إذا كان إصدار SageMath Python أقدم من 3.9.0، فقد لا تعمل بعض الميزات في بعض البرامج النصية.

## الاستخدام
توجد اختبارات الوحدة في دليل `test` ويمكن تنفيذها باستخدام وحدة `unittest` أو باستخدام `pytest`. لا ينبغي أن يستغرق هذا وقتًا طويلاً، ربما بضع دقائق حسب جهازك.

لتشغيل هجوم معين، يجب عليك إضافة الكود إلى الملف المناسب قبل تنفيذه.

### مثال

على سبيل المثال، تريد مهاجمة RSA باستخدام هجوم Boneh-Durfee، بالمعاملات التالية (مأخوذة من [test_rsa.py](https://github.com/jvdsn/crypto-attacks/blob/HEAD/test/test_rsa.py)):```python
N = 88320836926176610260238895174120738360949322009576866758081671082752401596826820274141832913391890604999466444724537056453777218596634375604879123818123658076245218807184443147162102569631427096787406420042132112746340310992380094474893565028303466135529032341382899333117011402408049370805729286122880037249
e = 36224751658507610673165956970793195381480143363550601971796688201449789736497322700382657163240771111376677180786660893671085854060092736865293791299460933460067267613023891500397200389824179925263846148644777638774319680682025117466596019474987378275216579013846855328009375540444176771945272078755317168511

بعدها يمكنك ببساطة تنفيذ الملف باستخدام Sage. لا يهم من أين تنفّذه، فمسار Python يُضبط تلقائيًا (يمكنك أيضًا استدعاء الهجمات من ملفات Python أخرى، لكن سيتعين عليك حينها إصلاح مسار Python بنفسك):```commandline
[crypto-attacks]$ sage -python attacks/rsa/boneh_durfee.py
INFO:root:Trying m = 3, t = 1...
DEBUG:root:Generating shifts...
DEBUG:root:Creating a lattice with 11 shifts (order = 'invlex', sort_shifts_reverse = False, sort_monomials_reverse = False)...
DEBUG:root:Reducing a 11 x 11 lattice...
DEBUG:root:Reconstructing polynomials (divide_original = True, modulus_bound = False, divide_gcd = True)...
DEBUG:root:Polynomial at row 8 is constant, ignoring...
DEBUG:root:Reconstructed polynomial has gcd 1312232632720549890113031660369306919929075823824696839212183146130434668203517349691252841557097914064120078389640402109017308806168467714230057403815071456395553717020189622129706447677967264344568789118172311850383406340547579993263937406518074980025897726255316031512238322022839331135299265704052474541497687419350763703993630899191179705015113329644753599872380152055902238937889027950089072598069861391599563222633064848996619752054685734260976071760984100109990150069201501748622288840900421607423175114026653242500476408861976142751384898489130281755466581359057847077651502734556259387442296763474369957121 with polynomial at 8, dividing...
DEBUG:root:Reconstructed 10 polynomials
DEBUG:root:Computing pairwise gcds to find trivial roots...
DEBUG:root:Using Groebner basis method to find roots...
DEBUG:root:Sequence length: 10, Groebner basis length: 1
DEBUG:root:Sequence length: 9, Groebner basis length: 1
DEBUG:root:Sequence length: 8, Groebner basis length: 1
DEBUG:root:Sequence length: 7, Groebner basis length: 2
DEBUG:root:Found Groebner basis with length 2, trying to find roots...
Found p = 7866790440964395011005623971351568677139336343167390105188826934257986271072664643571727955882500173182140478082778193338086048035817634545367411924942763 and q = 11227048386374621771175649743442169526805922745751610531569607663416378302561807690656370394330458335919244239976798600743588701676542461805061598571009923

* قم بتنفيذ نسخة تصحيح أخطاء (debug) من الهجوم الذي تحاول استخدامه (مع نتائج معروفة)، وحدد قيمة `m` التي تعطي متجهات شبكة جيدة. ثم استدعِ طريقة الهجوم مباشرةً مع قيمة `m` الصحيحة.


