1use base2k::{FFT64, Module, ScalarZnx, ScalarZnxAlloc, ScratchOwned, Stats, VecZnxOps, ZnxViewMut};
  2use sampling::source::Source;
  3
  4use crate::{
  5    elem::{GetRow, Infos},
  6    ggsw_ciphertext::GGSWCiphertext,
  7    glwe_ciphertext_fourier::GLWECiphertextFourier,
  8    glwe_plaintext::GLWEPlaintext,
  9    keys::{SecretKey, SecretKeyFourier},
 10    keyswitch_key::GLWESwitchingKey,
 11};
 12
 13#[test]
 14fn encrypt_sk() {
 15    (1..4).for_each(|rank_in| {
 16        (1..4).for_each(|rank_out| {
 17            println!("test encrypt_sk rank_in rank_out: {} {}", rank_in, rank_out);
 18            test_encrypt_sk(11, 8, 54, 3.2, rank_in, rank_out);
 19        });
 20    });
 21}
 22
 23fn test_encrypt_sk(log_n: usize, basek: usize, k_ksk: usize, sigma: f64, rank_in: usize, rank_out: usize) {
 24    let module: Module<FFT64> = Module::<FFT64>::new(1 << log_n);
 25    let rows = (k_ksk + basek - 1) / basek;
 26
 27    let mut ksk: GLWESwitchingKey<Vec<u8>, FFT64> = GLWESwitchingKey::new(&module, basek, k_ksk, rows, rank_in, rank_out);
 28    let mut pt: GLWEPlaintext<Vec<u8>> = GLWEPlaintext::new(&module, basek, k_ksk);
 29
 30    let mut source_xs: Source = Source::new([0u8; 32]);
 31    let mut source_xe: Source = Source::new([0u8; 32]);
 32    let mut source_xa: Source = Source::new([0u8; 32]);
 33
 34    let mut scratch: ScratchOwned = ScratchOwned::new(
 35        GLWESwitchingKey::encrypt_sk_scratch_space(&module, rank_out, ksk.size())
 36            | GLWECiphertextFourier::decrypt_scratch_space(&module, ksk.size()),
 37    );
 38
 39    let mut sk_in: SecretKey<Vec<u8>> = SecretKey::new(&module, rank_in);
 40    sk_in.fill_ternary_prob(0.5, &mut source_xs);
 41
 42    let mut sk_in_dft: SecretKeyFourier<Vec<u8>, FFT64> = SecretKeyFourier::new(&module, rank_in);
 43    sk_in_dft.dft(&module, &sk_in);
 44
 45    let mut sk_out: SecretKey<Vec<u8>> = SecretKey::new(&module, rank_out);
 46    sk_out.fill_ternary_prob(0.5, &mut source_xs);
 47
 48    let mut sk_out_dft: SecretKeyFourier<Vec<u8>, FFT64> = SecretKeyFourier::new(&module, rank_out);
 49    sk_out_dft.dft(&module, &sk_out);
 50
 51    ksk.encrypt_sk(
 52        &module,
 53        &sk_in,
 54        &sk_out_dft,
 55        &mut source_xa,
 56        &mut source_xe,
 57        sigma,
 58        scratch.borrow(),
 59    );
 60
 61    let mut ct_gglwe_fourier: GLWECiphertextFourier<Vec<u8>, FFT64> = GLWECiphertextFourier::new(&module, basek, k_ksk, rank_out);
 62
 63    (0..ksk.rank_in()).for_each(|col_i| {
 64        (0..ksk.rows()).for_each(|row_i| {
 65            ksk.get_row(&module, row_i, 0, &mut ct_gglwe_fourier);
 66            ct_gglwe_fourier.decrypt(&module, &mut pt, &sk_out_dft, scratch.borrow());
 67            module.vec_znx_sub_scalar_inplace(&mut pt, 0, row_i, &sk_in, col_i);
 68            let std_pt: f64 = pt.data.std(0, basek) * (k_ksk as f64).exp2();
 69            assert!((sigma - std_pt).abs() <= 0.2, "{} {}", sigma, std_pt);
 70        });
 71    });
 72}
 73
 74#[test]
 75fn keyswitch() {
 76    let module: Module<FFT64> = Module::<FFT64>::new(2048);
 77    let basek: usize = 12;
 78    let log_k_grlwe: usize = 60;
 79    let rows: usize = (log_k_grlwe + basek - 1) / basek;
 80
 81    let rank: usize = 1;
 82
 83    let sigma: f64 = 3.2;
