_one_bit;
+
+/// Additional noise generated by the keyswitch step.
+pub fn variance_keyswitch(
+    input_lwe_dimension: u64, //n_big
+    log2_base: u64,
+    level: u64,
+    ciphertext_modulus_log: u32,
+    variance_ksk: f64,
+) -> f64 {
+    input_lwe_dimension as f64
+        * variance_keyswitch_one_bit(log2_base, level, ciphertext_modulus_log, variance_ksk)
+}
+
+#[cfg(test)]
+mod tests {
+
+    use crate::gaussian_noise::conversion::variance_to_modular_variance;
+    use crate::gaussian_noise::security::minimal_variance_lwe;
+
+    use super::*;
+
+    #[test]
+    fn golden_python_prototype_security_variance_keyswitch_1() {
+        let golden_modular_variance = 5.997_880_135_602_194e68;
+        let internal_ks_output_lwe_dimension = 1024;
+        let ciphertext_modulus_log = 128;
+        let security = 128;
+
+        let actual = variance_keyswitch(
+            4096,
+            5,
+            9,
+            ciphertext_modulus_log,
+            minimal_variance_lwe(
+                internal_ks_output_lwe_dimension,
+                ciphertext_modulus_log,
+                security,
+            ),
+        );
+
+        approx::assert_relative_eq!(
+            variance_to_modular_variance(actual, ciphertext_modulus_log),
+            golden_modular_variance,
+            max_relative = 1e-8
+        );
+    }
+
+    #[test]
+    fn golden_python_prototype_security_variance_keyswitch_2() {
+        // let golden_modular_variance = 8.580795457940938e+66;
+        // the full npe implements a part of the full estimation
+        let golden_modular_variance = 7.407_691_550_271_225e48; // full estimation
+        let internal_ks_output_lwe_dimension = 512;
+        let ciphertext_modulus_log = 64;
+        let security = 128;
+
+        let actual = variance_keyswitch(
+            2048,
+            24,
+            2,
+            ciphertext_modulus_log,
+            minimal_variance_lwe(
+                internal_ks_output_lwe_dimension,
+                ciphertext_modulus_log,
+                security,
+            ),
+        );
+
+        approx::assert_relative_eq!(
+            variance_to_modular_variance(actual, ciphertext_modulus_log),
+            golden_modular_variance,
+            max_relative = 1e-8
+        );
+    }
+}
diff --git a/src/gaussian_noise/noise/keyswitch_one_bit.rs b/src/gaussian_noise/noise/keyswitch_one_bit.rs
new file mode 100644
index 000000000..dbaad3437
--- /dev/null
+++ b/src/gaussian_noise/noise/keyswitch_one_bit.rs
@@ -0,0 +1,32 @@
+use crate::gaussian_noise::conversion::modular_variance_to_variance;
+use crate::utils::square;
+
+/// Additional noise generated by the bit multiplication
+pub fn variance_keyswitch_one_bit(
+    log2_base: u64,
+    level: u64,
+    ciphertext_modulus_log: u32,
+    variance_ksk: f64,
+) -> f64 {
+    let variance_key_coefficient_binary: f64 =
+        modular_variance_to_variance(1. / 4., ciphertext_modulus_log);
+
+    let square_expectation_key_coefficient_binary: f64 =
+        modular_variance_to_variance(square(1. / 2.), ciphertext_modulus_log);
+
+    let base = 2_f64.powi(log2_base as i32);
+    let b2l = 2_f64.powi((log2_base * 2 * level) as i32);
+    let q_square = 2_f64.powi((2 * ciphertext_modulus_log) as i32);
+
+    // res 2
+    let res_2 = (q_square / (12. * b2l) - 1. / 12.)