## الهجمات المنفذة
### القاسم المشترك التقريبي
* [x] [هجوم كثيرات الحدود متعددة المتغيرات](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/acd/mp.py) [^acd_mp]
* [x] [الهجوم القائم على التعامد](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/acd/ol.py) [^acd_ol]
* [x] [هجوم التقريب الديوفانتي المتزامن](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/acd/sda.py) [^acd_sda]

### CBC
* [x] [هجوم قلب البتات](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/cbc/bit_flipping.py)
* [x] [هجوم استعادة IV](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/cbc/iv_recovery.py)
* [x] [هجوم أوراكل الحشو](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/cbc/padding_oracle.py)

### CBC + CBC-MAC
* [x] [هجوم إعادة استخدام المفتاح (encrypt-and-MAC)](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/cbc_and_cbc_mac/eam_key_reuse.py)
* [x] [هجوم إعادة استخدام المفتاح (encrypt-then-MAC)](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/cbc_and_cbc_mac/etm_key_reuse.py)
* [x] [هجوم إعادة استخدام المفتاح (MAC-then-encrypt)](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/cbc_and_cbc_mac/mte_key_reuse.py)

### CBC-MAC
* [x] [هجوم تمديد الطول](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/cbc_mac/length_extension.py)

### CTR
* [x] [هجوم قلب البتات](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/ctr/bit_flipping.py)
* [x] [هجوم CRIME](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/ctr/crime.py)
* [x] [هجوم أوراكل الفاصل](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/ctr/separator_oracle.py)

### ECB
* [x] [هجوم استعادة النص الصريح](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/ecb/plaintext_recovery.py)
* [x] [هجوم استعادة النص الصريح (نسخة أصعب)](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/ecb/plaintext_recovery_harder.py)
* [x] [هجوم استعادة النص الصريح (أصعب نسخة)](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/ecb/plaintext_recovery_hardest.py)

### تشفير المنحنيات الإهليلجية
* [x] [هجوم إعادة استخدام nonce في ECDSA](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/ecc/ecdsa_nonce_reuse.py)
* [x] [هجوم Frey-Ruck](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/ecc/frey_ruck_attack.py) [^ecc_frey_ruck_attack]
* [x] [هجوم MOV](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/ecc/mov_attack.py) [^ecc_mov_attack]
* [x] [استعادة المعاملات](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/ecc/parameter_recovery.py)
* [x] [هجوم المنحنى المفرد](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/ecc/singular_curve.py)
* [x] [هجوم Smart (مع منحنيات فوق حقول الامتداد)](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/ecc/smart_attack.py) [^ecc_smart_attack1] [^ecc_smart_attack2]

### تشفير ElGamal
* [x] [هجوم إعادة استخدام nonce](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/elgamal_encryption/nonce_reuse.py)
* [x] [هجوم المولد غير الآمن](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/elgamal_encryption/unsafe_generator.py)

### توقيع ElgGamal
* [ ] هجوم Bleichenbacher
* [ ] هجوم Khadir
* [x] [هجوم إعادة استخدام nonce](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/elgamal_signature/nonce_reuse.py)

### التحليل إلى عوامل
* [x] [التحليل عبر تحويل الأساس](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/factorization/base_conversion.py)
* [x] [هجوم التفرع والتقليم](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/factorization/branch_and_prune.py) [^factorization_branch_and_prune]
* [x] [التحليل بالضرب المركب (منحنيات إهليلجية)](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/factorization/complex_multiplication.py) [^factorization_complex_multiplication]
* [x] [تحليل Coppersmith](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/factorization/coppersmith.py)
* [x] [تحليل فيرما](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/factorization/fermat.py)
* [x] [هجوم Ghafar-Ariffin-Asbullah](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/factorization/gaa.py) [^factorization_gaa]
* [x] [التحليل الضمني](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/factorization/implicit.py) [^factorization_implicit]
* [x] [التحليل عند معرفة phi](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/factorization/known_phi.py) [^factorization_known_phi]
* [x] [ROCA](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/factorization/roca.py) [^factorization_roca]
* [x] [خوارزمية Shor (الكلاسيكية)](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/factorization/shor.py) [^factorization_shor]
* [x] [تحليل الأعداد الأولية التوأم](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/factorization/twin_primes.py)
* [x] [تحليل المعاملات غير المتوازنة](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/factorization/unbalanced.py) [^factorization_unbalanced]