 84
 85    let mut ct_grlwe_s0s1: GLWESwitchingKey<Vec<u8>, FFT64> =
 86        GLWESwitchingKey::new(&module, basek, log_k_grlwe, rows, rank, rank);
 87    let mut ct_grlwe_s1s2: GLWESwitchingKey<Vec<u8>, FFT64> =
 88        GLWESwitchingKey::new(&module, basek, log_k_grlwe, rows, rank, rank);
 89    let mut ct_grlwe_s0s2: GLWESwitchingKey<Vec<u8>, FFT64> =
 90        GLWESwitchingKey::new(&module, basek, log_k_grlwe, rows, rank, rank);
 91
 92    let mut source_xs: Source = Source::new([0u8; 32]);
 93    let mut source_xe: Source = Source::new([0u8; 32]);
 94    let mut source_xa: Source = Source::new([0u8; 32]);
 95
 96    let mut scratch: ScratchOwned = ScratchOwned::new(
 97        GLWESwitchingKey::encrypt_sk_scratch_space(&module, rank, ct_grlwe_s0s1.size())
 98            | GLWECiphertextFourier::decrypt_scratch_space(&module, ct_grlwe_s0s2.size())
 99            | GLWESwitchingKey::keyswitch_scratch_space(
100                &module,
101                ct_grlwe_s0s2.size(),
102                ct_grlwe_s0s1.size(),
103                ct_grlwe_s1s2.size(),
104            ),
105    );
106
107    let mut sk0: SecretKey<Vec<u8>> = SecretKey::new(&module, rank);
108    sk0.fill_ternary_prob(0.5, &mut source_xs);
109
110    let mut sk0_dft: SecretKeyFourier<Vec<u8>, FFT64> = SecretKeyFourier::new(&module, rank);
111    sk0_dft.dft(&module, &sk0);
112
113    let mut sk1: SecretKey<Vec<u8>> = SecretKey::new(&module, rank);
114    sk1.fill_ternary_prob(0.5, &mut source_xs);
115
116    let mut sk1_dft: SecretKeyFourier<Vec<u8>, FFT64> = SecretKeyFourier::new(&module, rank);
117    sk1_dft.dft(&module, &sk1);
118
119    let mut sk2: SecretKey<Vec<u8>> = SecretKey::new(&module, rank);
120    sk2.fill_ternary_prob(0.5, &mut source_xs);
121
122    let mut sk2_dft: SecretKeyFourier<Vec<u8>, FFT64> = SecretKeyFourier::new(&module, rank);
123    sk2_dft.dft(&module, &sk2);
124
125    // GRLWE_{s1}(s0) = s0 -> s1
126    ct_grlwe_s0s1.encrypt_sk(
127        &module,
128        &sk0.data,
129        &sk1_dft,
130        &mut source_xa,
131        &mut source_xe,
132        sigma,
133        scratch.borrow(),
134    );
135
136    // GRLWE_{s2}(s1) -> s1 -> s2
137    ct_grlwe_s1s2.encrypt_sk(
138        &module,
139        &sk1.data,
140        &sk2_dft,
141        &mut source_xa,
142        &mut source_xe,
143        sigma,
144        scratch.borrow(),
145    );
146
147    // GRLWE_{s1}(s0) (x) GRLWE_{s2}(s1) = GRLWE_{s2}(s0)
148    ct_grlwe_s0s2.keyswitch(&module, &ct_grlwe_s0s1, &ct_grlwe_s1s2, scratch.borrow());
149
150    let mut ct_rlwe_dft_s0s2: GLWECiphertextFourier<Vec<u8>, FFT64> =
151        GLWECiphertextFourier::new(&module, basek, log_k_grlwe, rank);
152    let mut pt: GLWEPlaintext<Vec<u8>> = GLWEPlaintext::new(&module, basek, log_k_grlwe);
153
154    (0..ct_grlwe_s0s2.rows()).for_each(|row_i| {
155        ct_grlwe_s0s2.get_row(&module, row_i, 0, &mut ct_rlwe_dft_s0s2);
156        ct_rlwe_dft_s0s2.decrypt(&module, &mut pt, &sk2_dft, scratch.borrow());
157        module.vec_znx_sub_scalar_inplace(&mut pt, 0, row_i, &sk0, 0);
158
159        let noise_have: f64 = pt.data.std(0, basek).log2();
160        let noise_want: f64 = noise_grlwe_rlwe_product(
161            module.n() as f64,
162            basek,
