Kuantum dolanıklık alışık olmayanlar için karışık görünen ve üzerine yazılmış pek çok popüler yazı bulunabilecek bir konu. Ne yazık ki bazen fazla mistifiye edilmesi ve garip anlamlar yüklenmesi nedeniyle insanlarda, sanki anlaması zor bir kavram olduğu fikrini uyandırmakta. Bu yazıda kuantum dolanıklık fenomenini kullanarak herhangi iki nokta arasında, en azından kuramsal düzeyde, yüzde yüz güvenli iletişim kurma fikrinin temellerine değineceğiz.
Şifreli İletişim
Eldeki yazı içinde kriptolojinin temellerine ve türlerine girmeyeceğiz, bunlar için sitemizde daha önce üzerine yazdığımız “Şifrelerin Matematiği: Kriptografi” ve/veya “Kriptografiye Giriş: Şifreleme Teorisi” isimli kitaplara göz atabilirsiniz. Bunun yerine olabildiğince basit örnekler üzerinden bir anlatı kurmayı tercih edeceğiz.
Yukarıda iki tane ikiye ikilik, toplam dört kareden oluşan bir mesaj var. Eğer bu mesaj size dördü de beyaz gelirse 0000, sağdaki gibi gelirse de 0101 olarak kodlandığını düşünelim. Yani beyaz kareler 0, siyah kareler 1 demek. Toplamda 4 bit veriden oluşan bu resmin 2 üzeri 4 hesabından 0000, 0001… 1110, 1111’e kadar uzanan 16 farklı şekilde bulunabileceğini biliyoruz (2x2x2x2 = 16).
Karelerde toplama işlemini de ikili kurallara göre yukarıdaki gibi yapabileceğimizi düşünelim. Yani 01 + 01 = 10 edecek şekilde, burada ikili sistem kullandığımız için 2 rakamı 10, 3 rakamı 11, 4 rakamı 100 şeklinde yazılabiliyor. Dolayısıyla 1111 aslında 1000 + 0100 + 0010 + 0001, yani 8 + 4 + 2 + 1 = 15 ediyor, 0000 = 0 olarak başladığımız için de yukarıdaki şekil üzerindeki boyalı kareler üyeleri {0,1,2…,14,15} olan 16 elemanlı bir kümeye tekabül ediyor.
Eğer ben size 1010 mesajını gönderip anahtarımın da 0100 olduğunu söylemiş olsaydım size aslında göndermeye çalıştığım mesajın 0110 olduğunu anlayabilirdiniz. Ancak size yalnızca 1010 mesajını gönderip bunun şifreli bir mesaj olduğunu söyleseydim, 0000 ila 1111 arasındaki 16 olasılıktan herhangi birisini anahtar olarak kullanmış olabileceğimden ötürü elinizde 16 farklı ihtimali olan bir mesaj olurdu. [Burada ufak bir not olarak 1111 + 0001 = 10000 olduğunu ancak elimizde yalnızca dört kare olduğundan bunun size 0000 olarak görüneceğini hatırlatalım, aşağıdaki resim daha açıklayıcı olabilir.]
Kuantum Bunun Neresinde?
Şimdiye kadar yalnızca kare boyayarak gösterdiğimiz hikayenin kuantum mekaniğiyle hiçbir ilgisi yok, yaptığımız her şey sadece toplama ve çıkarma. Aynı anahtarı iki tarafın da şifreleme ve çözme işlemi için kullandığı sisteme kısaca “simetrik anahtarlı şifreleme” diyebiliriz, mesajı şifreleyen de çözen de aynı anahtar (aksi durumlar için kitaplara bakabilirsiniz). Mesajla aynı uzunlukta bir simetrik anahtarın yalnızca tek seferlik kullanıldığı durumlara ise ‘tek kullanımlık şerit’ (one-time-pad) anahtarlar deniyor, yani anahtarınızı yalnızca kendisiyle aynı boydaki bir mesajı şifrelemek ve çözmek için tek seferlik kullanıyorsunuz.