### GCM
* [x] [الهجوم المحظور](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/gcm/forbidden_attack.py) [^gcm_forbidden_attack]

### مشكلة العدد المخفي
مع تطبيقات على الكشف الجزئي لـ nonce في (EC)DSA.
* [x] [مشكلة العدد المخفي الموسعة](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/hnp/extended_hnp.py) [^hnp_extended_hnp]
* [ ] هجوم تحليل فورييه
* [x] [هجوم قائم على الشبكات](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/hnp/lattice_attack.py)

### IGE
* [x] [هجوم أوراكل الحشو](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/ige/padding_oracle.py)

### أنظمة التشفير بحقيبة الظهر
* [x] [هجوم الكثافة المنخفضة](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/knapsack/low_density.py) [^knapsack_low_density]

### المولدات الخطية التطابقية

* [x] [استعادة معاملات LCG](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/lcg/parameter_recovery.py)
* [x] [استعادة معاملات LCG المبتور](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/lcg/truncated_parameter_recovery.py) [^lcg_truncated_parameter_recovery]
* [x] [استعادة حالة LCG المبتور](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/lcg/truncated_state_recovery.py) [^lcg_truncated_state_recovery]

### التعلم مع الأخطاء

* [x] [هجوم Arora-Ge](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/lwe/arora_ge.py) [^lwe_arora_ge]
* [ ] هجوم Blum-Kalai-Wasserman
* [ ] هجوم اختزال الشبكات

### Mersenne Twister

* [x] [استعادة الحالة](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/mersenne_twister/state_recovery.py)

### لوحة الاستعمال الواحد

* [x] [إعادة استخدام المفتاح](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/otp/key_reuse.py)

### الأعداد شبه الأولية

* [x] [توليد أعداد Miller-Rabin شبه الأولية](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/pseudoprimes/miller_rabin.py) [^pseudoprimes_miller_rabin]

### RC4

* [x] [هجوم Fluhrer-Mantin-Shamir](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rc4/fms.py)

### RSA

* [x] [هجوم Bleichenbacher](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rsa/bleichenbacher.py) [^rsa_bleichenbacher]
* [x] [هجوم تزوير التوقيع لـ Bleichenbacher](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rsa/bleichenbacher_signature_forgery.py)
* [x] [هجوم Boneh-Durfee](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rsa/boneh_durfee.py) [^rsa_boneh_durfee]
* [x] [هجوم Cherkaoui-Semmouni](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rsa/cherkaoui_semmouni.py) [^rsa_cherkaoui_semmouni]
* [x] [هجوم المعامل المشترك](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rsa/common_modulus.py)
* [x] [هجوم الأخطاء في CRT](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rsa/crt_fault_attack.py)
* [x] [هجوم الأخطاء على d](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rsa/d_fault_attack.py)
* [x] [هجوم Desmedt-Odlyzko (تزوير انتقائي)](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rsa/desmedt_odlyzko.py) [^rsa_desmedt_odlyzko]
* [x] [هجوم Wiener الموسع](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rsa/extended_wiener_attack.py) [^rsa_extended_wiener_attack]
* [x] [هجوم البث لـ Hastad](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rsa/hastad_attack.py)
* [x] [هجوم الأسس المعروفة لـ CRT](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rsa/known_crt_exponents.py) [^rsa_known_crt_exponents]
* [x] [هجوم الأسس المعروفة جزئيًا لـ CRT](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rsa/known_crt_exponents.py) [^rsa_partial_known_crt_exponents]
* [x] [هجوم الأس الخاص المعروف](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rsa/known_d.py)
* [x] [هجوم الأس العام الصغير](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rsa/low_exponent.py)
* [x] [هجوم أوراكل LSB (أوراكل التكافؤ)](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rsa/lsb_oracle.py)
* [x] [هجوم Manger](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rsa/manger.py) [^rsa_manger]
* [x] [هجوم Nitaj لـ CRT-RSA](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rsa/nitaj_crt_rsa.py) [^rsa_nitaj_crt_rsa]
* [x] [هجوم الأس العام غير الأولي نسبيًا](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rsa/non_coprime_exponent.py) [^rsa_non_coprime_exponent]
* [x] [الكشف الجزئي للمفتاح](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rsa/partial_key_exposure.py) [^rsa_partial_key_exposure1] [^rsa_partial_key_exposure2] [^rsa_partial_key_exposure3] 
* [x] [هجوم الرسائل المرتبطة](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rsa/related_message.py)
* [x] [هجوم الرسائل النمطية](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rsa/stereotyped_message.py)
* [x] [هجوم Wiener](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rsa/wiener_attack.py)
* [x] [هجوم Wiener لـ RSA بالأعداد الأولية المشتركة](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rsa/wiener_attack_common_prime.py) [^rsa_wiener_attack_common_prime]
* [x] [هجوم Wiener (نسخة الشبكات الاستدلالية)](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/rsa/wiener_attack_lattice.py) [^rsa_wiener_attack_lattice] [^rsa_wiener_attack_lattice_extended] [^small_roots_aono]