163            0.5,
164            0.5,
165            0f64,
166            sigma * sigma,
167            0f64,
168            log_k_grlwe,
169            log_k_grlwe,
170        );
171
172        assert!(
173            (noise_have - noise_want).abs() <= 0.1,
174            "{} {}",
175            noise_have,
176            noise_want
177        );
178    });
179}
180
181#[test]
182fn keyswitch_inplace() {
183    let module: Module<FFT64> = Module::<FFT64>::new(2048);
184    let basek: usize = 12;
185    let log_k_grlwe: usize = 60;
186    let rows: usize = (log_k_grlwe + basek - 1) / basek;
187
188    let rank: usize = 1;
189    let rank_out: usize = 1;
190
191    let sigma: f64 = 3.2;
192
193    let mut ct_grlwe_s0s1: GLWESwitchingKey<Vec<u8>, FFT64> =
194        GLWESwitchingKey::new(&module, basek, log_k_grlwe, rows, rank, rank_out);
195    let mut ct_grlwe_s1s2: GLWESwitchingKey<Vec<u8>, FFT64> =
196        GLWESwitchingKey::new(&module, basek, log_k_grlwe, rows, rank, rank_out);
197
198    let mut source_xs: Source = Source::new([0u8; 32]);
199    let mut source_xe: Source = Source::new([0u8; 32]);
200    let mut source_xa: Source = Source::new([0u8; 32]);
201
202    let mut scratch: ScratchOwned = ScratchOwned::new(
203        GLWESwitchingKey::encrypt_sk_scratch_space(&module, rank, ct_grlwe_s0s1.size())
204            | GLWECiphertextFourier::decrypt_scratch_space(&module, ct_grlwe_s0s1.size())
205            | GLWESwitchingKey::keyswitch_inplace_scratch_space(&module, ct_grlwe_s0s1.size(), ct_grlwe_s1s2.size()),
206    );
207
208    let mut sk0: SecretKey<Vec<u8>> = SecretKey::new(&module, rank);
209    sk0.fill_ternary_prob(0.5, &mut source_xs);
210
211    let mut sk0_dft: SecretKeyFourier<Vec<u8>, FFT64> = SecretKeyFourier::new(&module, rank);
212    sk0_dft.dft(&module, &sk0);
213
214    let mut sk1: SecretKey<Vec<u8>> = SecretKey::new(&module, rank);
215    sk1.fill_ternary_prob(0.5, &mut source_xs);
216
217    let mut sk1_dft: SecretKeyFourier<Vec<u8>, FFT64> = SecretKeyFourier::new(&module, rank);
218    sk1_dft.dft(&module, &sk1);
219
220    let mut sk2: SecretKey<Vec<u8>> = SecretKey::new(&module, rank);
221    sk2.fill_ternary_prob(0.5, &mut source_xs);
222
223    let mut sk2_dft: SecretKeyFourier<Vec<u8>, FFT64> = SecretKeyFourier::new(&module, rank);
224    sk2_dft.dft(&module, &sk2);
225
226    // GRLWE_{s1}(s0) = s0 -> s1
227    ct_grlwe_s0s1.encrypt_sk(
228        &module,
229        &sk0.data,
230        &sk1_dft,
231        &mut source_xa,
232        &mut source_xe,
233        sigma,
234        scratch.borrow(),
235    );
236
237    // GRLWE_{s2}(s1) -> s1 -> s2
238    ct_grlwe_s1s2.encrypt_sk(
239        &module,
240        &sk1.data,
241        &sk2_dft,
242        &mut source_xa,
243        &mut source_xe,
244        sigma,
245        scratch.borrow(),
246    );
247
248    // GRLWE_{s1}(s0) (x) GRLWE_{s2}(s1) = GRLWE_{s2}(s0)
249    ct_grlwe_s0s1.keyswitch_inplace(&module, &ct_grlwe_s1s2, scratch.borrow());
250
251    let ct_grlwe_s0s2: GLWESwitchingKey<Vec<u8>, FFT64> = ct_grlwe_s0s1;
252
253    let mut ct_rlwe_dft_s0s2: GLWECiphertextFourier<Vec<u8>, FFT64> =
254        GLWECiphertextFourier::new(&module, basek, log_k_grlwe, rank);
255    let mut pt: GLWEPlaintext<Vec<u8>> = GLWEPlaintext::new(&module, basek, log_k_grlwe);