Alternatif olarak iki haneli bir anahtar da kullanabilirdiniz, örneğin 10’ı anahtar olarak seçmiş olsaydınız 0101 mesajını şifrelemek için iki kere 10’i tekrarlayıp 1010’ı anahtar yapabilirdiniz. Ancak bu tek kullanımlık şerit kadar güvenli olmazdı zira 16 değil yalnızca 8 olası anahtarınız olurdu (örneğin asla 0110 gibi bir anahtar üretemezdiniz). Bu da anahtarınızın boyutunu bilen bir şifre kırıcının işini yarı yarıya kolaylaştırırdı.
Tek kullanımlık şerit şifreleme dünyasında kırılamayacağı garantilenmiş bir şifreleme metodu, örneğin yalnızca 32 bit veri içeren (yani 4 byte) bir mesaj için 4,294,967,296 potansiyel anahtar olabilir. Ortalama bir MMS mesajının bile 600 kilobyte, yani 600 x 1000 x 8 = 4800000 bit veri içerdiğini göz önünde bulundurursak potansiyel anahtar sayısının saçma bir sayı olduğunu görebiliriz (4800000 tane 2’yi birbiriyle çarpın). Peki neden sürekli bunu kullanmıyoruz sorusunun yanıtıysa bir o kadar komik, anahtarı güvenli dağıtabilecek yöntemimiz yok!
Bazı büyük ve güvenliğe önem veren firmalar bu tek kullanımlık şeritleri fiziksel olarak birkaç terabyte’lık (yaklaşık 8,000,000,000,000 bit) harddisklere yükleyip bazen kıtalar arası güvenilir kuryeler aracılığıyla taşıma yoluyla bunu gerçekleştiriyorlar, ancak bunun fazlasıyla maliyetli olduğu açık. Üstüne üstlük kuryeye (veya çalıştığınız firmaya) güvenmenizi gerektiriyor. İşte kuantum dolanıklık burada devreye giriyor.
Kuantum Dolanık Durumlar
Yukarıdaki özel kuantum durumlarına alanın en büyük isimlerinden olan John Stewart Bell’in anısına Bell durumları deniyor. Bunlar olası iki durumu olan (0 ve 1) iki parçacığın maksimum dolanık durumda olabildiği muhtemel dört senaryoyu gösteriyor.
Örneğin ilk durumu ele alalım, 00 + 11. Baştaki bir bölü kök iki bize 00’ın da 11’inde ihtimalinin %50-%50 olduğunu gösteriyor. Yani bizim elimizde bu kuantum durumu olsa ve ölçüm yaparsak %50 ihtimalle 00 veya %50 ihtimalle 11 göreceğiz demek. Peki ya bu durum bir kişinin değil de iki kişinin eline dağıtılmış olsa?
İlk Bell durumundan (00+11) elinizde olduğunu düşünün ve bundaki parçacıkları iletişim kurmak istediğiniz bir arkadaşınızla paylaştınız. Sadece sizin elinizde olan (yani ilk parçacık) sizin bakış açınızdan aşağıdaki durumda olacaktır.
Arkadaşınız kendi parçacığına (ikinci parçacığa) baktığında o da yukarıdaki durumun aynısını görecektir. Dolayısıyla ikiniz de, eğer kendi parçacığınızı diğerini göz ardı ederek ölçerseniz %50 ihtimalle 0, %50 ihtimalle 1 bulacaksınız. Ancak kuantum dolanıklık işte tam bu noktada devreye giriyor. Belki fark etmişsinizdir, eğer sizin elinizde + varsa ve arkadaşınızın da elinde + varsa ortak durum aslında ++ durumu demektir. Bu da aşağıdaki kuantum duruma tekabül eder.