### مشاركة سر Shamir
* [x] [المعاملات الحتمية](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/shamir_secret_sharing/deterministic_coefficients.py)
* [x] [تزوير الحصص](https://github.com/jvdsn/crypto-attacks/blob/HEAD/attacks/shamir_secret_sharing/share_forgery.py)

## تنفيذات أخرى مثيرة للاهتمام
* [x] [طريقة Adleman-Manders-Miller لاستخراج الجذور](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/__init__.py) [^adleman_manders_miller]
* [x] [CRT سريع باستخدام فرق تسد](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/crt.py)
* [x] [معكوسات معيارية سريعة](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/__init__.py)
* [x] [رفع Hensel الخطي](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/hensel.py)
* [ ] رفع Hensel التربيعي
* [x] [خوارزمية أقرب مستوى لـ Babai](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/lattice.py)
* [x] [اللوغاريتم المتقطع للمصفوفات](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/matrices.py)
* [x] [اللوغاريتم المتقطع للمصفوفات (معادلة)](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/matrices.py)
* [x] [PartialInteger](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/partial_integer.py)
* [x] [GCD سريع لكثيرات الحدود باستخدام نصف GCD](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/polynomial.py)

### توليد المنحنيات الإهليلجية
* [x] [الضرب المركب](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/ecc.py)
* [x] [المنحنيات الشاذة](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/ecc.py)
* [x] [منحنيات MNT](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/ecc.py)
* [x] [الرتبة المحددة مسبقًا](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/ecc.py)
* [x] [الأثر المحدد مسبقًا](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/ecc.py)
* [x] [المنحنيات فائقة التفرد](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/ecc.py)

### الجذور الصغيرة
* [x] [جذور كثيرات الحدود باستخدام أسس Groebner](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/small_roots/__init__.py)
* [x] [جذور كثيرات الحدود باستخدام المحصلات](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/small_roots/__init__.py)
* [x] [جذور كثيرات الحدود باستخدام تنوع Sage (التحليل المثلثي)](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/small_roots/__init__.py)
* [x] [طريقة Aono (شبكة مجموع Minkowski)](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/small_roots/aono.py) [^small_roots_aono]
* [x] [طريقة Blomer-May](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/small_roots/blomer_may.py) [^small_roots_blomer_may]
* [x] [طريقة Boneh-Durfee](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/small_roots/boneh_durfee.py) [^rsa_boneh_durfee]
* [x] [طريقة Coron](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/small_roots/coron.py) [^small_roots_coron]
* [x] [طريقة Coron (مباشرة)](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/small_roots/coron_direct.py) [^small_roots_coron_direct]
* [x] [طرق Ernst وآخرون](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/small_roots/ernst.py) [^rsa_partial_key_exposure2]
* [x] [طريقة Herrmann-May (الخطية المفككة)](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/small_roots/herrmann_may.py) [^small_roots_herrmann_may]
* [x] [طريقة Herrmann-May (معيارية متعددة المتغيرات)](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/small_roots/herrmann_may_multivariate.py) [^small_roots_herrmann_may_multivariate]
* [x] [طريقة Howgrave-Graham](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/small_roots/howgrave_graham.py) [^small_roots_howgrave_graham]
* [x] [طريقة Jochemsz-May (جذور معيارية)](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/small_roots/jochemsz_may_modular.py) [^small_roots_jochemsz_may_modular]
* [x] [طريقة Jochemsz-May (جذور صحيحة)](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/small_roots/jochemsz_may_integer.py) [^small_roots_jochemsz_may_integer]
* [x] [طريقة Nitaj-Fouotsa](https://github.com/jvdsn/crypto-attacks/blob/HEAD/shared/small_roots/nitaj_fouotsa.py) [^small_roots_nitaj_fouotsa]