256
257    (0..ct_grlwe_s0s2.rows()).for_each(|row_i| {
258        ct_grlwe_s0s2.get_row(&module, row_i, 0, &mut ct_rlwe_dft_s0s2);
259        ct_rlwe_dft_s0s2.decrypt(&module, &mut pt, &sk2_dft, scratch.borrow());
260        module.vec_znx_sub_scalar_inplace(&mut pt, 0, row_i, &sk0, 0);
261
262        let noise_have: f64 = pt.data.std(0, basek).log2();
263        let noise_want: f64 = noise_grlwe_rlwe_product(
264            module.n() as f64,
265            basek,
266            0.5,
267            0.5,
268            0f64,
269            sigma * sigma,
270            0f64,
271            log_k_grlwe,
272            log_k_grlwe,
273        );
274
275        assert!(
276            (noise_have - noise_want).abs() <= 0.1,
277            "{} {}",
278            noise_have,
279            noise_want
280        );
281    });
282}
283
284#[test]
285fn external_product() {
286    let module: Module<FFT64> = Module::<FFT64>::new(2048);
287    let basek: usize = 12;
288    let log_k_grlwe: usize = 60;
289    let rows: usize = (log_k_grlwe + basek - 1) / basek;
290
291    let rank: usize = 1;
292    let rank_out: usize = 1;
293
294    let sigma: f64 = 3.2;
295
296    let mut ct_grlwe_in: GLWESwitchingKey<Vec<u8>, FFT64> =
297        GLWESwitchingKey::new(&module, basek, log_k_grlwe, rows, rank, rank_out);
298    let mut ct_grlwe_out: GLWESwitchingKey<Vec<u8>, FFT64> =
299        GLWESwitchingKey::new(&module, basek, log_k_grlwe, rows, rank, rank_out);
300    let mut ct_rgsw: GGSWCiphertext<Vec<u8>, FFT64> = GGSWCiphertext::new(&module, basek, log_k_grlwe, rows, rank);
301
302    let mut pt_rgsw: ScalarZnx<Vec<u8>> = module.new_scalar_znx(1);
303    let mut pt_grlwe: ScalarZnx<Vec<u8>> = module.new_scalar_znx(1);
304
305    let mut source_xs: Source = Source::new([0u8; 32]);
306    let mut source_xe: Source = Source::new([0u8; 32]);
307    let mut source_xa: Source = Source::new([0u8; 32]);
308
309    let mut scratch: ScratchOwned = ScratchOwned::new(
310        GLWESwitchingKey::encrypt_sk_scratch_space(&module, rank, ct_grlwe_in.size())
311            | GLWECiphertextFourier::decrypt_scratch_space(&module, ct_grlwe_out.size())
312            | GLWESwitchingKey::external_product_scratch_space(
313                &module,
314                ct_grlwe_out.size(),
315                ct_grlwe_in.size(),
316                ct_rgsw.size(),
317            )
318            | GGSWCiphertext::encrypt_sk_scratch_space(&module, rank, ct_rgsw.size()),
319    );
320
321    let k: usize = 1;
322
323    pt_rgsw.raw_mut()[k] = 1; // X^{k}
324
325    pt_grlwe.fill_ternary_prob(0, 0.5, &mut source_xs);
326
327    let mut sk: SecretKey<Vec<u8>> = SecretKey::new(&module, rank);
328    sk.fill_ternary_prob(0.5, &mut source_xs);
329
330    let mut sk_dft: SecretKeyFourier<Vec<u8>, FFT64> = SecretKeyFourier::new(&module, rank);
331    sk_dft.dft(&module, &sk);
332
333    // GRLWE_{s1}(s0) = s0 -> s1
334    ct_grlwe_in.encrypt_sk(
335        &module,
336        &pt_grlwe,
337        &sk_dft,
338        &mut source_xa,
339        &mut source_xe,
340        sigma,
341        scratch.borrow(),
342    );
343
344    ct_rgsw.encrypt_sk(
345        &module,
346        &pt_rgsw,
347        &sk_dft,
348        &mut source_xa,