+        * (variance_key_coefficient_binary + square_expectation_key_coefficient_binary);
+
+    // res 3
+    let res_3 = 1. / 4. * variance_key_coefficient_binary;
+
+    // res 4
+    let res_4 = (level as f64) * variance_ksk * (square(base) + 2.) / 12.;
+
+    res_2 + res_3 + res_4
+}
diff --git a/src/gaussian_noise/noise/modulus_switching.rs b/src/gaussian_noise/noise/modulus_switching.rs
new file mode 100644
index 000000000..27557e4d6
--- /dev/null
+++ b/src/gaussian_noise/noise/modulus_switching.rs
@@ -0,0 +1,16 @@
+use crate::gaussian_noise::conversion::modular_variance_to_variance;
+use crate::utils::square;
+
+pub fn estimate_modulus_switching_noise_with_binary_key(
+    internal_ks_output_lwe_dimension: u64,
+    glwe_log2_polynomial_size: u64,
+    ciphertext_modulus_log: u32,
+) -> f64 {
+    let nb_msb = glwe_log2_polynomial_size + 1;
+
+    let w = 2_f64.powi(nb_msb as i32);
+    let n = internal_ks_output_lwe_dimension as f64;
+
+    (1. / 12. + n / 24.) / square(w)
+        + modular_variance_to_variance(-1. / 12. + n / 48., ciphertext_modulus_log)
+}
diff --git a/src/gaussian_noise/noise/private_packing_keyswitch.rs b/src/gaussian_noise/noise/private_packing_keyswitch.rs
new file mode 100644
index 000000000..2a78b6bdc
--- /dev/null
+++ b/src/gaussian_noise/noise/private_packing_keyswitch.rs
@@ -0,0 +1,39 @@
+use crate::gaussian_noise::conversion::modular_variance_to_variance;
+use crate::utils::square;
+
+// packing private keyswitch for WoP-PBS, described in algorithm 3 of https://eprint.iacr.org/2018/421.pdf (TFHE paper)
+pub fn estimate_packing_private_keyswitch(
+    var_glwe: f64,
+    var_ggsw: f64,
+    log2_base: u64,
+    level: u64,
+    output_glwe_dimension: u64,
+    output_polynomial_size: u64,
+    ciphertext_modulus_log: u32,
+) -> f64 {
+    let variance_key_coefficient_binary: f64 = 1. / 4.;
+    let expectation_key_coefficient_binary: f64 = 1. / 2.;
+
+    let l = level as f64;
+    let b = (1 << log2_base) as f64;
+    let n = (output_glwe_dimension * output_polynomial_size) as f64; // param.internal_lwe_dimension.0 as f64;
+    let b2l = f64::powi(b, 2 * level as i32);
+    let var_s_w = 1. / 4.;
+    let mean_s_w = 1. / 2.;
+    let res_1 = l * (n + 1.) * var_ggsw * (square(b) + 2.) / 12.;
+
+    #[allow(clippy::cast_possible_wrap)]
+    let res_3 = (f64::powi(2., 2 * ciphertext_modulus_log as i32) - b2l) / (12. * b2l)
+        * modular_variance_to_variance(
+            1. + n * variance_key_coefficient_binary + square(expectation_key_coefficient_binary),
+            ciphertext_modulus_log,
+        )
+        + n / 4.
+            * modular_variance_to_variance(variance_key_coefficient_binary, ciphertext_modulus_log)
+        + var_glwe * (var_s_w + square(mean_s_w));
+
+    let res_5 = modular_variance_to_variance(var_s_w, ciphertext_modulus_log) * 1. / 4.
+        * square(1. - n * expectation_key_coefficient_binary);
+
+    res_1 + res_3 + res_5
+}
diff --git a/src/gaussian_noise/security.rs b/src/gaussian_noise/security.rs
new file mode 100644
index 000000000..61bbde09f
--- /dev/null
+++ b/src/gaussian_noise/security.rs
@@ -0,0 +1,65 @@
+mod security_weight;
+
+pub use security_weight::{security_weight, supported_security_levels};
+
+/// Noise ensuring security
+pub fn minimal_variance_lwe(
+    lwe_dimension: u64,
+    ciphertext_modulus_log: u32,
+    security_level: u64,
+) -> f64 {
+    minimal_variance_glwe(lwe_dimension, 1, ciphertext_modulus_log, security_level)
+}
+
+/// Noise ensuring security
+pub fn minimal_variance_glwe(
+    glwe_dimension: u64,
+    polynomial_size: u64,
+    ciphertext_modulus_log: u32,
+    security_level: u64,
+) -> f64 {
+    let equiv_lwe_dimension = glwe_dimension * polynomial_size;
+    let security_weights = security_weight(security_level)
+        .unwrap_or_else(|| panic!("{security_level} bits of security is not supported"));
+
+    let secure_log2_std =
+        security_weights.secure_log2_std(equiv_lwe_dimension, ciphertext_modulus_log as f64);
+    let log2_var = 2.0 * secure_log2_std;
+    f64::exp2(log2_var)
+}
+
+#[cfg(test)]
+mod tests {
+    use super::super::conversion::variance_to_std_dev;
+    use super::minimal_variance_glwe;
+
+    #[test]
+    fn golden_python_prototype_security_security_glwe_variance_low() {
+        // python securityFunc(10,14,64)= 0.3120089883926036
+        let integer_size = 64;
+        let golden_std_dev = 2.168_404_344_971_009e-19;
+        let security_level = 128;
+
+        let actual = minimal_variance_glwe(10, 1 << 14, integer_size, security_level);
+        approx::assert_relative_eq!(
+            golden_std_dev,
+            variance_to_std_dev(actual),
+            epsilon = f64::EPSILON
+        );
+    }
+
+    #[test]
+    fn golden_python_prototype_security_security_glwe_variance_high() {