Bu durum görebileceğiniz üzere başta elimizde olduğunu iddia ettiğimiz Bell durumu değil. Dolayısıyla, kendi parçacığımıza ve arkadaşımızın parçacığına ayrı ayrı bakarak yaptığımız çıkarım yanlış! Doğrusunu görebilmek için ikisine aynı anda bakabilmemiz gerekiyor. İşte bu özellik, klasik mekanikte karşılığı olmayan ve kuantum mekaniksel parçacıklara özgü bir davranış, parçacıkların arasında yerel olmayan korelasyonların var olması.
Klasik mekanikte her parçacık kendi özelinde ele alınıp sonra sistem tüm bu parçacıkların toplamı olarak ortaya konulur, ancak kuantum mekaniğinde sistemin kendisi parçalarının toplamından ‘farklı’ bir bilgi içerebilir, yalnızca kendi elinizdeki parçacığa bakarak erişemeyeceğiniz bir bilgi. Kuantum mekaniğinde bütün, teker teker parçalarının toplamından daha çok bilgi içerebilir.
Şimdi elimizde aşağıdaki gibi bir anahtar olduğunu ve bunu arkadaşınızla aranızda paylaştığınızı varsayalım.
Sizin elinizdeki anahtar 16’da 1 ihtimalle her olasılık (0000, 0001, 0010,… ,1111) gelebilir, arkadaşınızın elindeki anahtar da 16’da 1 ihtimalle her olasılık (0000, 0001, 0010,… ,1111) gelebilir. Ancak sonradan kontrol edecek olursanız size 0100 geldiyse mutlaka arkadaşınıza da 0100 geldiğini göreceksiniz! Çünkü yukarıdan da hatırlayacağınız üzere, her bir basamak için Bell durumları ya 00 ya 11 olmak zorunda, siz 0 gördüyseniz arkadaşınız da mutlaka sıfır görmüş olacak.
Eğer verili bir mesajı siz elinizdeki birinci parçacığı kullanarak şifrelerseniz arkadaşınız da kendi elindeki ikinci parçacığı kullanarak çözebilir, aslında kuramsal olarak aradaki anahtarı sizin dahi bilmenize gerek kalmaz. Fakat bilmeniz gereken durumlarda bile hiç kimse araya girip sizi dinleyemez, araya birisi girip arkadaşınızla aranızdaki dolanıklığa eklemlenmeye çalışırsa başarısız olacaktır çünkü maksimum dolanıklık ancak iki parçacık arasında olabilir (buna dolanıklığın tek eşliliği adı verilir). Aranıza girmeye çalışan kişi kuantum dolanıklık durumunu bozmuş olacağından artık anahtarlarınız simetrik olmayacaktır, bu nedenle de şifre çözme işinde işlevini yitirecektir.
Dolanıklık Elde Etme
Peki bu özel dolanık durumları nasıl elde edebiliriz diye sorabilirsiniz. Daha önce yazdığımız “Kuantum Bilgisayarların Matematiği Üzerine” yazısını okuyanlar yukarıdaki kuantum durumlarında tanıdık işaretler görmüş olabilir. Kullanmayı tercih ettiğimiz birinci Bell durumunu kuantum mantık kapılarını kullanarak elde edilişini aşağıdaki şekilde görebilirsiniz.
Yani bir Hadamard ve bir C-NOT kapısıyla bu kuantum durumunu elde edebiliriz. Ancak kuantum mantık kapılarını kullanmadan, fiziksel başka yöntemlerle de dolanık parçacıklar elde etmek mümkün. Örneğin aşağıda, bir kaynak lazeri özel bir kristal türünden geçirerek enerjisi düşürülmüş ancak birbiriyle dolanık hale sokulmuş iki ışık huzmesi elde etmeyi gösteren bir mekanizma görülebilir (bu yönteme ‘spontaneous parametric down conversion’ / ‘kendiliğinden parametrik aşağı çevrim’ denir).
Şeytan Bunun Neresinde
Şimdiye kadar her şey kulağa kuantum dolanıklık bazlı iletişim güvenlik problemimize mucizevi bir çözümmüş gibi geliyor, o zaman neden hala bu sisteme geçmedik?