349        &mut source_xe,
350        sigma,
351        scratch.borrow(),
352    );
353
354    // GRLWE_(m) (x) RGSW_(X^k) = GRLWE_(m * X^k)
355    ct_grlwe_out.external_product(&module, &ct_grlwe_in, &ct_rgsw, scratch.borrow());
356
357    let mut ct_rlwe_dft_s0s2: GLWECiphertextFourier<Vec<u8>, FFT64> =
358        GLWECiphertextFourier::new(&module, basek, log_k_grlwe, rank);
359    let mut pt: GLWEPlaintext<Vec<u8>> = GLWEPlaintext::new(&module, basek, log_k_grlwe);
360
361    module.vec_znx_rotate_inplace(k as i64, &mut pt_grlwe, 0);
362
363    (0..ct_grlwe_out.rows()).for_each(|row_i| {
364        ct_grlwe_out.get_row(&module, row_i, 0, &mut ct_rlwe_dft_s0s2);
365        ct_rlwe_dft_s0s2.decrypt(&module, &mut pt, &sk_dft, scratch.borrow());
366        module.vec_znx_sub_scalar_inplace(&mut pt, 0, row_i, &pt_grlwe, 0);
367
368        let noise_have: f64 = pt.data.std(0, basek).log2();
369
370        let var_gct_err_lhs: f64 = sigma * sigma;
371        let var_gct_err_rhs: f64 = 0f64;
372
373        let var_msg: f64 = 1f64 / module.n() as f64; // X^{k}
374        let var_a0_err: f64 = sigma * sigma;
375        let var_a1_err: f64 = 1f64 / 12f64;
376
377        let noise_want: f64 = noise_rgsw_product(
378            module.n() as f64,
379            basek,
380            0.5,
381            var_msg,
382            var_a0_err,
383            var_a1_err,
384            var_gct_err_lhs,
385            var_gct_err_rhs,
386            log_k_grlwe,
387            log_k_grlwe,
388        );
389
390        assert!(
391            (noise_have - noise_want).abs() <= 0.1,
392            "{} {}",
393            noise_have,
394            noise_want
395        );
396    });
397}
398
399#[test]
400fn external_product_inplace() {
401    let module: Module<FFT64> = Module::<FFT64>::new(2048);
402    let basek: usize = 12;
403    let log_k_grlwe: usize = 60;
404    let rows: usize = (log_k_grlwe + basek - 1) / basek;
405
406    let rank: usize = 1;
407    let rank_out: usize = 1;
408
409    let sigma: f64 = 3.2;
410
411    let mut ct_grlwe: GLWESwitchingKey<Vec<u8>, FFT64> = GLWESwitchingKey::new(&module, basek, log_k_grlwe, rows, rank, rank_out);
412    let mut ct_rgsw: GGSWCiphertext<Vec<u8>, FFT64> = GGSWCiphertext::new(&module, basek, log_k_grlwe, rows, rank);
413
414    let mut pt_rgsw: ScalarZnx<Vec<u8>> = module.new_scalar_znx(1);
415    let mut pt_grlwe: ScalarZnx<Vec<u8>> = module.new_scalar_znx(1);
416
417    let mut source_xs: Source = Source::new([0u8; 32]);
418    let mut source_xe: Source = Source::new([0u8; 32]);
419    let mut source_xa: Source = Source::new([0u8; 32]);
420
421    let mut scratch: ScratchOwned = ScratchOwned::new(
422        GLWESwitchingKey::encrypt_sk_scratch_space(&module, rank, ct_grlwe.size())
423            | GLWECiphertextFourier::decrypt_scratch_space(&module, ct_grlwe.size())
424            | GLWESwitchingKey::external_product_inplace_scratch_space(&module, ct_grlwe.size(), ct_rgsw.size())
425            | GGSWCiphertext::encrypt_sk_scratch_space(&module, rank, ct_rgsw.size()),
426    );
427
428    let k: usize = 1;
429
430    pt_rgsw.raw_mut()[k] = 1; // X^{k}
431
432    pt_grlwe.fill_ternary_prob(0, 0.5, &mut source_xs);
433
434    let mut sk: SecretKey<Vec<u8>> = SecretKey::new(&module, rank);