+        // python securityFunc(3,8,32)= 2.6011445832514504
+        let integer_size = 32;
+        let golden_std_dev = 4.392_824_146_816_922_4e-6;
+        let security_level = 128;
+
+        let actual = minimal_variance_glwe(3, 1 << 8, integer_size, security_level);
+        approx::assert_relative_eq!(
+            golden_std_dev,
+            variance_to_std_dev(actual),
+            epsilon = f64::EPSILON
+        );
+    }
+}
diff --git a/src/gaussian_noise/security/security_weight.rs b/src/gaussian_noise/security/security_weight.rs
new file mode 100644
index 000000000..74c1a294a
--- /dev/null
+++ b/src/gaussian_noise/security/security_weight.rs
@@ -0,0 +1,116 @@
+#[derive(Clone, Copy)]
+pub struct SecurityWeights {
+    slope: f64,
+    bias: f64,
+    minimal_lwe_dimension: u64,
+}
+
+impl SecurityWeights {
+    pub fn secure_log2_std(&self, lwe_dimension: u64, ciphertext_modulus_log: f64) -> f64 {
+        // ensure to have a minimal on std deviation covering the 2 lowest bits on modular scale
+        let epsilon_log2_std_modular = 2.0;
+        let epsilon_log2_std = epsilon_log2_std_modular - (ciphertext_modulus_log);
+        // ensure the requested lwe_dimension is bigger than the minimal lwe dimension
+        if self.minimal_lwe_dimension <= lwe_dimension {
+            f64::max(
+                self.slope * lwe_dimension as f64 + self.bias,
+                epsilon_log2_std,
+            )
+        } else {
+            ciphertext_modulus_log
+        }
+    }
+}
+
+// Security curves generated using the lattice-estimator
+// (https://github.com/malb/lattice-estimator) on the 24th of June 2022
+const SECURITY_WEIGHTS_ARRAY: [(u64, SecurityWeights); 9] = [
+    (
+        80,
+        SecurityWeights {
+            slope: -0.040_426_331_193_645_89,
+            bias: 1.660_978_864_143_672_2,
+            minimal_lwe_dimension: 450,
+        },
+    ),
+    (
+        96,
+        SecurityWeights {
+            slope: -0.034_147_803_608_670_51,
+            bias: 2.017_310_258_660_345,
+            minimal_lwe_dimension: 450,
+        },
+    ),
+    (
+        112,
+        SecurityWeights {
+            slope: -0.029_670_137_081_135_885,
+            bias: 2.162_463_714_083_856,
+            minimal_lwe_dimension: 450,
+        },
+    ),
+    (
+        128,
+        SecurityWeights {
+            slope: -0.026_405_028_765_226_22,
+            bias: 2.482_642_269_104_317_7,
+            minimal_lwe_dimension: 450,
+        },
+    ),
+    (
+        144,
+        SecurityWeights {
+            slope: -0.023_821_437_305_989_134,
+            bias: 2.717_778_944_063_667_3,
+            minimal_lwe_dimension: 450,
+        },
+    ),
+    (
+        160,
+        SecurityWeights {
+            slope: -0.021_743_582_187_160_36,
+            bias: 2.938_810_548_493_322,
+            minimal_lwe_dimension: 498,
+        },
+    ),
+    (
+        176,
+        SecurityWeights {
+            slope: -0.019_904_056_582_117_684,
+            bias: 2.816_125_280_154_224_7,
+            minimal_lwe_dimension: 551,
+        },
+    ),
+    (
+        192,
+        SecurityWeights {
+            slope: -0.018_610_403_247_590_085,
+            bias: 3.299_623_684_839_900_8,
+            minimal_lwe_dimension: 606,
+        },
+    ),
+    (
+        256,
+        SecurityWeights {
+            slope: -0.014_606_812_351_714_953,
+            bias: 3.849_362_923_469_300_3,
+            minimal_lwe_dimension: 826,
+        },
+    ),
+];
+
+pub fn supported_security_levels() -> impl std::iter::Iterator<Item = u64> {
+    SECURITY_WEIGHTS_ARRAY
+        .iter()
+        .map(|(security_level, _)| *security_level)
+}
+
+pub fn security_weight(security_level: u64) -> Option<SecurityWeights> {
+    let index = SECURITY_WEIGHTS_ARRAY
+        .binary_search_by_key(&security_level, |(security_level, _weights)| {
+            *security_level
+        })
+        .ok()?;
+
+    Some(SECURITY_WEIGHTS_ARRAY[index].1)
+}
diff --git a/src/lib.rs b/src/lib.rs
new file mode 100644
index 000000000..8db3e23b0
--- /dev/null
+++ b/src/lib.rs
@@ -0,0 +1,23 @@
+#![warn(clippy::nursery)]
+#![warn(clippy::pedantic)]
+#![warn(clippy::style)]
+#![allow(clippy::cast_lossless)]
+#![allow(clippy::cast_precision_loss)] // u64 to f64
+#![allow(clippy::cast_possible_truncation)] // u64 to usize
+#![allow(clippy::missing_panics_doc)]
+#![allow(clippy::module_name_repetitions)]
+#![allow(clippy::must_use_candidate)]
+#![allow(clippy::suboptimal_flops)]
+#![allow(clippy::cast_possible_wrap)]
+#![warn(unused_results)]
+
+pub mod gaussian_noise;
+
+pub(crate) mod utils {
+    pub fn square<V>(v: V) -> V
+    where
+        V: std::ops::Mul<Output = V> + Copy,
+    {
+        v * v
+    }
+}