[^acd_mp]: Galbraith D. S. et al., "Algorithms for the Approximate Common Divisor Problem" (Section 5)
[^acd_ol]: Galbraith D. S. et al., "Algorithms for the Approximate Common Divisor Problem" (Section 4)
[^acd_sda]: Galbraith D. S. et al., "Algorithms for the Approximate Common Divisor Problem" (Section 3)

[^ecc_frey_ruck_attack]: Harasawa R. et al., "Comparing the MOV and FR Reductions in Elliptic Curve Cryptography" (Section 3)
[^ecc_mov_attack]: Harasawa R. et al., "Comparing the MOV and FR Reductions in Elliptic Curve Cryptography" (Section 2)
[^ecc_smart_attack1]: Smart N. P., "The Discrete Logarithm Problem on Elliptic Curves of Trace One"
[^ecc_smart_attack2]: Hofman S. J., "The Discrete Logarithm Problem on Anomalous Elliptic Curves"

[^factorization_branch_and_prune]: Heninger N., Shacham H., "Reconstructing RSA Private Keys from Random Key Bits"
[^factorization_complex_multiplication]: Sedlacek V. et al., "I want to break square-free: The 4p - 1 factorization method and its RSA backdoor viability"
[^factorization_gaa]: Ghafar AHA. et al., "A New LSB Attack on Special-Structured RSA Primes"
[^factorization_implicit]: Nitaj A., Ariffin MRK., "Implicit factorization of unbalanced RSA moduli"
[^factorization_known_phi]: Hinek M. J., Low M. K., Teske E., "On Some Attacks on Multi-prime RSA" (Section 3)
[^factorization_roca]: Nemec M. et al., "The Return of Coppersmith’s Attack: Practical Factorization of Widely Used RSA Moduli"
[^factorization_shor]: M. Johnston A., "Shor’s Algorithm and Factoring: Don’t Throw Away the Odd Orders"
[^factorization_unbalanced]: Brier E. et al., "Factoring Unbalanced Moduli with Known Bits" (Section 4)

[^gcm_forbidden_attack]: Joux A., "Authentication Failures in NIST version of GCM"

[^hnp_extended_hnp]: Hlavac M., Rosa T., "Extended Hidden Number Problem and Its Cryptanalytic Applications" (Section 4) 

[^knapsack_low_density]: Coster M. J. et al., "Improved low-density subset sum algorithms"

[^lcg_truncated_parameter_recovery]: Contini S., Shparlinski I. E., "On Stern's Attack Against Secret Truncated Linear Congruential Generators"
[^lcg_truncated_state_recovery]: Frieze, A. et al., "Reconstructing Truncated Integer Variables Satisfying Linear Congruences"

[^lwe_arora_ge]: ["The Learning with Errors Problem: Algorithms"](https://people.csail.mit.edu/vinodv/6876-Fall2018/lecture2.pdf) (Section 1)

[^pseudoprimes_miller_rabin]: R. Albrecht M. et al., "Prime and Prejudice: Primality Testing Under Adversarial Conditions"