435    sk.fill_ternary_prob(0.5, &mut source_xs);
436
437    let mut sk_dft: SecretKeyFourier<Vec<u8>, FFT64> = SecretKeyFourier::new(&module, rank);
438    sk_dft.dft(&module, &sk);
439
440    // GRLWE_{s1}(s0) = s0 -> s1
441    ct_grlwe.encrypt_sk(
442        &module,
443        &pt_grlwe,
444        &sk_dft,
445        &mut source_xa,
446        &mut source_xe,
447        sigma,
448        scratch.borrow(),
449    );
450
451    ct_rgsw.encrypt_sk(
452        &module,
453        &pt_rgsw,
454        &sk_dft,
455        &mut source_xa,
456        &mut source_xe,
457        sigma,
458        scratch.borrow(),
459    );
460
461    // GRLWE_(m) (x) RGSW_(X^k) = GRLWE_(m * X^k)
462    ct_grlwe.external_product_inplace(&module, &ct_rgsw, scratch.borrow());
463
464    let mut ct_rlwe_dft_s0s2: GLWECiphertextFourier<Vec<u8>, FFT64> =
465        GLWECiphertextFourier::new(&module, basek, log_k_grlwe, rank);
466    let mut pt: GLWEPlaintext<Vec<u8>> = GLWEPlaintext::new(&module, basek, log_k_grlwe);
467
468    module.vec_znx_rotate_inplace(k as i64, &mut pt_grlwe, 0);
469
470    (0..ct_grlwe.rows()).for_each(|row_i| {
471        ct_grlwe.get_row(&module, row_i, 0, &mut ct_rlwe_dft_s0s2);
472        ct_rlwe_dft_s0s2.decrypt(&module, &mut pt, &sk_dft, scratch.borrow());
473        module.vec_znx_sub_scalar_inplace(&mut pt, 0, row_i, &pt_grlwe, 0);
474
475        let noise_have: f64 = pt.data.std(0, basek).log2();
476
477        let var_gct_err_lhs: f64 = sigma * sigma;
478        let var_gct_err_rhs: f64 = 0f64;
479
480        let var_msg: f64 = 1f64 / module.n() as f64; // X^{k}
481        let var_a0_err: f64 = sigma * sigma;
482        let var_a1_err: f64 = 1f64 / 12f64;
483
484        let noise_want: f64 = noise_rgsw_product(
485            module.n() as f64,
486            basek,
487            0.5,
488            var_msg,
489            var_a0_err,
490            var_a1_err,
491            var_gct_err_lhs,
492            var_gct_err_rhs,
493            log_k_grlwe,
494            log_k_grlwe,
495        );
496
497        assert!(
498            (noise_have - noise_want).abs() <= 0.1,
499            "{} {}",
500            noise_have,
501            noise_want
502        );
503    });
504}
505pub(crate) fn noise_gglwe_product(
506    n: f64,
507    basek: usize,
508    var_xs: f64,
509    var_msg: f64,
510    var_a_err: f64,
511    var_gct_err_lhs: f64,
512    var_gct_err_rhs: f64,
513    rank_in: f64,
514    a_logq: usize,
515    b_logq: usize,
516) -> f64 {
517    let a_logq: usize = a_logq.min(b_logq);
518    let a_cols: usize = (a_logq + basek - 1) / basek;
519
520    let b_scale = 2.0f64.powi(b_logq as i32);
521    let a_scale: f64 = 2.0f64.powi((b_logq - a_logq) as i32);
522
523    let base: f64 = (1 << (basek)) as f64;
524    let var_base: f64 = base * base / 12f64;
525
526    // lhs = a_cols * n * (var_base * var_gct_err_lhs + var_e_a * var_msg * p^2)
527    // rhs = a_cols * n * var_base * var_gct_err_rhs * var_xs
528    let mut noise: f64 = (a_cols as f64) * n * var_base * (var_gct_err_lhs + var_xs * var_gct_err_rhs);
529    noise += var_msg * var_a_err * a_scale * a_scale * n;
530    noise *= rank_in;
531    noise = noise.sqrt();
532    noise /= b_scale;
533    noise.log2().min(-1.0) // max noise is [-2^{-1}, 2^{-1}]
534}