İlk baştaki dört kareden oluşan resimlerimizi hatırlayın, aslında onlar bir nevi küçük kare kodlar olarak düşünülebilir. QR kodlar versiyonlarına bağlı olarak farklı miktarlarda veri barındırabilirler, ancak düz düşünecek olursak yukarıda 30×30’dan 900 kare bulunmakta, bu da 900 tane Bell durumu demek, basit bir qr kodu için.
Peki şimdi bunu arkadaşınızla yaptığınız öylesine bir WhatsApp konuşması temelinde, paylaştığınız emojileri ve mesaj kalabalığını göz önüne alarak hayal edin. Veya milyonlarca insanın su, elektrik, doğalgaz faturası ödemesinden kaynaklanan ‘gündelik’ bankacılık işlemlerinin hacmini. Bu kadar verinin hepsinin kuantum şifrelemeyle güvenli hale getirilmesinin neredeyse imkansız olduğu aşikar.
Daha önce “Tüm Sırların İfşa Olacağı Gün Yaklaşırken” başlıklı yazımızda kuantum bilgisayarların Shor algoritmasını kullanarak internetin şifreleme altyapısı olan RSA yöntemini kıracağından bahsetmiştik. Belki kuantum bilgisayarlara gerek bile kalmadan, çok daha gelişmiş klasik bir algoritma bile bunu yapabilir (yapılamayacağının matematiksel kanıtı yok). Bir başka yazımızdaysa “Hassas Verilerin Korunması Neden Önemlidir” sorusuna odaklanmıştık. Uzun vadede değeri olan verilerin (sağlık verileri gibi) olabildiğince erkenden kuantum-sonrası (post-quantum) kriptografik yöntemlerle korunmaya başlanmasının önemine değinmiş, olası bir senaryo üzerinden böylesi verilerin kaybedilmesinin uzun vadede ülkelere verebileceği zararlara dair küçük bir örnek vermiştik.
Toparlarsak
Bu yazıda kısaca kuantum dolanık parçacıklarla simetrik anahtar yaratımı üzerine bir genel tema aktarmaya çalıştık. Alanın geçmişine baktığımızda iki protokolün tüm bu süreci başlattığını söylemek mümkün, BB84 ve E91. BB84 yalnızca süperpozisyonu, E91 ise dolanıklığı kullanan protokoller. Bunların üzerine yazılmış Türkçe eserler dağınık da olsa bulunmakta (bir örnek). Konuyla ilgili güzel genel bir özet içinse Kutlu Kutluer’in Twitter paylaşımlarına bakılabilir.
Kuantum bilgisayarlar ilgi çekici ve heyecan uyandırıcı olabilirler, ancak kuantum kriptografi çoktandır burada olan ve ticari olarak aktif bir alan. Üstüne üstlük bu alandaki bilgi ve tecrübeler ileride kurulacak her türlü kuantum endüstrisine de aktarılabilir. Kuantum bilgisayım için kuantum kriptografinin neden önemli olduğuna dair geçtiğimiz ay PBS Space Time’ın hazırladığı bir giriş videosuna aşağıdan ulaşabilirsiniz.
Sonuç olarak kabul etmek gerekiyor ki, kuantum teknolojileri çok geniş bir spektrumdan gelecek. Bunlara şimdiden hazırlanmak ve genel kuantum fenomenlerinin (süperpozisyon, dolanıklık, bağlamsallık, belirsizlik ilkesi gibi gibi) teknolojiyle nerelerde bağlandığını görmek önemli. Gelecek herkesin beklediğinden hızlı geliyor olabilir.
Bu içerik orijinal olarak Düzensiz.org üzerinde yayınlanmıştır.
справки
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Mood changes can include euphoria and extreme depression and may alter the way
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One thing is that one of the most common incentives for utilizing your credit cards is a cash-back or rebate supply. Generally, you will get 1-5 back upon various acquisitions. Depending on the credit card, you may get 1 back on most expenditures, and 5 again on buying made from convenience stores, filling stations, grocery stores and also ‘member merchants’.
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