[^rsa_bleichenbacher]: Bleichenbacher D., "Chosen Ciphertext Attacks Against Protocols Based on the RSA Encryption Standard PKCS #1"
[^rsa_boneh_durfee]: Boneh D., Durfee G., "Cryptanalysis of RSA with Private Key d Less than N^0.292"
[^rsa_cherkaoui_semmouni]: Cherkaoui-Semmouni M. et al., "Cryptanalysis of RSA Variants with Primes Sharing Most Significant Bits"
[^rsa_desmedt_odlyzko]: Coron J. et al., "Practical Cryptanalysis of ISO 9796-2 and EMV Signatures (Section 3)"
[^rsa_extended_wiener_attack]: Dujella A., "Continued fractions and RSA with small secret exponent"
[^rsa_known_crt_exponents]: Campagna M., Sethi A., "Key Recovery Method for CRT Implementation of RSA"
[^rsa_partial_known_crt_exponents]: May A., Nowakowski J., Sarkar S., "Approximate Divisor Multiples - Factoring with Only a Third of the Secret CRT-Exponents"
[^rsa_manger]: Manger J., "A Chosen Ciphertext Attack on RSA Optimal Asymmetric Encryption Padding (OAEP) as Standardized in PKCS #1 v2.0"
[^rsa_nitaj_crt_rsa]: Nitaj A., "A new attack on RSA and CRT-RSA"
[^rsa_non_coprime_exponent]: Shumow D., "Incorrectly Generated RSA Keys: How To Recover Lost Plaintexts"
[^rsa_partial_key_exposure1]: Boneh D., Durfee G., Frankel Y., "An Attack on RSA Given a Small Fraction of the Private Key Bits"
[^rsa_partial_key_exposure2]: Ernst M. et al., "Partial Key Exposure Attacks on RSA Up to Full Size Exponents"
[^rsa_partial_key_exposure3]: Blomer J., May A., "New Partial Key Exposure Attacks on RSA"
[^rsa_wiener_attack_common_prime]: Jochemsz E., May A., "A Strategy for Finding Roots of Multivariate Polynomials with New Applications in Attacking RSA Variants" (Section 5)
[^rsa_wiener_attack_lattice]: Nguyen P. Q., "Public-Key Cryptanalysis"
[^rsa_wiener_attack_lattice_extended]: Howgrave-Graham N., Seifert J., "Extending Wiener’s Attack in the Presence of Many Decrypting Exponents"

[^adleman_manders_miller]: Cao Z. et al., "Adleman-Manders-Miller Root Extraction Method Revisited" (Section 5)

[^small_roots_aono]: Aono Y., "Minkowski sum based lattice construction for multivariate simultaneous Coppersmith's technique and applications to RSA" (Section 4)
[^small_roots_blomer_may]: Blomer J., May A., "New Partial Key Exposure Attacks on RSA" (Section 6)
[^small_roots_coron]: Coron J., "Finding Small Roots of Bivariate Integer Polynomial Equations Revisited"
[^small_roots_coron_direct]: Coron J., "Finding Small Roots of Bivariate Integer Polynomial Equations: a Direct Approach"
[^small_roots_herrmann_may]: Herrmann M., May A., "Maximizing Small Root Bounds by Linearization and Applications to Small Secret Exponent RSA"
[^small_roots_herrmann_may_multivariate]: Herrmann M., May A., "Solving Linear Equations Modulo Divisors: On Factoring Given Any Bits" (Section 3 and 4)
[^small_roots_howgrave_graham]: May A., "New RSA Vulnerabilities Using Lattice Reduction Methods" (Section 3.2)
[^small_roots_jochemsz_may_modular]: Jochemsz E., May A., "A Strategy for Finding Roots of Multivariate Polynomials with New Applications in Attacking RSA Variants" (Section 2.1)
[^small_roots_jochemsz_may_integer]: Jochemsz E., May A., "A Strategy for Finding Roots of Multivariate Polynomials with New Applications in Attacking RSA Variants" (Section 2.2)
[^small_roots_nitaj_fouotsa]: Nitaj A., Fouotsa E., "A New Attack on RSA and Demytko's Elliptic Curve Cryptosystem"